transformation groups pozna„ 1985: proceedings of a symposium held in pozna„, july 5–9,...
TRANSCRIPT
Lecture Notes in Mathematics
1217
Transformation Groups Poznar~ 1985 Proceedings of a Symposium held in Poznar~, July 5-9, 1985
Edited by S. Jackowski and K. Pawa|owski
IIIII H I I I III
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Editors
Stefan Jackowski Instytut Matematyki Uniwersytet Warszawski Pafac Kultury i Nauki IXp. 00-901 Warszawa, Poland
Krzysztof Pawalowski Instytut Matematyki Uniwersytet ira. A. Mickiewicza w Poznaniu ul. Matejki 48/49 60-769 PoznaS, Poland
Mathematics Subject Classification (1980): 57 S XX; 57 S 10; 57 S 15; 57 S 17; 57S25; 57R67; 57R80; 20J05
ISBN 3-540-16824-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16824-9 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr, 2146/3140-543210
Dedicated to the memory of
A° Jankowski
and
W. Pulikowski
P R E F A C E
The Symposium on Transformation Groups supported by the Adam
Mickiewicz University in Pozna~ was held in Pozna~, July 5-9, 1985.
The symposium was dedicated to the memory of two of our teachers and
friends, Andrzej Jankowski and Wojtek Pulikowski on the tenth anni-
versary of their deaths.
These proceedings contain papers presented at the symposium and
also papers by mathematicians who were invited to the meeting but
were unable to attend. All papers have been refereed and are in
their final forms. We would like to express our gratitude to the
authors and the many referees.
The participants and in particular the lecturers contributed to
the success of the symposium and we are most grateful to all of
them. Special thanks are due to our colleagues Ewa Marchow, Wojtek
Gajda, Andrzej Gaszak, and Adam Neugebauer for their help with the
organizational work and to Barbara Wilczy~ska who handled the ad-
ministrative and secretarial duties.
The second editor thanks Sonderforschungsbereich 170 in G6ttingen
for its hospitality which was very helpful in the preparation of
the present volume. Finally, we would like to thank Marrie Powell
and Christiane Gieseking for their excellent typing.
Pozna~/Warszawa, 20.O6.1986
ANDRZEJ JANKOWSKI (1938-1975) WOJCIECH PULIKOWSKI (1947-1975)
Andrzej graduated in 1960 from the Nicolaus Copernicus University
in Torud. Topology was his passion and his interests were very broad.
Andrzej worked on algebraic and differential topology, his main papers
being concerned with operations in generalized cohomology theories
and with formal groups. His was not an easy task. Andrzej worked essen-
tially alone. Polish topologists were at that time continuing the
tradition of their pre-war school. Andrzej's friend and Ph.D. student
wrote*): "He wanted to understand the deepest and most difficult the-
orems found by his contemporaries. At that beautiful time of great dis-
coveries Andrzej faced the difficult obstacle of being alone. He put
a lot of effort into overcoming this difficulty, and also conveying
his knowledge to others." Andrzej began to lecture on algebraic topo-
logy and to organize seminars as soon as he joined the University of
Warsaw in 1962. For nine years, from 1967, he was the spiritusmovens of
the Summer School on Algebraic Topology held annually in Gda~sk. He
moved to Gda~sk in 1971. From 1969 until his death he led a seminar
on transformation groups. Wojtek Pulikowski was one of the partici-
pants.
Wojtek graduated in 1969 from Pozna~ and moved to Warszawa and then
to Gda~sk. In 1973 Wojtek obtained his Ph.D. for the work on equivari-
ant bordism theories indexed by representations and returned to Pozna~.
He invested great effort into organizing seminars, summer schools and
meetings on various topics in algebraic and differential topology. At
the same time he continued teaching his students and before long di-
rected their research towards transformation groups. Wojtek was a born
teacher, able to convey not only his knowledge but also his passion,
enthusiasm, and interest in the subject. He wrote a number of papers
on equivariant homology theories, but he spent most of his time in
teaching - which he did with joy and love. His friends and students all
owe him a great deal.
Besides mathematics, both Andrzej and Wojtek had another passion -
mountains. And in the mountains both of them met their death in August
1975, Andrzej climbing the Tirach Mir peak in the Hindu Kush mountains
and Wojtek in an accident in the Beskidy mountains in Poland.
*) R.Rubinsztein: "Andrzej Jankowski (1938-1975)", Wiadomo~ci Mate-
matyczne, vol. XXIII (1980), pp. 85-91.
TABLE OF CONTENTS
Chronological list o f talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
Current addresses of authors and participants .......................................................... XII
Allday,C., and Puppe,V.: Bounds on the torus rank .............. 1
Andrzejewski,P.: The equivariant Wall finiteness obstruction and Whitehead torsion ....................................... ii
Assadi,A.: Homotopy actions and cohomology of finite groups .... 26
Assadi,A.: Normally linear Poincar6 complexes and equivariant splittings .................................................. 58
Carlsson,G.: Free (Z/2)k-actions and a problem in commutative algebra ..................................................... 79
tom Dieck,T. und L6ffler,P.: Verschlingungszahlen von Fixpunkt- mengen in Darstellungsformen. II ........................... 84
Dovermann,K.H., and Rothenberg,M.: An algebraic approach to the generalized Whitehead group ................................. 92
Battori,A.: Almost complex Sl-actions on cohomology complex projective spaces ........................................... 115
Illman,S.: A product formula for equivariant Whitehead torsion and geometric applications .................................. 123
Jaworowski,J.: Balanced orbits for fibre preserving maps of S 1 and S 3 actions .............................................. 143
Kania-Bartoszy~ska,J.: Involutions on 2-handlebodies ........... 151
Katz,G.: Normal combinatorics of G-actions on manifolds ........ 167
Kawakubo,K.: Topological invariance of equivariant rational Pontrjagin classes .......................................... 183
Ko{niewski,T.: On the existence of acyclic F complexes of the lowest possible dimension .................................. 196
Laitinen,E.: Unstable homotopy theory of homotopy representa- tions ....................................................... 210
Liulevicius,A., and Ozaydin,M.: Duality in orbit spaces ........ 249
Marciniak,Z.: Cyclic homology and idempotents in group rings... 253
Masuda,M.: ~2 surgery theory and smooth involutions on homo- topy complex projective spaces .............................. 258
Matumoto,T., and Shiota,M.: Proper subanalytic transformation groups and unique triangulation of the orbit spaces ......... 290
May,J.P.: A remark on duality and the Segal conjecture ........ 303
Pedersen,E.K.: On the bounded and thin h-cobordism theorem parametrized by ~k ....................................... 306
Ranicki,A.: Algebraic and geometric splittings of the K- and L-groups of polynomial extensions .......................... 321
Schw~nzl,R., and Vogt,R.: Coherence in homotopy group actions 364
Szczepa6ski,A.: Existence of compact flat Riemannian manifolds with the first Betti number equal to zero .................. 391
Weintraub,S.H.: Which groups have strange torsion? ............ 394
CHRONOLOGICAL LIST OF TALKS
A.Liulevicius (Chicago):
S.Illman (He]sinki):
P.L6ffler (G6ttingen):
W.Marzantowicz (Gda~sk):
A.Szczepa~ski (Gda6sk):
W.Browder (Princeton):
V.Puppe (Konstanz):
Z.Marciniak (Warszawa):
E.Laitinen (Helsinki):
J.Kania-Bartoszy~ska (Warszawa):
R.Vogt (Osnabr~ck):
K.H.Dovermann (West Lafayette):
M.Lewkowicz (Wroclaw):
A.Assadi (Charlottesville):
E.K.Pedersen (Odense):
M.Sadowski (Gda~sk):
J.Tornehave (Aarhus):
S.Weintraub (Baton Rouge):
K.Pawa~owski (Pozna~):
R.Oliver (Aarhus):
S.Jackowski (Warszawa):
J.Ewing (Bloomington):
Duality of symmetric powers of cycles
Product formula for equivariant White- head torsion
Realization of exotic linking numbers of fixed point sets in representation forms
The Sl-equivariant topology and periodic solutions of ordinary differential equa- tions
Euclidean space forms with the first Betti number equal to zero
Actions on projective varieties
Bounds on the torus rank
Idempotents in group rings and cyclic homology
Unstable homotopy theory of homotopy representations
Classification of involutions on 2-handlebodies
Coherence theory and group actions
Symmetries of complex projective spaces
Nonabelian Lie group actions and posi- tive scalar curvature
Homotopy actions and G-modules
The bounded and thin h-cobordism theo- rems
Injective sl-actions on manifolds covered by ~n
Units in Burnside rings and the Kummer theory pairing
Group actions and certain algebraic varieties
Smooth group actions on disks and Euclidean spaces
A transfer map for compact Lie group actions
A fixed point theorem for p-group actions
Symmetries of surfaces and homology
CURRENT ADDRESSES OF AUTHORS AND PARTICIPANTS
Christopher Allday Department of Mathematics University of Hawaii at Manoa Honolulu, HI 96822, USA
Pawe[ Andrzejewski Instytut Matematyki Uniwersytet Szczeci~ski ul. Wielkopolska 15 70-451 Szczecin, Poland
Amir H. Assadi Department of Mathematics University of Wisconsin Madison, WI 53706, USA
Grzegorz Banaszak Instytut Matematyki Uniwersytet Szczeci6ski ul. Wielkopolska 15 70-451 Szczecin, Poland
Agnieszka Bojanowska Instytut Matematyki Uniwersytet Warszawski PKiN, IX p. OO-901 Warszawa, Poland
William Browder Department of Mathematics Princeton University Princeton, NJ 08544, USA
Gunnar Carlsson Department of Mathematics Princeton University Princeton, NJ 08544, USA
Tammo tom Dieck Mathematisches Institut Universit~t G6ttingen BunsenstraSe 3-5 3400 G6ttingen, West Germany
Ryszard Doman Instytut Matematyki Uniwersytet im. A.Mieckiewicza ul. Matejki 48/49 60-769 Pozna£, Poland
Karl Heinz Dovermann Department of Mathematics University of Hawaii at Manoa Honolulu, HI 96822, USA
John Ewing Department of Mathematics Indiana University Bloomington, IN 47405, USA
Wojciech Gajda Instytut Matematyki Uniwersytet im. A.Mickiewicza ul. Matejki 48/49 60-769 Pozna~, Poland
Andrzej Gaszak Instytut Matematyki Uniwersytet im. A.Mickiewicza ul. Matejki 48/49 60-769 Pozna£, Poland
Jean-Pierre Haeberly Department of Mathematics University of Washington Seattle, WA 98195, USA
Akio Hattori Department of Mathematics Faculty of Science University of Tokyo Hongo, Tokyo, 113 Japan
S6ren Illman Department of Mathematics University of Helsinki Hallituskatu 15 OOIOO Helsinki iO, Finland
Stefan Jackowski Instytut Matematyki Uniwersytet Warszawski PKiN, IX p. OO-901 Warszawa, Poland
Tadeusz Januszkiewicz Instytut Matematyki Uniwersytet Wroclawski Pl. Grunwaldzki 2/4 50-384 Wroc!aw, Poland
XIIl
Jan Jaworowski Department of Mathematics Indiana University Bloomington, IN 47405, USA
Joanna Kania-Bartoszylska Department of Mathematics University of California Berkeley, CA 94720, USA
Gabriel Katz Department of Mathematics Ben Gurion University Beer-Sheva 84105, Israel
Katsuo Kawakubo Department of Mathematics Osaka University Toyonaka, Osaka, 560 Japan
Tadeusz Ko~niewski Instytut Matematyki Uniwersytet Warszawski PKiN, IX p. OO-901 Warszawa, Poland
Piotr Kraso6 Instytut Matematyki Uniwersytet Szczecilski ul. Wielkopolska 15 70-451 Szczecin, Poland
Erkki Laitinen Department of Mathematics University of Helsinki Hallituskatu 15 OOIOO Helsinki I0, Finland
Marek Lewkowicz Instytut Matematyki Uniwersytet Wroclawski PI. Grunwaldzki 2/4 50-384 Wroclaw, Poland
Arunas Liulevicius Department of Mathematics University of Chicago Chicago, IL 60637, USA
Peter L6ffler Mathematisches Institut Universit~t G6ttingen Bunsenstr. 3-5 3400 G6ttingen, West Germany
Ewa Marchow Instytut Matematyki Uniwersytet im. A.Mickiewicza ul. Matejki 48/49 60-769 Pozna~, Poland
Zbigniew Marciniak Instytut Matematyki Uniwersytet Warszawski PKiN, IX p. 00-901 Warszawa, Poland
Waclaw Marzantowicz Instytut Matematyki Uniwersytet Gdalski ul. Wita Stwosza 57 80-952 Gda~sk, Poland
Mikiya Masuda Department of Mathematics Osaka City University Osaka 558, Japan
Takao Matumoto Department of Matheamtics Faculty of Science Hiroshima University Hiroshima 730, Japan
J. Peter May Department of Mathematics University of Chicago Chicago, IL 60637, USA
Janusz Migda Instytut Matematyki Uniwersytet im. A.Mickiewicza ul. Matejki 48/49 60-769 Poznal, Poland
Adam Neugebauer Instytut Matematyki Uniwersytet im. A.Mickiewicza ul. Matejki 48/49 60-769 Pozna6, Poland
Krzysztof Nowilski Instytut Matematyki Uniwersytet Warszawski PKiN, IX p. OO-901 Warszawa, Poland
Robert Oliver Matematisk Institut Aarhus Universitet Ny Munkegade 8000 Aarhus C, Denmark
XIV
Murad Ozaydin Department of Mathematics University of Wisconsin Madison, WI 53706, USA
Michal Sadowski Instytut Matematyki Uniwersytet Gda~ski ul. Wita Stwosza 57 80-952 Gda~sk, Poland
Krzysztof Pawalowski Instytut Matematyki Uniwersytet im. A.Mickiewicza ul. Matejki 48/49 60-769 Pozna~, Poland
Jan Samsonowicz Instytut Matematyki Politechnika Warszawska PI. Jedno£ci Robotniczej 1 00-661 Warszawa, Poland
Erik Kjaer Pedersen Matematisk Institut Odense Universitet Campusvej 55 5230 Odense M, Denmark
Roland SchwAnzl Fachbereich Mathematik Universit~t Osnabr~ck AlbrechtstraSe 28 4500 OsnabrQck, West Germany
Jerzy Popko Instytut Matematyki Uniwersytet Gda6ski ul. Wita Stwosza 57 80-952 Gda~sk, Poland
Masahiro Shiota Department of Mathematics Faculty of General Education Nagoya University Nagoya 464, Japan
J6zef Przytycki Instytut Matematyki Uniwersytet Warszawski PKiN, IX p. OO-901 Warszawa, Poland
Jolanta Sgomi~ska Instytut Matematyki Uniwersytet im. M.Kopernika ul. Chopina 12 87-1OO Toru~, Poland
Volker Puppe Fakult~t fur Mathematik Universit~t Konstanz Postfach 5560 7750 Konstanz, West Germany
Andrzej Szczepa~ski Instytut Matematyki Politechnika Gda~ska ul. Majakowskiego 11/12 80--952 Gda~sk, Poland
Andrew Ranicki Department of Mathematics Edinburgh University King's Buildings, Mayfield Rd. Edinburgh EH9 3JZ, Scotland, UK
J~rgen Tornehave Matematisk Institut Aarhus Universitet Ny Munkegade 8000 Aarhus C, Denmark
Martin Raussen Pawel Traczyk Institut for Elektroniske Systemer Instytut Matematyki Aalborg Universitetscenter Uniwersytet Warszawski Strandvejen 19 PKiN, IX p. 9000 Aalborg, Denmark OO-901 Warszawa, Poland
Melvin Rothenberg Department of Mathematics University of Chicago Chicago, IL 60637, USA
Rainer Vogt Fachbereich Mathematik Universitit Osnabr~ck AlbrechtstraSe 28 4500 Osnabr~ck, West Germany
Slawomir Rybicki Instytut Matematyki Politechnika Gda£ska ul. Majakowskiego 11/12 80-952 Gda6sk, Poland
Steven H. Weintraub Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA
Bounds on the torus rank
C. Ailday and V. Puppe
For a topological space X let rko(X) := max{dim T , where T is a torus
which can act on X almost freely (i.e. with only finite isotropy sub-
groups)} be the torus rank of X . Stephen Halperin has raised the
following question (s.[11]) :
rk (X) (HD) Is it true that dim~ H~(X;~) ~ 2 o for any simply connected
reasonable space X ?
In this context "reasonable" (s. [11]) is a technical condition which
assures that one can apply the A. Borel version of P.A. Smith theory
(s. [4],[5],[12]) and Sullivan's theory of minimal models (s.[13],[IO],
[14]). In particular any connected finite CW-complex is certainly
"reasonable", but X being connected, paracompact, finitistic (s.[5]
p. 133) and of the rational homotopy type of a CW-complex would also
suffice.
In the first section we give some lower bounds for dim~ H~(X;~)
if X allows an almost free action Of an n-dimensional torus G = T n
These results are obtained using only the additive structure in H~(X;Q)
(and a version of the localization theorem (s. [2])) and hold for
rather general spaces X e.g. simply connecte~ness is not needed; but ' r o~X)
the bounds we get are far below the desired 2
The second section gives bounds on the torus rank in terms of the
cohomology of X , where a very special structure of the cohomology
ring H~(X;~), i.e. X being a rational cohomology K~hler space, is
used.
The third section is concerned with relations between properties
of the minimal model M(X) of X (in particular the rational homotopy
Lie algebra L~(X)), rko(X) and dim~ H~(X;~). Halperin observed
(s.[11], 1.5) that the results of [I], in particular the inequality
rko(X) ~ - X~(X), where Xz(X) is the rational homotopy Euler charac-
teristic (s.[1], Theorem I), implies an affirmative answer to his
question if X is a homogenous space G/K, K c G compact, connected Lie
groups. Among other th~g~x~e~ ~ . describe another class of spaces for
which dim~ H~(X;~) ~ 2 o holds, but the bound on the torus rank
given by the rational homotopy Euler characteristic is not sharp in
many cases (compare also [11], 4.4) and does not suffice to answer
(H~). Indeed, for this class the knowledge of the additive structure
of ~(X) ® Q is not enough; it is essential to use the Lie algebra
structure of L.(X) ~ ~,(~X) ® Q .
I. Let X be a connected, paracompact, finitistic space which has the
rational homotopy type of a CW-complex and on which a torus G = T n of
dimension n acts almost freely. If M(X) is the minimal model of X over
the field • of complex numbers and R := H*(BG;~) ~ ~[t I .... ,tn], then the
R-coohain algebra CG(X) := R ~ M(X) (where the twisting of the bound-
ary, indicated by "N,', reflects the G-action) is a model for the Borel
construction X G-
For any ~ = (~1,...,en) 6 ~n we denote by ~e the field • together
with the R-algebra structure given by the evaluation map
ee: R = ~[t I, .... tn] ~ ~' ti -'~ ei for i = I ..... n. The cochain alge-
bra CG(X) ~ (over 6) is defined to be the tensor product CG(X)a :=
~ ~ CG(X). Theorem (4.1) of [2] implies that H~(CG(X) ~) = O for all
• 0 (since the G-action is assumed to be almost free). It follows
from a theorem of E.H. Brown (s• [7], (9.1), compare [2], (2.3)) that
there exists a twisted boundary on DG(X) := R ~ H*(X;~) which makes
R ~ H~(X;~) homotopy equivalent to R ~ M(X) as R-cochain complexes. We
therefore get (for an almost free action) that H(DG(X) ~) = O for all
¢ 0 and we shall use this information to obtain the following propo-
sition:
!IoI) Proposition: Under the above hypothesis one has
a) dimQ H~(X;Q) ~ 2n for all n = 1,2,...
b) dimQ H*(X;Q) ~ 2(n+I) for all n ~
Proof: We can of course assume that dimQ H~(X;Q) is finite, and the
fact that the action has no fixed point implies that the Euler charac-
teristic X(X) is zero. Let Xl,...,x k be a homogenous Q-basis of
HeV(X;Q) with [Xll ~...> IXk[ = O and Yl ..... Yk a homogenous Q-basis
of H°dd(x;Q) with [y1[ Z .... ~ [Yk I > O ([ I denotes degree)• Since
Iti[ = 2 for i = I, .... n the twisted boundary d on R @ H*(X;~) is giv-
en by two kxk-matrices P = (Pij) and Q = (qij), where the entries
Pij' qij are homogenous polynomials in the variables tl,...,t n of de-
gree % O, i.e.
~F I = P11X1 +--.+ PlkXk ?Xl = q11Y1 +...+ qlkYk
~Yk = PklXl +'''+ PkkXk ~Xk = qklYl +'''+ qkkYk "
If Pij * O (rasp. qij % O) then lyil > Ixjl and IPi9 I = lyil-lxjI+l N
(rasp. Ixil > lyjl and lqi9 I = IxiI-lyjl+1), in particular dx k ~ O
(i.e. qkj ~ 0 for all j = I ..... k).
The equation ~o~ = O is equivalent to PQ = QP = O and the van-
ishing of H(DG(X)~) for any e 6 cn~{o] then means that rkP(~) + rk~d)
= k for all ~ 6 ~n~{o}, where rkP(~) denotes the rank of the k×k-ma -
trix over { obtained from P by evaluating the polynomials Pij at the
point ~ E ~n (similar for rk~x)). The semi-continuity of rk P(e) and
rk Q(e) (as a function of e) (together with rk P(e) + rk Q(e) = k for
• o) then implies that rk P(e) and rk Q(e) have to be constant on
end{o}.
To prove part a) on only needs to observe that the variety
V(Plk,...,Pkk) can only consist of the point O E ~n If the polyno-
mials Pik' i = 1,...,k would have a common zero e 6 ~n~{o} then "at
the point e" the cycle x k could not be a boundary and hence H(DG(X) e)
would not vanish. Since the Pik' i = 1,...,k are k polynomials in n
variable one gets k ~ n (otherwise V(Plk, .... Pkk) D ~n~{0} ~ ~).
To get the slight improvement b) a considerably more involved
argument is necessary:
We assume k = n and will show that this implies n < 2.
Case I: Let lyll > Ixil for all i, i.e. the top dimensional classes
have odd degree. Again V(Pl n .... ,Pnn ) = 0 and it now follows that
Pln .... 'Pnn is a regular sequence in R = C[t I .... ,tn]. Therefore the
~S condition QP = O implies that all the qij are contained i~ the ide-
al <Pln .... ,Pnn > c R generated by Pin' i : 1,...,n. (From [ qijpjn=O
it follows that the equivalence class of qij Pjn in j=1 A
R/(Pln'''''Pjn'''''Pnn ) is zero. The regularity of the sequence
Pln'''''Pnn then implies that the class of qij is already zero in
^ • ^ . . . . . . Pnn ) R/(Pln' .... Pjn ..... Pnn ). Hence qi ~ E (Pln .... Pjn "''Pnn) C(Pln '
for all i,j = I, .,n.) Since lqij I = Ixi]-iYjl+1 < ly11+1 = IPln I one
actually has qij 6 (P2n ..... Pnn ) for all i,j = 1, .... n. (This is where
we use the assumption that the top classes have odd degree and - as
one sees from the above inequality - the weaker assumption
"]YIi+lYkl>Ix11" would suffice.) Choosed 6 V(P2 n ..... Pnn ) fl (~n~{o}) .
Then qij (e) = 0 for i,j = I .... ,n and hence rk Q(~) = O. Since rk Q is
constant on {n~{o} we get Q ~ 0 and rk P must therefore be maximal
(= n) on ~n~{o}. Since det P is a polynomial in the variables
tl,...,t n this can only happen if n = I.
Case 2: Let IXll > lyil for all i, i.e. the top classes have even de-
gree. We have qnj E 0 for j = 1,...,n; V(Pl n .... ,Pnn ) = O, i.e.
Pln,...,pn n is a regular sequence (as before), and in addition Pil ~ O
for i = 1,...,n (for degree reasons); V(q11,...,q1 n) = O, i.e.
q11t...,qln is a regular sequence, since otherwise x I would give a non-
zero element in H(DG(X) e ) for any ~ E V(q11,...,ql n) ~ (~n~{o}). Anal-
ogous to case I we get from QP = PQ = O that qi~ E (Pln,...,Pnn) and
Pij £ (q11' .... qln ) for all i,j = 1,...,n. In particular (Pln' .... Pnn )
= (q11 ..... qln ). Since IPln I > IPij I for all (i,j) with j < n it fol-
lows that Pij 6 (P2n ..... Pnn ) if (i,j) # (1,n). For n > 1 choose
6 V(P2n, .... Pnn) n ({n~{o}), then rk P(~) = I and therefore rk Q(~)=
n-1. This implies rk Q = n-1 on ~n~{o}. Since qnj ~ 0 for all j =
1,...,n the (n-l) x (n-l) minors QI .... 'Qn of the matrix
q11 .... qln have to form a regular sequence
• . (Qj is obtained by skipping the j-th column) P
qn-11 .... qn-ln
The expension formula for the determinant (with respect to the first
row) of the matrix
q11 " '' qln
q11 " "" qln
q21 "'" q2n
qn-1 I ' ° q:n-ln
gives q11Q1-q12 Q2 + + (-I)n+I "'" qln Qn = O.
As above one gets qij 6 (QI .... 'Qn ) for all i,j = I ..... n. This is on-
ly possible if (n-l) = I (otherwise Iq111 < IQjl for all j, which
gives a contradiction). This finishes the proof of (1.1).
~<!.2) Corollary: If X is a paracompact, finitistic space which has the
rational homotopy type of a CW-complex, then
a) dim~ H~(X;~) ~ 2 rko(X)
b) dim~ H~(X;~) ~ 2(rko(X)+1), if rko(X) ~ 3.
(1.3) Remark: G. Carlsson has asked the analogous question to (H
concerning spaces X on which an elementary abelian 2-group G = (~/2)n
acts freely (such that X becomes a finite G-CW complex). Results of
Carlsson [8] and - using different methods - of W. Browder [6] in this
direction imply in particular, that if G acts trivially in cohomology
with Z/2 coefficients s. [8] (resp. ~(2) = ~ localized at 2 s. [6])
then the cohomology of X (with the corresponding coefficients) is non-
zero in at least n+l different dimensions.
The methods used to prove proposition (1.1) above can be applied
in a similar fashion to free (~/2)n-actions (compare [2], 2.). One
then obtains for a finite, free (~/2)n-cw complex X with trivial ac-
tion on H*(X;Z/2):
a) dimz/2 H~(X;2/2) ~ n+1 for all n
b) dim~/2 H~(X;Z/2) ~ n+2 for n ~ 2
C0~bining our approach with Carlsson's result leads to
c) dimz/2 H~(X;~/2) ~ 2n for all n.
Browder's methods also work for G (~/p)n = , p an odd prime and he
proves results analogous to the case p = 2 also for odd primes. The
above approach would also work for p an odd prime (compare [2], 3.)
but there are some technical complication arising from the more com-
plicated structure of H~(B(~/p)n; ~/p) in case p is odd.
2. Let X be a connected paracompact finitistic space which is a ra-
tional Poincar6 duality space of formal dimension 2m. In this sec-
tion H ~ denotes sheaf (or Alexander-Spanier or Cech) cohomology and
H~ denotes singular cohomology. X will be called an agreement space
if the natural transformation H~(X;Q) ~ H~(X;~) is an isomorphism
(e.g. X reasonable, as above).
(2.1) Definition: (compare [4]) The space X is said to be a rational
cohomology K~hler space (CKS) if
(i) there exists ~ 6 H2(X;~) such that m is non-zero in H2m(x;~)~;
and
(ii) the cup-product with ~J: H m-j (X;~) ~ H m+j (X;~) is an isomorphism
for 0 < j < m.
(2.___2! Theore______~m: If X is a CKS, then rko(X) ~ el(X) := the maximal num-
ber of algebraically independent elements in HI (X;~). In particular,
dim~ H*(X;~) ~ 2 rk°(x) . Furthermore if X is an agreement space and
if a torus G acts almost-freely on X, then H*(X;~) and H*(X/G;~) ®
H~(G;~) are isomorphic as graded ~-algebras.
Proof: Suppose G : T n acts almost-freely on X. Let EG ~ BG be a uni-
versal principal G-bundle, and let X G = (XxEG)/G be the Borel con-
struction. Let E P'q be the rational cohomology Leray-Serre spectral r
sequence.of X G ~ BG; and let s be the rank of the linear map _2,0 ~ .2 H I d2 :- E2o,1 = HI(x;~) ~ ~2 = n (BG;~). Choose YI' .... Ys £ (X;~) such
that d2(Y i) = ai, I < i < s, are linearly independent. Then, for -- - -j+1 ^ .
I ~ i I <...< ij+ I < s , d2(Yil...y i ) = ~ ~ ai®Yi "''Yi "''Yi ' -- j+1 k=1 k I k j+1
Hence it follows by induction that yl...ys # O. I.e. s ~ ~I(X). Now let
K be the subtorus such that the ideal (a I .... ,a s ) = ker[H~(BG;~) ~
H~(BK;~)]. In particular, dim K = n-s. In the Leray-Serre spectral se-
quence of X K BK, then d2: _o,I 2,0 ~2 ~ E 2 is zero. Thus, by Blanchard
([3]), the spectral sequence collapses; and so X K % ~. So K is trivial,
and n = s ~ el(X).
If X is an agreement space, then it follows from the fibre bundle
X ~ X ~ BG that X G is an agreement space also. Now, above,
d2: H~(X;~) ~ H2(BG;~) is onto, since G is acting almost-freely: hence
H2(BG;~) ~ H2(XG;~ ) is zero. On the other hand G ~ X×EG ~ X G is the
pull-back of G ~ EG ~ BG via X G ~ BG; in particular it is orientable
with respect to H~ . Thus X (homotopy equivalent to XxEG) has a K.S.-
model of the form M(X G) ® A(s I .... ,Sn), where deg(s i) = I, lJi<n, and
d(s i) = O, 1~iJn (since H2(BG;~) ~ H2(XG;~) is zero). Hence H%(X;~)
H~(XG;~) ® H~(G;~) as algebras. So H~(X;~) = H~(X/G;~) ® H~(G;~) , as
algebras, by the Vietoris-Begle mapping theorem.
(2.3) Remarks: (i) An argument similar to the above, applied to a sim-
ple closed connected subgroup, shows that no non-abelian compact con-
nected Lie group can act almost-freely on a CKS: and only condition
(i) of definition (2.1) is needed for this.
(ii) Again if we assume only condition (i) of definition (2.1),
then we get rko(X) ~ BI(X) := dim~ HI(x;~).
(iii) Theorem (2.2) is "best possible", since T 2m is a CKS, and it can
act freely on itself.
3. Let X be a simply connected reasonable space and let L~(X) =
z~(~X) ® ~ denote its rational homotopy Lie algebra. The following re-
sult is proved in [2], Theorem (4.6)':
If Li(X) = O for all odd i, then rko(X) ~ dim~ Z L~(X), where Z L~(X)
denotes the centre of the Lie algebra L~(X).
This improves the bound given by -X~(X) = dim L~(X) for this type
of spaces and it is clear that one does need some improvement in this
direction to get an affirmative answer to (H~) since dim~ H*(X;~)
2dim L~(X) in the case at hand and equality holds only if the minimal
model of X has trivial boundary (compare [2],(4.5)). In fact, the min-
imal model for a space X with Lodd(X) = O is the exterior algebra
A~(V) over the vector space V = Hom(L~(X),O) dual to L~(X) with a de-
gree shift and a boundary which is a derivation on A(V). The quadratic
part of the boundary corresponds (under duality) to the Lie multipli-
cation of L,(X). The next simplest case to having a trivial boundary
on the minimal model M(X) of X would be to have the boundary complete-
ly determined by the Lie product of L,(X). These are the so-called ~-
formal (or co-formal) spaces. Their rational homotopy type is deter-
mined by L,(X) and in particular the cohomology H*(X;~) is just the
algebraica~ydefined cohomology H*(L,(X)) of the Lie algebra L,(X).
Together with the above bound on rk (X) one would get an affirmative
answer to (HQ) if dim H(L,) ~ 2 dim ~ L, for graded, connected Lie al-
gebras with Lod d = O . We do not know whether this holds in general,
but Deninger and Singhof (s.[9]) have given lower bounds for the di-
mension of the cohomology of a nilpotent Lie algebra which imply the 3 above inequality if L, = O (i.e. all three fold Lie brackets are zero).
Putting all this together we get:
o~(3"I) ..... Proposition-. Let X be a simply-connected, reasonable, T-formal 3
space such that Lodd(X) = O and L,(X) = O, then
dim~ H*(X;~) ~ 2 dim Z L,(X) ~ 2rko(X)" "
If X is not z-formal (i.e. the boundary of M(X) is not determined
by the Lie product of L,(X)), then - similar to the situation described
above for Xz(X) - the upper bound on rko(X) given by dim Z L,(X) will
not suffice to provide the desired lower bound on dim~ H*(X;~). But
non-vanishing higher order Whitehead products (i.e. non-trivial higher
order terms in the boundary of M(X)) will reduce the torus rank of X
below dim Z L,(X) in general. This is illustrated by the following
example:
(3.2) Example: Let X be a finite CW-complex such that M(X) :
A(yl,Y2,Y3,Y4,y), where deg(y i) = 3,1 ~ i ~ 4, deg(y) = 11, dy i = O,
I~i~4 and dy = yiAY2AY3^Y4 • Then dim~ ZL,(X) = dim~ L,(X) = 5. But, by
[2], Theorem (4.1), rko(X) J 1.
One might ask whether the bound given by dim Z L,(X) is "best pos-
sible" for reasonable, simply connected, q-formal spaces X with
Lodd(X) = ~ev(X) ® ~ = O , or more general ask for lower bounds (in
terms of M(X)) on the torus rank of X . Since M(X) depends only on the
rational homotopy type of X, "best possible" is to be interpreted as
"within the rational homotopy type of X there exists a simply-con ~
nected, reasonable space X which has the desired torus rank" (compare
[11], 4.3). The following propositions provide an answer to this ques-
tion.
(3.3) Proposition: Let M(X) = (A*(V),6) be the minimal model (over ~)
of a simply-connected, reasonable space X with Lodd(X) = ~ev(X)®~ = O
and dim~ H~(X;~) < ~ If the boundary 6: V ~ A~(V) factors through
A~(W), where W c V is a (graded) linear subspace of the (graded) vee ~
tor space V of codimension n < ~ , then there exists a simply con-
nected, finite CW-complex X which is rationally homotopy equivalent to
X and carries a free action of an n-dimensional torus G = T n (n =
dim(V/W)).
Proof: In view of [11], 4.2 it suffices to define a twisting ~ of 6 on
~[t I ..... t n] ~ A~(V) (which makes (~[t I ..... t n] ~ i~(V),~)a
~[t I ..... tn]-Cochain algebra) such that the cohomology of
(~[tl,...,t n] ~ A~(V),~) is finite dimensional. We choose a splitting
of V into a direct sum of graded vector spaces V = W e Z and a homo-
genous basis Zl,...,z n of Z. We now define:
~(t i) = O , i = 1,...,n
~(W) := ~ (W) for w 6 W
rziI+1 2
Z(Zi) ~= ~(Z i) + t i , i = 1,...,n .
Since ~ (V] c A~ (W) one gets ~ = 6~6 -= 0 ,
Hence ~IV extends to a unique derivation on ~[tl, .... t n] ~ A~(V)
and is a twisting (n-parameter family of deformations) of ~ .
It remains to show that dim~ H(~[tl,...,t n] ~ A~(V),~) is finite.
Since A~(V) is an exterior algebra (Vev = O) one can find k i £ ~ such
Izil+1 k . k. that (6(zi)) i = - t i = O for i = I, .... n. Since the
t.'s are cycles it follows (using the fact that ~ is a derivation)
that <t i 2 > i is a boundary in ~[t I ..... t n] ~ A~(V) (compare [12],
Chap VII, Len~a (1.1)). A well known spectral sequence argument (using
the "Serre" spectral sequence of the "fibration" ~[t] ..... t n] ~-~
~[tl,...,t n] ~ A~(V) ~ A~(V) and the fact that ~[t I ..... t n] is
noetherian) shows that H(~[tl, .... t n] ~ A~(V),~) is finitely gener-
ated over ~[tl,...,tn] (since dim~ H(A~(V),6) < ~, i.e. H(A~(V),6)
finitely generated over ~). Together with the fact that sufficiently
high powers of the ti's are zero in H(~[tl,...,t n] ~ A~(V) , ~) we get
that dim~ H(~[t I ..... t n] ~ A~(V),~) <
i3.4! Proposition: Let M(X) = (A~(V),6) be a minimal model (over ~) of
a simply connected CW-complex X of finite ~-type (H~(X;~) of finite
type).
a) If 6: V ~ A~(V) factors through A~(W), where W c V is a graded lin-
ear subspace, then dim~ ZL~(X) ~ dim~(V/W).
b) If z(X) := dimQ ZL~(X) then there exists a graded linear subspace
W c V of codimension z(X) such that the quadratic part g of the
boundary 61V (i.e. the composition q: V ~ A~(V) ~ A2(V)) factors
through A 2 (W) (A~V : • AIV and A*V ~ A2V is the canonical projection). i
Proof: a) Clearly q: V 6 A~(V ) ~ A2(V) factors through A2(W). For the
dual multiplication on Hom(V,~) = ~.(X) ® ~ it follows immediately
that all products where one of the factors is contained in Hom(V/W,~)
vanish. Hence Hom(V/W,~) corresponds (after dimension shift) to a sub-
space of the centre ZL~(X) of the Lie algebra L~(X) = ~(~X) ® Q .
b) Let Z c Hom(V,~) be the subspace of ~(X) ® ~ = Hom(V,~) which cor-
responds to ZL~(X) under the dimension shift. The Lie product
L~(X) ® L~(X) ~ L~(X) factors through L~(X)/zL~(X ) ® L~(X)/zL~(X ) and
therefore the dual map ~: V ~ V®V factors through W®W, where W is de-
fined such that Hom(W,~) = Hom(V,~)/z. Since ~ is anti-commutative the
dual ~ actually factors through £2w and the map V ~ A2W ~ A2V ob-
tained this way coincides with the quadratic part q of 6 .
(3.5) Corollary: Let X be a simply connected, finite, n-formal CW-com-
plex with Lodd(X) = ~ev(X) ® ~ = O . Then the torus rank of the ratio-
nal homotopy type of X (s.[11], 4.3) is equal to the dimension of the
centre ZL~(X) of the rational homotopy Lie algebra L~(X) = ~(QX) ®
of X . In other words:
rko(X) ~ dim~ ZL~(X) and this bound is "best possible".
References
[I] ALLDAY, C. and HALPERIN, S.: Lie group actions on spaces of finite rank. Quart. J. Math. Oxford (2) 29, 69-76 (1978)
[2] ALLDAY, C. and PUPPE, V.: On the localization theorem at the co- chain level and free torus actions. (preprint)
[3] BLANCHARD, A.: Sur les vari~t~s analytiques complexes. Annales Ec. Norm. Sup. 73, 157-202 (1957)
[4] BOREL, A.: Seminar on Transformation Groups. Annals of Math. Studies, No. 46, Princeton, New Jersey: Princeton Univ. Press 1960
10
[5] BREDON, G.E.: Introduction to Compact Transformation Groups. New York - London: Academic Press 1972
[6] BROWDER, W.: Cohomology and group actions. Invent Math. 71, 599-607 (1983)
[7] BROWN, E.H.: Twisted tensor products, I. Ann. of Math. 69, 223-246 (1959)
[8] CARLSSON, G.: On the homology of finite free (~/2)n-complexes, Invent Math. 74, 139-147 (1983)
[9] DENNINGER, C. and SINGHOF, W.: On the cohomology of nilpotent Lie algebras. (preprint)
[10] HALPERIN, S.: Lectures on Minimal Models. Memoirs de la Soc. Math. France 0984)
[11] HALPERIN, S.: Rational homotopy and torus actions. (preprint) [12] HSIANG, W.Y.: Cohomology Theory of Topological Transformation
Groups. Berlin-Heidelberg-New York: Springer 1975 [13] SULLIVAN, D.: Infinitesimal computations in topology.
.Inst. Hautes Etudes Sci. Publ. Math. No. 47, 269-331 (1977) [14] TANRE, D.: Homotopie rationelle: Mod61es de Chen, Quillen,
Sullivan. Lect. Notes Math. 1025, Berlin, Heidelberg, New York: Springer !983
THE EQUIVARIANT WALL FINITENESS
OBSTRUCTION AND WHITEHEAD TORSION
Pawel Andrzejewski Szczecin, Poland
Dedicated to the memory of Andrzej Jankowski and Wojtek Pulikowski
Let G be a compact Lie group and X a G-CW-complex G-dominated by a finite one.
Then it is natural to ask whether X has the G-homotopy type of a finite G-CW-com-
plex. As in the non-equivariant case [16] one can expect that the answer to this
question will depend on some algebraic invariants. The aim of this paper is to describe
the equivariant version of the finiteness obstruction from the following two points
of view.
The first one goes along the classical Wall's line. Namely, for any closed sub-
group H of G and any component X H of X H one defines [i0] the group
= E w(x ) = _ its lifting which is a Lie group and acts
on the universal covering X H . These groups are used to define the family of
e l e m e n t s w (X) ~ Ko(Z[~o(~I-I )~]) and t o show t h a t t h e f i n i t e l y d o m i n a t e d G-CW-
complex X is G-homotopy equivalent to a finite G-CW-complex iff all wH(x) are
zero.
On the other hand, it is not difficult to generalize the construction of the
finiteness obstruction given by S. Ferry [7] to the equivariant case. Precisely,
under the above assumption on X there exists a single invariant CG(X)eWH h (X × SI),
such that OG(X) = 0 iff X is G-finite (up to G-homotopy type).
Moreover, there is a natural relaion between these obstructions. By the results
of Illman [i0] and Bass-Heller-Swan [5] the equivariant Whitehead group WhG(X × S I)
maps onto the direct sum
~ Ko(Z[~o(WH)~]) H
of (reduced) projective class groups and we are able to prove that the image of
OG(X) decomposes exactly into the family of elements wH(x) .
In the case when G is a finite group and any fixed point set X H is connected
and non-empty, J. Baglivo [4] has defined an algebraic Wall-type obstruction to
finiteness. The equivariant version of the finiteness obstruction for finite group
actions was also established by D. Anderson [i] . (Unfortunately, the definition of
the obstructions in [i] is not quite correct). The generalization of Ferry's work
is due to S. Kwasik [12] and we briefly recall his results. Recently W. L~ck [13]
has presented another geometrical approach to the finiteness obstruction.
12
A short survey of the contents of the paper is as follows. Section 1 contains the
description of the relative equivariant Wall-type obstruction WG(X,A) for relatively
free actions which plays a crucial role in the next section that deals with the general
construction of the invariants wH(x) . As stated above, we shortly recall the
generalization of Ferry's results [12] and this is done in section 3 while section 4
contains the comparison of these obstructions via the Bass-Heller-Swan isomorphism.
Finally, applying Illman's result [ii] we obtain in section 5 a product formula for
finiteness obstructions and its geometric application.
Our notations are the standard ones. For any closed subgroup H of G we define
X >H = {x ~X : G× $ H} ,
Furthermore, we denote X (H) = GX H and X >(H) = GX >H , We define a partial order in
the set of all conjugacy classes by (H) ~ (K) iff there exists g ~ G such that
gHg -I m K , and by (H) > (K) we mean (H) ~ (K) and (H) ~ (K) . We also assume
familiarity with the first part of Illman's paper [I0].
I wish to thank the referee for helpful suggestions which allowed to improve the
final version of this paper,
i. The case of a relatively free action
Let G be a compact Lie group and X a G-space. The space X is G-dominated by
the G-space K if there exist G-maps ~ : K --, X and s : X --~ K such that
• s G id X . Then ~ is called the domination map and s its section ~.
If now X is a connected G-CW-complex and p : X --, X denotes its universal
covering then we can consider the lifting of the action of the group G on X to
the covering action of a group G on ~ (see sect. 5 in [10] for details). The
group G is a Lie group and fits into the exact sequence
0 ~ ~I(X) ~ G ~ G --~ 0 .
Moreover, X is a G -CW-complex ([i0] th. 6.6). Let further A
subcomplex of X such that the inclusion induces an isomorphism
of fundamental groups. Then one can define an action of no(G × )
logy groups ~ (X,A,x) , H (X,A) such that it makes them into modules over the n , o n
group ring Z[~o(G )] (see sect. 7 in [I0]). We say that the action of G on the
pair (X,A) is relatively free if G acts freely on X-A . We say that the G-CW-
pair (X,A) is relatively finite if X-A has finite number of G-cells. By a relatSv e
G-domination~ we mean the G~map ~ : (K,L) --~ (X,A) along with its section
s : (X,A) --~ (K,L) such that ~ s G . ~ id(x,A ) •
Let now the relatively free G~CW-pair (X,A) be G-dominated by a relatively free,
relatively finite G-CW-pair (K,L) via the map ~ : (K,L) ~ (X,A) and let
be a G-invariant
~ I ( A , X o ) z ~ I ( X , X o )
on h o m o t o p y a n d h o m o -
13
A q : K ~ K be the pull-back of p : ~ --~ X by ~ i.e.
= {(~.k) : p(~) = 9(k)} .
The group G acts on K by the formula g,~(x,k) = (g*(~),~(g*)(k)) . Then K is
* * _q- I ( Cn ( Z[~o(G* ) ] a G -CW-complex and G acts freely on K-~ = K ~) . Since X.A) ~
C ~"J the cellular chain complexes C,(K,~) and ,(X.A) are complexes of free Z[~To(G*)]-
modules and C,(K.L) is finite. Moreover the map ~ induces the domination map
C,(K,L) ~ C,(X,A) of chain complexes. We define the relative equivariant Wall
finiteness obstruction as
WG(X,A) = w(C,(~,~))E ~o(Z[~o(G*)] ) ,
where w(C,) is the algebraic finiteness obstruction [17], [4] . The following pro-
perty will serve as an inductive step in the next section.
Proposition I.I. Let a relatively free G-CW-pair (X,A) be G-dominated by a re-
latively free, relatively finite G-CW-pair (K,L) and suppose that L is of finite
type; then there exist a relatively free, relatively finite G-CW-pair (Y,A) and a
G-homotopy equivalence h : (Y,A) --~ (X,A) with hIA = id A iff WG(X,A) = 0 in
~ o ( Z [ ~ o ( G * ) ] ) •
Remarks.
I. The relative equivariant Wall finiteness obstruction was also defined in [2]
by different (geometrical) methods.
2. A G-CW-complex is of finite type if it contains a finite number of G-cells in
each dimension.
In order to prove the above proposition we need some auxiliary facts. Let K,X be
connected G-CW-complexes and ~ : K --+ X a G-map such that
1) f o r a n y s u b g r o u p H o f G ~H d e t e r m i n e s t h e b i j e c t i o n o f t h e s e t s o f com-
p o n e n t s K H and X H and
2) for any H and corresponding components, ~H : KH___~ X H induces the iso-
morphism of fundamental groups.
Let M = M(~) denote the mapping cylinder of ~ and r : M---~ M its universal
covering. We set ~n(~) = Wn(M~,K~,Xo) and Hn(~ ~) = Hn ~ ~(~'KH) " Let us fix the
connected component X~ of X H and take the elements ai~ ~n(~) (i=1,2 .... k)
r e p r e s e n t e d by t h e p a i r s o f maps ( s i , t i ) , s i : D n - - ~ X~, t i : S n - 1 - - ~ K H~ s u c h
i sn-I n denote a G-CW-complex obtained from that ~H~ . ti = s . Let L=K0 e~ U...U e k
K by a d d i n g k G - n - c e l l s o f t y p e (H) v i a t h e G-maps d e t e r m i n e d by t i . E x t e n d
to a G-map ~ : L --~ X by means of the maps s. . Such pair (L,~) is said to I
be obtained from (K,O) by a t t a c h i n g G - n - c e l l s o f t y p e (H) t__o K v i ~ a i . The
14
following is an immediate consequence of the construction.
Lemma 1.2. If i < n then ~i(~ H) = ~i(, H) . If n > 1 then in the exact sequence
of Z[~o(WH)~] -modules
"'" n+l ~) --~ ~-(LH'KH'x~ ~ ~ o ) ~ ~ n(~0c~ --~ 0--~ ...
"~n(L , K ,x o) i s a f r e e 2~[~o(!~rI)c~]-module w i t h g e n e r a t o r s b i such t h a t
d ( b i ) = a i ( i = 1 , 2 . . . . . k ) .
Now we a p p l y t h e c e l l - a t t a c h i n g t e c h n i q u e t o g e n e r a l i z e t h e r e s u l t s o f [ 4 ] .
Lemms 1.3. Let G be a compact Lie group and X a G-CW-complex. If X is G-do-
minated by a finite G-CW-complex then X is G-homotopy equivalent to a G-CW-com-
plex Y of finite type.
Proof. We shall show inductively that:
If ~ : K --~ X is a G-(n-l)~connected domination map and K is finite then there
exists a finite G-CW-complex L containing K and G-n-connected extension
: L --~ X of ~ .
There are three cases to consider.
H H ~ H Case I (n = 0) . Since }, : z (K) -- ~ (X) is an epimorphism so let us
o o H H map to Lhe component X H . We attach a G-l-cell suppose the components K 1 , K 2
of type (H) to K (H) to obtain an extension ~ : L--~ K of #H which induces
an isomorphism on the ~ -level. o
Case 2 (n = i). Let ~H : KH~ X H be the restriction to corresponding com- %H
ponents. The group ~2(~) is finitely generated in ~I(K~)~ so we can extend
to 9 by attaching G-2-cells of type (H) via the generators of the group
~2(~) • Case 3 (n > I). By assumption ~n(~)_ ~ Hn(~ ~)_ is finitely generated
Z[~o(WH)~]-module, where (~)~ = {w • WH ; wX = } . If L denotes a G-CW-com-
plex obtained from K by attaching G-n-cells of type (H) via the generators of H
~n(~ ) then lepta 1.2 shows ~n(~) = 0 . Lemma 1.3 is now obvious (cf. [4] p. 312).
Now we are ready to prove the proposition I.I.
If (X,A) is G-homotopy equivalent to relatively finite pair then WG(X,A) = 0
by the homotopy type invariance of algebraic Wall obstruction.
Suppose now that (X,A) is G-dominated by (K,L) and that WG(X,A) = 0 . Let
: (K,L) ~ (X,A) be a domination map. It follows from the proof of lemma 1.3
that we can assume ¢IL: L --~ A to be a G-homotopy equivalence and ~ : K --~ X
to be G-n-connected where n = max(dim(K-L),2) . By lemma 2.1 in [16]
~n+l (~) m Hn+l(~ ) is projective and finitely generated Z[~o(G )]-module and it
15
represents we(X,A) ([8] p. 340). By assumption there exist finitely generated,
free Z[~o(G )]-modules C, D such that ~n+l(~) C = D. Let rank C = m and let
#I : KI--~ X be a G-map obtained from @ by attaching m free G-n-cells to K
via trivial maps a i ~ ~n(#) . Then lemma 1.2 shows that ~n+l(~l) = ~n+l ® C = D .
Attach now to K 1 free G-(n+l)-cells via free generators of the module ~n+l(~l)
to obtain a G-map ~2 : (K2'L) --~ (X,A) such that ~2 : K2 ---+ X is a homotopy
equivalence and ~21L : L --+ A is a G-homotopy equivalence of pairs (cf. [3],
prop. 1.2) with (K2,L) relatively finite. Now, extending the G-homotopy inverse of
~2 L one can obtain the required G-homotopy equivalence h : (Y,A) --~ (X,A) .
2. The equivariant Wall-type obstruction to finiteness
Throughout this section G will denote a compact Lie group and X a G-CW-com-
plex. Suppose that X is G-dominated by a finite G-CW-complex K and let ~:K -~ X
be a domination with the section s : X --~ K . In this section we will define the
family of Wall obstructions which determine if the G-CW-complex X has the G-homo-
topy type of a finite one.
For any closed subgroup H of G the fixed point set X H is an NH-space as
well as a WH-space where WH = NH/H . If X is a G-CW-complex then Illman ([i0]
sect. 4) observed that X H is a WH-CW-complex and it is finite if X is. We will
need the following observation, the proof of which is completely straightforward.
Lemma 2.1. If X is an H-CW-complex then the twisted product G ×H X is a G-CW-
complex and it is finite if X is.
Let further XH be a connected component of xH and denote (~) ={n~NH:nX~=X~}
and (WH)~ = {w ~ WH : wX~ = ~} . Both, (NH) and (WH) , are compact Lie groups
and ~{ is a (WH) -CW-complex, The set (WH)~ = (NH)X~ is called the WH-component
of X H .
Let now X H be a connected component of X H such that N occurs as an isotropy
subgroup in X H , i. e. xH-x >H ~ @ . We define an equivalence relation ~ in the
set of such components ~ , by setting Ai ~ ~ iff there exists an element n ~ G
n~ K We denote the set of equivalence classes such that nHn -I = K and = X~ .
of this relation by CI(X) . Note that CI(X) is a subset of the set C(X) intro-
duced by Illman [i0] .
Lemma 2.2. Suppose that a G-CW-compIex X is G-dominated by a G-CW-complex K
and let # : K --~ X denote the domination map with the section S:X --~ K . Let
H be components of X H and K H respectively, such that s( ) c K~ and K~ , •
H (WH)-dominates X H Then (WH)~ = (W~)~ and K~ ~ •
16
If X H is a component of X H which represents an element of the set CI(X) c~ H H The group (WH)~ acts then let KsH be a component of K H such that s(X ) c K~
on the pairs (~,X~)and (~,K~ H) in such away that (~X~ H) is relatively
H >H free and (Ks,Ks) is relatively free and relatively finite. By the relative version
H>H of lemma 2.2 we have that (K~,K~) (WH) -dominates ( ,X ) . We define an in-
variant wH(x) to he w (X) (X ,X H) ~ Ko(Z[Zo(WH)~ ] .We wish to show that = w(WH)
this is independent of the choice of representative ~ from the equivalence class
[~] in CI(X) . Let ~ be a component of X K such that ~ ~ ~. This means
that there exists n ~G such that nHn-I = K and n~ = ~. The map n:~ --~ ~
is a ¥(n)-isomorphism from the (WH) -CW-complex ~ to the (WK)~-CW-complex ~.
Here ¥(n) : (WH) ~ (WK)$ is an isomorphism defined by ¥(n)(n H)=(nn n-l)K •
Furthermore, ~'(n) induces the canonical isomorphism
F : Ko(Z[~o(WH)])~ ~o(Z[~o(WK) ])
which is independent of n . The isomorphism n : X H ~ X~ induces an isomorphism
~ of chain complexes and from this it follows that
F(wH(x)) = w~(X) .
We can now state the following result.
Theorem 2.3. Let a G-CW-complex X be G-dominated by a finite G-CW-complex K .
Suppose X has a finite number of isotropy types. Then X has the G-homotopy type
of a finite G-CW-complex iff all the invariants wH(x) vanish.
Proof. Since the necessity part is clear, we only have to prove the sufficiency.
Suppose that wH(X) = 0 for any equivalence class [~]-- in CI(X) . Note that the
set CI(X) consists of one connected component from each WH-component (W~)~i for
(WH)X~ --- (WH)X~ H + ~ i.e. X H - X~ H- + ~ , Here H runs through a complete which
set of representatives for all the isotropy types (H) which occur in X .
By assumption on X the set CI(X) is finite.
Let (HI),...,(Hr) be isotropy types occurring on X ordered in such a way that H. H.
if (H i ) > (Hj) then i < j Let X I, .. X I . . . . , ~ denote the representatives of WH i- 1 S.
H, 1 components of X 1 . Order the set of pairs {(p,q) : i , < p ~ r, 1 ~ q = < Up} lexico-
graphically.
The proof goes by induction. We shall construct for each pair (p,q) a G-CW-com-
plex Y and a G-homotopy equivalence f : Y ~ X such that P,q P,q P,q
17
H I) (Yp,q) is WH-finite for any subgroup H of G with H ~ (H i ) for some
i-_< i < p .
H H H 2) G(Y )^P is G-finite for any component (Y )^P of (Yp,q) p correspond-
H p'q ~j P,q ~j
ing to X p under f for 1 =< j <-- q. c~, p,q
J
Then Y will be a finite G-CW-complex G-homotopy equivalent to X r,s r
We begin with Y0,0 = X and f0,0 = idx" Suppose now that Yp,q satisfying the
above conditions has been constructed. We will identify X with Y via f p,q p,q'
and will assume that X H is WH-finite for any H conjugate to some H. (i ~ i < p) H 1
and that GX p is G-finite for 1 =< j _-< q . c~. J
There are two cases to consider.
Case 1 (q < Sp). We simplify the notation by setting H = Hp and ~ = ~q+l"
H >H ~ _ Since (X ,X ) is (NH) -dominated by a finite pair, w (X) = 0 and X: H is (NH)
finite by inductive assumption, the proposition i.i implies that there exists (NH) -
homotopy equivalence of pairs f : (Z,X: H) --~ (X~,X: H) with Z a finite (NH) -CW-
complex and flx:H an identity. Now f induces a G-homotopy equivalence of twisted
products
We define an equivalence relation ~ on G ×(NH) X: H by setting [g,x]~[g',x']
iff gx = g'x' in X . We extend this relation to G X(NH) Z and G ×(NH) X~
by identifying no points outside G ×(NH) X: H . Let Y = G X(NH) Z/~ and note
that G xH/~ = GX H c X . By lemma 2.1 Y is a finite G-CW-complex and f' X(NH) ~
induces the G-homotopy equivalence fH : y --~ GX~ . Now we can use the techniques
of [9] , section 4 to extend f" to a G-homotopy equivalence fp,q+l: Yp,q+l --~ X.
Case 2 (q = Sp) . The proof of it is completely analogous to that of case i.
3. The equivariant version of Ferry's construction
In [12] S. Kwasik has generalized the construction of the finiteness obstruction
presented by Ferry to the equivariant case. We briefly recall his description
especially as the proof the the theorem 3.4 in [12] is not totally clear. I am in-
debted to $.Illman and S. Kwasik for helpful remarks concerning the proof of the
theorem 3.2 below.
Let # : K ~ X be a domination map with the section s : X ---+ K . If
A = s-~ : K---~ K then denote by T(A) the mapping torus of A obtained from
18
the mapping cylinder M(A) by identification of the top and bottom of M(A) by
means of the identity map.
Proposition 3.1. [12] If the G-space X is G-dominated by a finite G-CW-complex K
then the mapping torus T(A) is a finite G-CW-complex and has the G-homotopy type of
the G-space X x S 1 (with trivial G-action on S I) .
Let B : T(A) --~ X x S 1 denote the G-homotopy equivalence of proposition 3.1.
The natural infinite cyclic covering of X × S 1 induces an infinite cyclic covering
I(A) of T(A) . The G-space I(A) is an infinite G-CW-complex with two ends g+, e
and the G-homotopy equivalence B gives rise to a G-homotopy equivalence between
X and I(A) .
Let u : S 1 --~ S I be the homeomorphism given by the complex conjugation and
B -I the homotopy inverse to B . Denote by ~(h) e WHG(T(A)) the torsion of the
G-homotopy equivalence h = B -I- (id X x u)~B : T(A) --~ T(A) . We define the equivariant
obstruction to finiteness as OG(X) = B,(~(h)) ~ W~G(X x S I) .
One can show that this obstruction is well-defined (see [7] th. 2.3).
Theorem 3.2. The finitely dominated G-space X has the equivariant homotopy type
of a finite G-CW-complex iff CG(X) = 0 .
Proof. If X has the G-homotopy type of a finite G-CW-complex K we may assume
that the domination map ~ : K --~ X is the G-homotopy equivalence and A ~ id K .
Then T(A) is G-homotopy equivalent to K x S 1 and B = ~ x id Hence S 1 "
• (h) = ~(i~ × u) ~ WHG(K x S I) and we show that this torsion vanishes.
By the product formula for equivariant Whitehead torsion [ii] we have for the
(H × Q,~ × ~)-component (K x SI)~Q
• (id x u) HxQ = (K) • j.J(u) e Wa(Wo(WH)~ x Zo(WQ)~)
Since the action of G on S 1 is trivial we get ~(u)~ = 0 and OG(X) = 0 .
If OG(X) = 0 then ~(h) = 0 and it means that h : T(A) ~ T(A) is an equi-
variant simple-homotopy equivalence. Making use of the equivariant version of [6],
exercise 4. D.~p. 16, we can find a finite G-CW-eomplex W and two equivariant
collapses fi : W --~ T(A) such that the diagram
W
h T(A) , T(A)
commutes up to G-homotopy.
19
Now, passing to infinite cyclic coverings we have a diagram
K_ 2 K_ I K o K 1 K 2
K_ 2 K_ 1 K o K 1 K 2
I(A)
f 2
where I(A)
For sufficiently large m the region of
an equivariant strong deformation retract of
type of X .
4. The relation between OG(X) __and wH(x)
Supose that a G-CW-complex X is G-dominated by a finite G-CW-complex
[i0] Illman showed that there is a natural isomorphism
WHG(X × S I) = ~ Wh(~o(~7{) ~ × Z) . c(x)
Furthermore, for an arbitrary short exact sequence of groups
0--~ R--~ P--~ Z--~ 0
we have the natural Bass-Heller-Swan decomposition of the Wh-functor [5]
Wh(P) = Wh(R) ~ ~o(Z[R]) ~ N
In particular, we have an epimorphism
s : ~(P)--~ ~o([R])
whose definition is given below. Hence we obtain the natural decomposition
c ) c(x)
and t h e n a t u r a l e p i m o r p h i s m
S: WhG(X × S I) ~ ~o(Z[~o(WH)~]) .
denotes the reversed (with respect to the ends) copy of I(A) and K.=K. 1
between (~I)'I(K m ) and (~2)-l(Km) is
and therefore it has the G-homotopy
K . In
20
The aim of this section is to prove the following result.
Theorem 4.1.. The equivariant finiteness obstruction OG(X) decomposes into the
family of obstructions wH(x). Precisely, the image of the (H,~)-component OG(X) ~
wH(X) .
Z[P] = r~Z Z[R]tr where t~P maps to l~Z
D' = ~ Z[R]t r and let x E Wh(P) be represented r < o
of the obstruction OG(X) under epimorphism
S : Wh(~o(WH) ~ × Z) --, Ko(Z[Vo(WH)~])
is equal to the equivariant Wall-type obstruction
We start with the definition of the homomorphism
s : Wh(p) -~ ~o(Z[R])
Decompose the group ring Z[P] as
Denote by D = ~ Z[R]t r and r > o
by a Z[P]-isomorphism
d : ZIP] k --~ Z[P] k
Choose s > 0 so large that d(Dkt s) c D k and D 'k c d(D'kt s) . Then a Z[R]-module
Dk/d(Dkt s) is finitely generated and projective ([15] prop. 10.2) and, by definition,
it represents S(x) . One can show that S is a well-defined group homomorphism
([15] th. 8.1).
Now the G-homotopy equivalence h : T(A) --~ T(A) induces the G-homotopy equi-
valence ~ between I(A) and its reversed copy I(A) . Taking the mapping cylinder
of ~ we may assume that ~ : I(A) --~ I(A) is an equivariant strong deformation
retraction of I(A) (see prop. 1.3 in [9]). In I(A) consider a G-invariant sub-
complex L such that L is a neighborhood of E+ and (i(A) -L) U I(A) is a neigh ~
borhood of e . Let L 1 = L N I(A) . We will need the following observation.
Lemma 4.2: The pair (L,LI) is G-dominated by a relatively finite pair (LoU LI,LI).
Proof. The G-homotopy equivalence h : T(A) --~ T(A) induces proper homotopy equi-
valence h' : T(A)/G --~ T(A)/G and h : I(A) --~ I(A) induces the proper strong
deformation retraction of I(A)/G , [14], lemma 4.7.
Let h t : I(A) --, I(A) be the G-homotopy between id and ~ • Passing to the
orbit spaces one can find a G-subcomplex L 2 of L such that ht(L2) c L for
all t . Extend now the G-homotopy h t : L 2 U L 1 --, L to the G-homotopy
k t : L --~ L constant on L 1 . The complex L - (L 1U L 2) is G-finite so there
exists a G-finite subcomplex Lo c L with kt(L-L2) c L ° U L 1 . Now the inclusion
(L O U LI,L I) --~ (L,LI) is a G-domination map .
H H LI)~ H) (WH)-dominates Hence by lemma 2.2 the pair ((L ° U LI)~, (LI) ~ U (L ° U
the pair (L~,(LI)~ U L2)and we can define the obstruction
21
wH(I(A),I-'(-~-,g+) = w(C,(LH,(LI)H U L>H))~::o(Z[~:o(W:(I(A)))~]) .
This obstruction is independent of the choice of subcomplex L , because for another
L' c L there are only finitely many G-cells in L-(L' U I(A)) .
Choose now neighborhoods L+ , L_ o f g+ , e so t h a t I ( A ) - L + , I ( A ) - L _ a r e
neighborhoods of ~_ and ~+ , respectively, and L+ U L_ = I(A) . Then the sub-
complex L+ U L_ is G-finite and since I(A) is G-dominated by K the Mayer-Vietoris
sequence
- " >;"> co<( >;">. --, 0 ~ C,((L+ : L )~,(L+ N L --~ _ _
C,(I(A)~,I(A)~ H ) - --~ 0
shows that C,((L+)~,(L+)~ H)"- and C,((L_)~,(L_)~ H)'- are dominated by finitely gene-
rated free complexes. Thus we can define the obstructions
w~(I(A),~+) = w(C,((L+)~,(L+)~H))
w~(I(A),e_) = w(C,((L )~,(L >~H))
which do not depend on the choice of L+ and L . Similarly, the neighborhoods
h + -- '+ ~ I(A), :: = ~_ ~ :-CA: ~ive the obstructions w~a-:),~+), w~(I(A~_) a,,d
and we have
wH(I(A) e+)--wH(I-~),g+) + wH(I(A),I(A),:+)
+ and L have the G-homotopy type of In our situation L:
wH(l(A),e+) = wH(!(A),e ) = 0 ,
K so
and again the Mayer-Vietoris sequence yields
wH(I(A)) = wH(I(A),e+) + wH(I(A),s_) = wH(I(A),~+)
= w[(I(A),I(A),e+) .
The crucial step in the proof of the theorem 4.1 lies in the following.
ProDosition 4.3. if. S 1 Wh(~o(WH(T(A))):) ~o(Z[~o(W2d(I(A)))e] is the B-H-S-
and • (h) n denotes the (H,~)-component of the equivariant Whitehead epimorphism
torsion of h then
S:(~(h)~> = w~(:(A),~(--77,c+)
Proof. First of all one can observe that w~(I(A),I(A),E+) does not change under
the equivariant formal deformations of the mapping cylinder M(h) mod T(A) . Hence,
22
by corollary 4.4 in [9] we may assume that the pair (M(h),T(A)) is in simplified
form i.e.
3 M(h) = T(A) U U b~ U U c, .
l I
Let V = M(h) be the mapping cylinder of
its universal covering. Then we have
= H A ) u u b . u U c , . 1 1
: I(A) ~ I(A) and p : ~ ~ V
By the second part of the corollary 4.4 of [9] the cellular chain complex
has the form
d
• . . • 0 --~ C 3 ~ C 2 --~ 0 ~ . . .
C 2 m C 3 ~ (ZE~o(WH(T(A)))*])k with preferred bases derived from where the lifted
equivariant 2- and 3-cells, respectively. Denote by T the generating covering trans-
lation of V over M(h) and by t its lifting to ~ . Then pt = Tp .
Now we choose large s > 0 and let L(s) be a G-subcomplex of V obtained
from I(A) by attaching G-2-cells Trp(~ i) and G-3-cells Trp(tS~i)= r+s r~ T p t c i ) f o r
r ~ o and all i .
Then L(s) Is a neighborhood of g+ and (M(~) -L(s)) U I(A) is a neighborhood
of E , so by definition
On the other hand, the cellular chain complex C ( L ( s ) H , I ( A ) H U L ( s ) : H i s a complex
o f f r e e Z [ ~ e ( g r H ( I ( A ) ) ) ~ ] - m o d u l e s and a g a i n by [9] c o r o l l a r y 4 .4 we have
C~(L(s)H'I (A)Hz a ~ U L ( s ) : H) = D k c C 2
and ~ ~----~. ....... /
C~(L(s)H'I(A)Hj ~ a U L(s): H) = Dkt s c C 3
For large s the quotient module B = Dk/d(Dkt s) is projective and by definition s
Sl(~(h) ~) = [Bs]e~o(Z[~o(WH(I(A))):]) •
The p r o j e e t i v i t y o f B s i m p l i e s t h a t C ~ (L( s ) H I (A) H U L ( s ) >H) i s c h a i n homotopy
equivalent to the complex of the form
.°°
with B s
0--~ 0 B s
in dimension 2, Thus w~(I(A),I(A),e+) = [Bs] .
23
Now we have the commutative diagram
Wh(~o(WH(T(A))): ) "m Wh(~o(WH): x Z)
J, si i s ~o(Z[~o(WH(I(A))):]) B, ~ , --~ Ko(Z[~o(WH)~])
which yields finally
S(OG(X)H)=sB,(~(h)H ) = B,SI(~(h)H ) : B,(wH(I(A),i(A),e+))
:B,(H(I(A))) = wH(x)
5. A product formula for equivariant finiteness obstruction
and its application
In this section G and P denote arbitrary compact Lie groups, unless otherwise
is stated. Recently S. Illman [Ii] has given the product formula for the equivariant
Whitehead torsion ~(f×h) in terms of the equivariant Whitehead torsions of f
and h and various Euler characteristics. We use his formula to derive the correspond-
ing formula for the obstructions OG(X) and wH(x)a and its geometric application.
Let X be a G-CW-complex G-dominated by a finite G-CW-complex K and L a
finite P-CW-complex. Then the product L×X is finitely (P×G)-dominated by L×K and
we have the obstruction
OpxG(LXX) ~WhpxG(LxX×Sl).
Now the domination map defines the (PxG)-homotopy equivalence
: T(idL×A) ~ L×XxS I .
But we have T(id×A) = LxT(A), B = idL×B and our finiteness obstruction is given by
OpxG(L×X) : (id×B),(~(idxh))~Whp×G(LXXxS-l) .
Since for the (QxH,$×~)-component of
we obtain for the (QxH,~x~)-component
~(idxh) we have
(LxX)Q ×H ~x~
where
and
( I )
24
i : ~o(WH)~ ~ ~o(WQ)$ × ~o(W~)a
denotes the inclusion.
By naturality of the B-H-S decomposition and theorem 4.1 we also obtain
w : LxX) Moreover, any obstruction wS(L×X) where (S,y) is not of a product form, equals
7 zero.
As an immediate corollary of the formula (i) or (2) we have the following geometric
result (cf.[13] cor. 6.4)
Theorem 5.1. Let G be a finite group and X a G-CW-complex G-dominated by a
finite one. Let V be any unitary complex representation of the group G and S(V)
its unit sphere. Then the product X × S(V) with the diagonal G-action has the G-
homotopy type of a finite G-CW-complex.
Remark. The above theorem is not true for arbitrary compact Lie groups.
References.
[I]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9] S.
[i0] S.
[ii] S.
[12] S.
[13] W.
[143 L.C
D.R. Anderson: Torsion invariants and actions of finite groups, Michigan Math. J. 29 (1982), 27-42.
P. Andrzejewski: On the equivariant Wall finitenes obstruction, preprint.
S. Araki, M. Muruyama: G-homotopy types of G-complexes and representation of G-homotopy theories, Publ. RIMS Kyoto Univ. 14 (1978), 203-222.
J:A. Baglivo: An equivariant Wall obstruction theory, Trans. Amer. Math. Soc. 256 (1979), 305-324.
H. Bass, A. Heller, R. Swan: The Whitehead group of a polynomial extension, Publ. IHES 22 (1964), 67-79.
M.M. Cohen: A course in simple-homotopy theory, Graduate Texts in Math. Springer- Verlag, 1973.
S. Ferry: A simple-homotopy approach to the finiteness obstruction Shape Theory and Geometric Topology, Lecture Notes in Math. 870 (1981), 73-81.
S.M. Gersten: A product formula for Wall's obstruction, Amer. J. Math. 88(1966), 337-346.
lllman: Whitehead torsion and group actions, Ann. Acad. Sci. Fennicae, Ser. AI 588 (1974), 1-44.
Illman: Actions of compact Lie groups and equivariant Whitehead torsion, preprint, Purdue Univ. (1983).
Illman: A product formula for equivariant Whitehead torsion and geometric applications, these proceedings.
Kwasik: On equivariant finiteness, comp. Math. 48 (1983), 363-372
L~ek: The geometric finiteness obstruction, Mathematica Gottingensis, Heft 25 (1985).
• Siebenmann: On detecting Euclidean space homotpically among topological manifolds, Invent. Math. 6 (1968), 245-261.
25
[15] L.C. Siebenmann: A total Whitehead torsion obstruction to fibering over the circle, Comment. Math. Helv. 45 (1970), 1-48.
[16] C.T.C. Wall: Finiteness conditions for CW-complexes, Ann, Math. 81 (1965), 55-69.
[17] C.T.C. Wall: Finiteness conditions for CW-complexes, II, Proc. Royal Soc. London, Ser. A, 295 (1966), 129-139.
Homotopy Actions and Cohomology of Finite Groups
Amir H. Assadi *)
University of Virginia
Charlottesville, Virginia 22903
Max-Planck-Institut fur Mathematik, Bonn
Introduction
Let X be a connected topological space, and let H(X) be the
monoid of homotopy equivalences of x . The group of self-equivalen-
ces of X , E(X) , is defined to be z H(X) . A homomorphism o
: G ~ E(X) is called a homotopy action of G on X . Equivalently,
the assignment of a self-homotopy equivalence ~(g) : X ~ X to each
g 6 G such that ~(glg2 ) ~ ~(gl)e(g2) and ~(I) % I X is also
called a homotopy action. Since it is easier to construct self-homo-
topy equivalences rather than homeomorphisms of X , it is natural to
consider the questions of existence of actions first on the homotopy
level, (i.e. homotopy actions) and then try to find an equivalent
topological action. A topological G-action ~ on Y is said to be
equivalent to a homotopy action ~ on X , if there exists a homo-
topy equivalence f : Y ~ X which commutes with ~ and e up to
homotopy, i.e. f is homotopy equivariant (for short, f is an
h-G-map). This is the point of view taken in [16] and the motivation
for G . Cooke's study of the question:
*) This work has been partially supported by an NSF grant, the Center for advanced
Study of University of Virginia, the Danish National Science Foundation, Matematisk
Institut of Aarhus University, and Forschungsinstitut fHr Mathematik of ETH,
ZHrich, and Max-Planck-lnstitut fHr Mathematik, Bonn, whose financial support and
hospitality is gratefully acknowledged. It is a pleasure to thank W. Browder,
N. Habegger, I. Madsen, G. Mislin, L. Scott, R. Strong, and A. Zabrodsky for help-
ful and informative conversations. Special thanks to Leonard Scott for explaining
the results of [8] to me which inspired some of the algebraic results, and to
Stefan Jaekowski for his helpful and detailed comments on the first version of
this paper.
27
Question I. Given a homotopy action ~ on X , when is (X,~) equi-
valent to a topological action?
The problem is quickly and efficiently turned into a lifting
problem: A homomorphism ~ : G ~ E(X) yields a map B~ : BG ~ BE(X)
On the other hand the exact sequence of monoids HI(X) ~ H(X) ~ E(X)
yields a fibration BHI(X) ~ BH(X) ~ BE(X)
Theorem (G. Cooke) [16]. (X,~)
if and only if B~ : BG ~ BE(X)
BH(X) ~ BE(X)
is equivalent to a topological action
lifts to BH(X) in the fibration
Note that if X does not have a "homotopically simple structure",
e.g. if X is not a K(n,n) and dim X<~ , then ~i(BHI(X)) is
exceedingly difficult to calculate, and the above lifting problem will
have infinitely many a priori non-zero obstructions. However, if G
is a finite group (and we will assume this throughout) and X is lo-
calized away from the prime divisors of IGI, e.g. if ~1(X) = I and
X is rational, then all the obstructions vanish, and any such (X,~)
is equivalent to a topological action. Algebraically, this can be in-
terpreted by the fact that all the relevant RG-modules (where R is a
ring of characteristic prime to IGI ) are semi-simple and consequently
cohomologically trivial. Thus the interest lies in the "modular case",
(i.e. when a prime divisor of IGI divides the characteristic of R )
and the inetgral case R = Z
In comparison with topological actions, homotopy actions have
very little structure in general. For instance, there are no analogues
of "fixed point sets", "orbit spaces" or "isotropy groups". This makes
a general study of homotopy actions a difficult task. Notwithstanding,
there has been some applications to problems in homotopy theory and
geometric (differential) topology (e.g. [5] [6] [16] [22] [34] [35]
for a sample).
Given a homotopy functor h and a homotopy action of G , say
(X,~) , we obtain a "representation of G" . E.g. if X ~ K(~,n) and
h = ~n ' then ~n(X) ~ ~ becomes a ~G-module. In this case, any ~G-
module ~ also gives rise to a homotopy G-action on X ~ K(z,n) ,
and in fact a topological G-action.
28
For spaces which are not homotopically easy to understand (such
as most manifolds and finite dimensional spaces) homology and cohomo-
logy provide a more useful representation module. From this point of
view, spaces with a single non-vanishing homology, known as Moore
spaces, are the simplest to study. For simplicity, suppose we are
given a ZG-module M which is ~-free. Then it is easy to see that
there exists a homotopy action ~ of G on a bouquet of spheres X
such that H,(X) ~ M as ~G-modules. We say that "(X,~) realizes
M" , or that M is realizable by (X,~) . An obstruction theory argu-
ment shows that the question of realizability of ZG-modules by homo-
topy G-actions on Moore spaces has a 2-torsion obstruction ([7] [22])
which can be identified with appropriate cohomological invariants of
the ZG-module M ([7] P. Vogel, unpublished), in relation with the
question of how close these homotopy actions are to topological
actions, one should mention the following well-known problem attri-
buted to Steenrod [26]:
Question 2. Is an integral representation of G realizable by a G-
action on a Moore space?
There has been some partial progress in answering the above
question and we refer the reader to [3] [9] [13] [22] [30] [32] [33]
and their references. In an attempt to understand homotopy actions, we
will specialize and apply the methods of this paper to the above prob-
lem. Thus constructions and the study of the counterexamples for
Question 2 in this paper should be regarded as a method of producing
and investigating "invariants of homotopy actions" for more general
spaces.
As mentioned above, the usual notion of transformation groups
such as fixed points, isotropy groups, and orbit spaces do not carry
over to homotopy actions as such. Therefore, we will try to attach
other invariants, mostly of cohomological nature, to both G-spaces
and homotopy G-actions, and compare them. For topological actions
these invariants are naturally (and expectedly) related to fixed
point sets and isotropy groups (whenever they are well-defined). Thus
we have placed special emphasis on topological actions with some
finiteness condition on the underlying space (e.g. finite cohomolo-
gical dimension) as well as G-actions with collapsing spectral
29
sequence in their Borel construction. On the algebraic side, our fee-
ling is that the category of integral (modular) representations of G
which arise as homology (cohomology) of G-spaces is an important part
of the category of all representations, and its algebraic study is
worthwhile in its own right. The projectivity criterion (Thm. 2.1) as
well as the complexity criterions (Sec. 3) and their consequences are
some steps in this direction.
In comparing homotopy and topological actions, we will study:
Question 3. When is a representation of G realizable by the homo-
logy of a G-space?
As we will see below, there are integral (and modular) represen-
tations of G which are not realizable via the homology of any
G-space (we do not restrict ourselves to Moore spaces). On the other
hand, there are representations which are not realizable by G-actions
on Moore spaces but they can still be realized by G-actions on other
spaces (Section 5). All these representations arise from homotopy
actions. These examples show that, even for homologically simple
spaces, such as bouquet of spheres, the collection of integral re-
presentation of G on H,(X) induced by a homotopy action
: G ~ [(X) does not by itself decide whether (X,~) is equivalent
to a topological action. It is the interrelationship of all Hi(X)
as ZG-modules which determines the realizability in this case (Sec-
tion 5 ) . In the applications of homotopy actions to differential
topological problems, one often needs to find finite dimensional
G-spaces which realize a given homotopy action. The solution to the
lifting problem mentioned earlier in the introduction, provides an
infinite dimensional free G-space. In this context, the following
problem is often necessary to answer:
Question 4. suppose X is homotopy equivalent to a finite dimensional
space and ~ : G × X ~ X is an action. When does there exist a finite
dimensional G-space K and a G-map f : X ~ K inducing homotopy
equivalence?
We study this problem and the related question Question 3 by
"reduction to p-groups". This is the subject of a future paper. In
particular, one has satisfactory characterizations for groups with
periodic cohomology and some other classes of groups which includes
30
nilpotent groups or some of the alternating groups.
Notation and conventions. All rings are commutative with unit. F P
is the field with p-element, where p always denotes a prime number,
and k is a field of characteristic p > 0 (often an algebraic
closure of Fp ). For a finite group G , H G denotes the r~ng
H2i(G;k) if p is odd and H G = @ Hi(G;k) if p = 2 H* de- @i i " notes Tate cohomology [14] and the terminologies in this context are
in [14] and [28]. Z ~ ~/p ~ ~ integers (mod p) . The localization P
of a ring R with respect to the multiplicative subset generated by
an element y 6 R is denoted by R[y -I] . For an ideal J in a ring
R , tad(J) is the radical of J and if M is an R-module, Ann(x)
is the annihilating ideal of x C M . The dual of a k-algebra A is
denoted by A* . For an RG-module M and a subgroup H , MIRH de-
notes the restriction to H . The terminology and conventions in
topological group actions are taken from [10] and [19] and those
related to homotopy actions are to be found in [16]. For example E G
is the contractible free G-space and E G × G X is the Borel construc-
tion of a G-space X . If a G-space X needs to have a base point in
the context, we replace X by its suspension ZX and take x6xG#~ ,
unless X is already endowed with a base point. Many of the state-
mets which are phrased in terms of cohomology have their counterparts
in homology and we have avoided repeating this fact. The spaces X
are not necessarily CW complexes unless otherwise specified. We may
use sheaf cohomology for more general situations and the proofs are
still valid (with some mild modification if necessary). The basic
reference is [27] part I in particular its appendix, and we have used
Quillen's terminology and notation when appropriate. E.g. Cdp(X)
means cohomological dimension of X (mod p)
The bibliography contains the references which have been available
to us, at least in some written form. Otherwise they have been men-
tioned in the context.
Section I. Localization and Projectivity
In this section we present a variation on P.A. Smith's theorem
as a consequence of Quillen's version of the localization theorem of
Borel (cf. [19] or [27]). The statements are not as general as they
could be because we will present different proofs when the cohomo-
31
logical finiteness of the G-spaces are not assumed. These finiteness
assumptions are necessary when applying the localization theorem.
There is an analogy between the finiteness assumptions of this section
on the level of orbit spaces and the weaker finiteness assumptions
for cohomology in the following sections. There is also a localiza-
tion-type argument implicit in the arguments of sections 2 and 3 which
are explicit in the context of this section. The special cases treated
differently in this section will hopefully serve to give motivation
and some insight into the more algebraic arguments of the following
sections. The basic reference for some details of the assertions of
this sections (as well as the terminology and the notation) is [10].
More general forms of the localization theorem are discussed in ~19].
1.1 Proposition. Let G be a finite group and let X be a connected
G-space which is either compact, or Cdp(X/G) <= for a fixed prime
p . Assume that for each subgroup C c G in order p , Hi(X;Fp) is
a cohomologically trivial F C-module for all i > 0 . Then the P X P p-singular set of X , Sp(X) ~ ~ , where P ranges over non-tri-
vial p-subgroups of G , satisfies H*(Sp(X) ;Fp) = 0 .
Proof: Let C c G and ICI = p , and let y 6 H2(C;Fp) be the poly-
nominal generator. Without loss of generality, we may assume that
X G # ~ , hence X C # ~ . Choose x E X G c X c . The Serre spectral
sequence of the Borel construction (X,x) ~ EcXc(X,x) ~ BC collapse
since Hi(BC;HJ(x,X;~p)) = 0 for i > 0 and all j by cohomological
triviality. Thus H~(X,X;Fp) ~ H0(BC;H*(X,X;Fp)) . Localization with
respect to y shows ([27]):
H~(X,X;~p) [7 -1 ] ~. H*(BC;H*(X,x)) [y-l]
= H*(C;H*(X,x)) = 0 ,
(by the hypothesis of cohomological triviality) where H* denotes
Tate cohomology. By the localization theorem
H$(xC,x;imp) [y-l] =~ H~(X,X;Fp) [y-l] = 0 .
Since H~(xC,x;Fp)[y -I] ~ H*(xC,x;Fp)® FpH*(C;I~p) ,
H*(xC,x;imp) = 0
it follows that
For any subgroup K c G , such that IK I = pr and K D C , it
32
follows that xK~ ~ and H*(xK;~p) = 0 by an induction. Since this
holds for every cyclic p-subgroup C ~ G , one has H*(xK;Fp) = 0
for all subgroups K c G , K ~ I . An inductive argument using Mayer-
Vietoris sequences yields the desired conclusion. •
We will be particularly interested in the class of G-spaces for
which the Serre spectral sequence of their Borel construction collap-
ses. This is formulated as condition (DSBC) (degenerate spectral se-
quence of Borel construction) below.
CONDITION (DSBC) : Let X be a G-space and let A c G be a subgroup.
We say that X satisfies the condition (DSBC) for A if the Serre
spectral sequence of the fibration X ~ EA×AX ~ BA (in the Borel
construction of the A-space X ) collapses.
1.2 Proposition. Let p be a prime divisor of order of G , and
suppose that X is a connected G-space such that either X is com-
pact or that Cdp(X/G) <~ . Assume that:
(I) X satisfies condition (DSBC) for each maximal elementary abelian
subgroup A c G .
(2) The p-singular set Sp(X) satisfies: Sp(X) # ~ and H*(S(X) ;Fp)=0.
Then H*(X;Fp) is cohomologically trivial as an ~pG-module.
Proof: Let A be any p-elementary abelian rank t subgroup, and let
e A 6 H2t(A;F ) be the product of the t 2-dimensional polynomial
generators i~ H2(A;~ ) , (cf. [27] Part I). Since S (x)A=x A and (2) A p G
implies that H*(X ,x;F ) = 0 (where x £ X ~ ~ is the base point), -I
it follows that H~(X,X;Fp) [e A ] = 0 , by the localization theorem
([27] Part I). Since the Serre spectral sequence of (X,x) ~ E A ×A
(X,x) ~ BA collapses by (I), we may localize the E2-term with respect
and conclude that H*(BA;H*(X,X;Fp)) [e~ I]= = 0 . But H*(BA;H* to e A
(X,X;Fp)) [e~ I ]~ ~ H*(A;H*(X,X;~p)) . Since this is true for all p-ele-
mentary abelian groups A ~ G , IAl=p r , it follows that H*(X,X;Fp)
is cohomologically trivial over all p-elementary abelian subgroups of
G . By Chouinard's theorem (cf. [15] and [20]) H*(X,X;Fp) is cohomo-
logically trivial over G (see the introduction to section 2). m
We obtain a special case of Theorem 2.1 as a corollary:
33
1.3 Corollary. Suppose that X
wing properties:
is a connected G-space with the follo-
(I) Either X is compact or Cdp(X/G) <~ for each p dividing order
G .
(2) X satisfies condition (DSBC) for each p-elementary abeiian sub-
group A ~ G . Then H*(X) is ZG-projective if and only if H*(X) IZC
is ZC-projective for each subgroup C ~ G of prime order. In parti-
cular, this conclusion holds if X is a Moore space which satisfies
(I) .
Proof: By 1.1 and 1.2, the cohomological triviality of H*(X) over G
is equivalent to the cohomological triviality of H*(X;Fp) for all
cyclic subgroups of order p . But a ~G-module is ZG-projective if and
only if it is ~-free and cohomologically trivial (cf. [28]). •
Section 2. The Pr0jectivit[ Criteria
Let G be a finite group. Sylow(G) denotes the set of Sylow sub-
groups, and G 6 Sylow(G) denotes a p-Sylow subgroup. Let R be a P
ring and RG be the group algebra over R . In studying the cohomo-
logical properties of RG-modules, it is necessary to have a good under-
standing of projective modules. The following two theorems have played
important roles in the "local-to-global" arguments.
(I) Rim [28]: A XG-module is ZG-projective if and only if MIZG p is
~Gp-projective for a l l Gp £ Sylow(G)
(21 Chouinard [15] (See also Jackowski [20]): A ~G-module M is ~G-
projective if and only if MIZE is ~E-projective for all p-elementary
abelian groups.
Chouinard's theorem is particularly useful in the problems related to
cohomo!ogical properties of M , since the cohomology of elementary
abelian groups are well-understood, whereas the cohomology ring of a
general p-group is far more complicated and has remained mysterious as
yet.
Thus, the projectivity of a ~G-module M is detected by its re-
strictions to the elementary abelian subgroups. Now suppose that M is
a kE-module, where E is p-elementary of rank n (i.e. of order pn),
and where k is a field of characteristic p .(For simplicity, assume
34
that k is algebraically closed, although for the most part this
assumption is not used.)
It is tempting to look for a projectivity criterion for M in terms
of a family of proper subgroups of E In general there is no such
criterion if we consider only subgroups of E . However, there is such
a characterization if we include a certain family of well-behaved sub-
groups of kE . This is basically the content of a result due to Dade
[17]. To describe this, let I be the augmentation ideal: 0 ~ I ~ kE
k ~ 0 and choose an Fp-basis for E , say {el, .... ,en}CE . Let
A = (aij) be a non-singular n × n matrix over k and define the
homomorphism ~A: kE ~ kE by:
n ~A(ei) = 1 ÷ Z a~i(ej-1)
3= 1 3
Then ~A is an automorphism since A is non-singular. In [11] J.
Carlson called subgroups of order pm in kE , m -_< n , generated by
{~A(el) , .... ,~A(em) } , "shifted subgroups" of kE . Such subgroups are
p-elementary abelian and for m = n , {~A(el),...,~A(en)} generate
kE as a k-algebra. A cyclic subgroup S of the shifted subgroup
<~A(el),...,~A(en)> is called a "shifted cyclic subgroup" and any ge-
nerator of S is called a "shifted unit". From now on we assume that
all kE-modules are finite dimensional over k .
(3) Dade [17]: A kE-module M is kE-projective if and only if M!kS
is kS-projective for every shifted cyclic subgroup of kE .
(Since kE is a local ring, projective, injective, cohomologically
trivial, and free modules coincide [28]). In fact, one can show that
MIkS is kS-projective if and only if M!kS' is kS'-projective provi-
ded that the shifted units generating S and S' are congruent mo-
dulo 12 . This leads to the following more intrinsic definition of
shifted subgroups and units [11] [8]. Let L be an n-dimensional
k-subspace of I such that I = L @ 12 . Then every element 1 6 L
satisfies I p = 0 , and a k-basis of L generates kE as a k-algebra.
Consequently, for any 1 6 L , I + Z is a shifted unit and for any
k-basis of L , say {/1,...,/n} , the p-elementary subgroup generated
by {I+/I,....I+/n } is a shifted subgroup. J. Carlson attached a glo-
bal invariant to a kE-module M , by taking the set V[(M) consisting
of all nonzero 1 £ L for which M!k<1+/> is not k<1+/>-free (where
<I+/> is the group generated by I+1 ) together with zero. He showed
that this is an affine algebraic variety and exhibited many beautiful
35
properties of V[(M) , called "the rank variety of M " (cf. [11]).
Carlson conjectured that V~(M) is isomorphic to the cohomology
variety of M , VE(M) (called the Quillen variety and inspired by
r(M) injects into Quillen's ideas in [27]), and he showed that V L
VE(M) . The Quillen variety VE(M) is the affine variety in k n de-
fined by the ideal of elements in the commutative graded ring HE~@ i
H2i (E;k) which annihilate the HE-module H*(E;M) (HE=SiHi(E;k) when
E is a 2-group. The conjecture of Carlson is proved by Avrunin-Scott
[8], and as a corollary V~(M) is independent of L up to isomor-
phism. Thus the projectivity criterion of Dade which can be detected
"locally" by shifted units, has the following "global formulation". r(M)
From now on we drop the subscript L in V L
(4) Carlson [11]: M is kE-free if and only if Vr(M) = 0 .
This motivates the search for a projectivity criterion for ~G-
modules which appear as (reduced) homology of G-spaces. It turns out
that the family of cyclic subgroups of order p of G detects the
projectivity (and cohomological triviality). Thus "the geometry of
M " is determined by a restricted class of subgroups of G in this
case, and gives an idea of how restricted the category of realizable
=G-modules is. This is not true for homology of all G-spaces, rather
a special class which includes Moore spaces. The projectivity crite-
rion for the homology of more general G-spaces should be described in
terms of "global invariants" attached to a G-space. The specific na-
ture of a G-action on a space X determines a certain interrealation-
ship between Hi(X ) and Hj(X) as =G-modules, and this fact is not
detectable by simply considering the graded module SiHi(X) . The
examples of the following sections will elaborate more on this point.
2.1 Theorem. Suppose X is a connected G-space which satisfies the
condition (DSBC) for each p-elementary abelian subgroup A ~ G . Let
M be the =G-module determined by the G-action on the total homology
of X in positive dimensions. Then M is =G-projective if and only
if MI~C is =C-projective for each subgroup C c G of prime order.
(Similarly for cchomological triviality).
2.2 Corollary. Suppose the =G-module M appears as the homology of
a Moore G-space. Then M is =G-projective if and only if M is
=C-projective for each cyclic subgroup of G .
We will give two proofs of the above theorem. The first is in the
36
spirit of transformation group theory and while it is quite elementary
it reveals the topological nature of this criterion. The second proof
is in a more general setting and hopefully will provide some motiva-
tion for introducing and emphasis on the global invariants of a G-
space.
2.3 Corollary. Suppose X I and X 2 are connected G-spaces, both of
which satisfy (DSBC) as in (2.1) and suppose f : X I ~ X 2 is a G-map.
Let M I and M 2 denote the total reduced homology of X 1 and X 2
as ZG-modules and let ~ : M I ~ M 2 be the ~G-homomorphism induced by
f . Then there are ZG-projective modules PI and^.P2 such that
MI 8 PI ~ M2 ~ P2 if and only if ~,:HI(C;M I) ~ HI(C;M 2) are isomor-
phisms for i = 0,1 , and all cyclic subgroups C ~ G of prime order.
Section 3. Varieties associated to a G-s~ace
Let k be an algebraically closed field of characteristic p>0 ,
and let G be a p-elementary abelian group of rank n . For a connec-
ted G-space X , we will assume X G ~ ~ (when needed) and x £ X G is
the base point. As far as homological invariants of X are concerned
at this point, this will be no restriction, since we acn always sus-
pend the action. For a kG-module M , the rank variety Vr(M) reveals
much about its cohomological invariants. Thus, we are tempted to con-
variety vr(@iHi(X,x;k) and investigate its sider the rank influence
on the topology of the G-space X . However, the more directly related
variety, (when we have sufficient knowledge about the G-action) is the
"support variety" VG(X) .
In [27], Quillen studied cohomological varieties arising from
equivariant cohomology rings H~(X;k) for a G-space X (cohomology
with constant coefficients), and he proved his celebrated stratifica-
tion theorem among other results. According to Quillen's stratifica-
tion theorem, the cohomological variety of a G-space X for a general
finite group G has a piecewise description in terms of varieties
arising from elementary abelian subgroups of G . Inspired by this
work of Quillen, ~vrunin-Scott in [8] defined the cohomological varie-
ty VG(M) for a finitely generated kG-module M and proved an anlo-
guous stratification theorem for VG(M) in terms of elementary abelian
subgroups of G . Here, VG(M) is the largest support (in Max H G ) of
the HG-moduie H*(G,N®M) where N ranges over all finitely generated
37
kG-modules. Avrunin-Scott's stratification theorem may be regarded as
generalizing the special case of Quillen's result for the G-space
X=point to the equivariant cohomology with local coefficients H~
(point;M) (the kG-module M replacing the constant coefficients k
of Quillen) . The stratification of support varieties in the case of
equivariant cohomology with local coeeficients H~(X;M) for a G-space
X (whose orbit space X/G has finite cohomelogical dimension over
k ) is carried out by Stefan Jackowski in [21] under the extra hypo-
thesis that M is a kG-algebra. Jackowski's theorem yields a topolo-
gical proof of Avrunin-Scott theorem in the spirit of Quillen's ori-
ginal approach.
Such stratification theorems describe the above mentioned cohomo-
logical varieties of a general finite group G in terms of elementary
abelian subgroups of G . When G is an elementary abelian group,
VG(X) is the affine algebraic variety defined by the annihilator ideal
in H G of H~(X,x;k) . For the rest of this section, we will assume
that G is an elementary abelian group. The corresponding results and
notions for the case of a general finite group is obtained from this
basic case and the appropriate stratification theorem. Elaboration of
these ideas will appear elsewhere.
While one hopes that VG(X) ~ V~(eiHi(X,x)) , this turns out to
be true only for a restricted, but nevertheless important class of
G-spaces. For a G-space with Hi(X) ~ 0 for only finitely many i (and
some mildly more general class), it turns out that one can define a
different, (but related) rank variety in a natural way. This is done by
associating to X a ZG-module defined up to a suitable stable equi-
valence. The V~(X) is defined to be the rank variety of this module
(tensored with k ). The isomorphism VG(X) = V~(X) will show that
the "cohomological support variety" is also a "rank variety" and as
such, it will enjoy the properties of rank varieties.
Following ~5], call two G-spaces X I and X 2 "freely equiva-
• c y , and Y-X. lent", if there exists a G-space Y such that X l l
are free G-spaces with Cdp(Y-X i) <~ for i = 1,2 . This defines an
equivalence relation between G-spaces. We may also consider the case
when Y/X i is compact if cd(y-xi)=~ with appropriate modifications.
3.1 Lemma. Suppose X I and X 2 are freely equivalent. Then VG(Xl )~=
V G (X 2 )
38
Proof: Compare the Leray spectral sequences for EG×GX i ~ Xi/G with
EG×GY ~ Y/G where X i and Y are as above, Y-X i = free G-space [27].
It follows that VG(X i) ~ VG(Y) . m
3.2 Proposition. Suppose Hi(X;k) ~ 0 for only finitely many i . Then
VG(X) c V~(@iHi(X,x;k)). . If X satisfies the condition (DSBC) for G,
then VG(X)~V~(~iHI(X,x;k))
Proof: Proceed by induction on ~(x) d~fnumbere {ilHi(X,x;k) # 0} . For
~(X) = I , X is a Moore space and the spectral sequence of (X,x)
EG×G!X,x) ~ BG degnerates to one line, which shows that VG(X) ~ V G
(~jH3(X,x;k) (~ its support variety). By Avrunin-Scott's~ proof of J.
Carlson's conjecture [8], the latter is isomorphic to V~(~jHJ(x,x;k))._
Suppose the assertion is true whenever ~(X) < m , m > I . Given X I
with ~(X I) = m , we add free G-cells to X I to obatin the G-space Y
so that Y-X is free, dim(Y-X) <~ , and ~(Y) < m . For example, kill
the first non-vanishing homology, say Hl(X,x;k) , using Serre's ver-
sion of the Hurewicz theorem, (after suspending X , if needed). Then
VG(X) ~ VG(Y) since X and Y are freely equivalent (Lemma 3.1) and
(@jH j r(x) c V~(X). VG(Y) ~ V~ (Y,x;k)) by induction. On the other hand, V G _ G This follows again because (Y/X) = point and dim(Y/X) <~ . Alterna-
tively, if we kill Hz(X,x;k) (the first non-vanishing) to obtain Y ,
we have the exact sequence:
0 ~ HZ+ I (X;k) ~ HI+ I (Y;k) ~ F ~ Hz(X;k) ~ 0
where F is a free kG-module, and
Hi(X;k) ~ Hi(Y;k) for i > £+I .
For every shifted cyclic subgroup S of kG for which Hi(X,x;k) IkS
is kS-free, Hi(y,x;k) IkS will also be kS-free by Schanuel's lemma.
Hence V~(SjHJ(Y,x;k)) c V~(SiHi(X,x;k)) as desired.
If X satisfies the condition (DSBC) for G , then in the Serre spec-
tral sequence of X ~ EG×GX ~ BG, E~ 'q = E p'q~ . Thus rad(Ann H~
(X,x;k)) ~ rad(Ann H*(G,H*(X,x;k))) by a simple calculation and a fil-
tration argument. Since rad(Ann H*(G,H*(X,x;k))) ~ D rad(Ann H*(G;H i
(X,x;k))) , it follows that i
VG ~ VG (@iHi = VG(Hi G (Hi (X,x;k) (X) = (X,x;k)) ~ U (X,x;k)) ~= U V ) ~ l i
39
r (@iHl (X,x ;k) ) V G
r of Hi(X,x;k) is due (where the isomorphism between V G and V G
Avrunin-Scott's theorem again). •
The second assertion of 3.2 is not true in general. The examples
in the following sections illustrate this point.
The above observations lead us to define a kG-module M(X) for
each G-space X with Hi(X;k) # 0 for only finitely many i , such
that VG(X) ~ V~(M(X)) Since for Moore spaces X , VG(X) ~ V~(H*
(X,x;k)) , we embed X in a "mod k " Moore G-space Y freely equi-
valent to it. This is possible since Hi(X;k) = 0 for large i and
we can add free G-cells inductively using Serre's Hurewicz theorem.
Let M(X) H H,(Y,x;k) . Although M(X) is not well-defined, H*(G;M
(X)*) and H~(X,x;k) are isomorphic modulo HG-torsion. Hence VG(X)~
VG(M(X)*) ~ V~(M(X)*) ~ V~(M(X)) and VG(X) has a description as a
rank variety.
The module M(X) is well-defined only in a "stable sense". For a 0 ~ w1 kG-module L , define ~ (L) = L , and (L) z ~(L) by the exact se-
quence 0 ~ ~(L) ~ F ~ L ~ 0 , where F is kG-free, and ~i+1(L)
~(~I(L)) . These modules are stably well-defined by Schanuel's lemma
(cf. e.g. Swan's Springer-Verlag LNM 76) .
3.3 Proposition. Suppose X is a G-space such that Hi(X;k) # 0 for
finitely many i . Let YI and Y2 be two mod k Moore G-spaces
freely equivalent to X . Then there are integers s and t a 0 ,
~S(H*(Y1,x;k)) is stably isomorphic to ~t(H*(Y2,x;k)) such that
(Call this w-stability for short.)
Proof: Choose a G-space Z freely equivalent to YI and Y2 and con-
taining Y1 and Y2 ' and such that Hi(Z,x;k) = 0 for i # £ , £7>
nonzero dimensions in H*(Yj;k) for j = 1,2 . Then C,(Z/Yi;k) are
free kG-modules except for * = 0 , where the base point naturally de-
fines a split augmentation C0(Z/Yi;k)¢ ~i) k ~ 0 . C.(Z/Yi;k) has
homology (mod k ) nonzero only in two dimensions above 0 , corres-
ponding to Hl(Z;k) and H,(Yi,x;k) . An appropriate application of
the Schanuel's lemma shows that ~t(H,(Y1,x;k)) ~ H£(z;k) ~ uS(H,
(Y2,x;k)) for some integers t,s ~ 0 . •
3.4 Corollary.Given a G-space X with Hi(X;k) = 0 for sufficiently
40
large i , there exists a kG-module M(X) which is well-defined up to r
~-stability and VG(X ) ~ VG(M(X)) .
The e-stable class of M(X) is in fact a "composite extension" of Si(H i various ~ (X;k)) for all i > 0 and appropriate integers si~0.
This means that if 0 < i(I) < i(2) < .... < i(m) are the dimensions
where Hi(X;k) # 0 , then there are integers s(1) ,...,s(m) and ex-
tensions:
0 ~ Hi(j+1)(X;k) ~ Li(j+1) ~ s(j)(Li(j) ) ~ 0 for j = 1,...,m , and
where Li(1) ~ Hi(1) (X;k) and M(X) ~ ~tLi(m) for some t ~ 0
Let us refer to this construction as "an ~-composite extension".
We have the following formal corollary:
3.5 Corollary. Suppose that Hi(X;k) = 0 for all sufficiently large
i , and suppose X has a homotopy G-action ~ : G ~ E(X) . Then (X,~)
is equivalent to a topological G-action only if some m-composite ex-
tension L of the kG-modules Hi(X;k) (as given by ~ ) is realizable
by a mod k Moore G-space. m
While this corollary seems to be a formal consequence of defini-
tions, it does lead to the following theorem which will be proved in
section 5.
3.6 Theorem. There exist decomposabl ~ kG-modules M which are reali-
zable by homotopy G-actions, but they are not realizable by the homo-
logy of any G-space X .
Next, we apply the above results to give a proof of Theorem 2.1.
Proof of Theorem 2.1: Let M = @ Hi(X) . Then, if M is ~G-projec- i>0
tive, clearly M is ZC-projective for any subgroup, in particular
cyclic subgroups of G . Conversely, suppose M is ZC-projective for
all such C ~ G as in the theorem. Let M' = i~0 Hi(X;k) . By Choui-
nard's theorem, it suffices to consider the case where G is p-ele-
mentary abelian, and we will assume this for the sequel. Since X
satisfies the condition (DSBC) for G , one has VG(X) ~ V~(M') , by
Proposition 3.2. At this point one has several (basically equivalent)
ways of finishing the proof. The first is somewhat longer, but more
illuminating, and we will refer to it in the applications.
41
First argument: VG(X) is defined via the radical of the annihilator
of H~(X,x;k) , say j , in H G , which is the intersection of asso-
ciated prime ideals AnnHG(~) , for ~ £ H~(X,x;k) . Since associated
primes are closed under the Steenrod algebra, a theorem of Landweber
[24] and [25] (generalizing a theorem of Serre [29]; see also [I])
shows that they are generated by two dimensional classes in • H 2i i>0
(G;Fp) c H G . Landweber's proof is for ~p-COefficients throughout, but
one can easily check that his arguments goes through with k-coeffi-
cients and the same conclusion. (The invariance of associated primes
under the Steenrod algebra has been observed by several authors [25]
[31] [18]). Thus J is defined by linear equations with Fp-COeffi-
r(M') are F -rational cients. Consequently VG(X) as well as V G P ,
(i.e., a union of subvarieties defined by linear equations with ~ -co-
r(M'qkS) ~ ~(M') efficients). For a shifted cyclic subgroup S c kG , V S =
n trs,G(V~(k))_ (cf. [8]) where trs, G is the transfer. It follows
that for each shifted cyclic subgroup which is not a subgroup of G ,
r(M') = 0 (Here we assume to have S n G = {I} and trG,s(V (k)) N V G
chosen a k-vector space L such that I = L 8 12 , I = augmentation
ideal, as described in Section 2.) Hence V~(M') is detected by the
shifted cyclic subgroups S such that S N {G} # {I} , i.e. cyclic
subgroups of G . By the hypothesis, M'IkS is kS-free for all such
c G . Thus, V~(M') = 0 and M' is kG-free. Since Hi(X,x) is S
~C-projective, it is ~-free. The long exact sequence of cohomology
associated to 0 ~ Z ~ X ~ F ~ 0 breaks into short exact sequences: P
0 ~ Hi(x;~) xp > Hi(x;~) ~ Hi(X;Fp) ~ 0 .
But for all A ~ G , H*(A;H*(X,X;~p)) = 0 (H* = Tate cohomology and
kG-projectivity implies ~pG-Cohomological triviality [14]). Hence
H*(A,H*(X,x)) is p-divisible, which means that it vanishes for all
A ~ G . Therefore H*(X,x) is ~G-projective, being Z-free and ~-coho-
mologically trivial [28].
Second argument: An inductive argument using Cartan's formula shows
that the annihilating ideal of H~(X,x;k) is invariant under the
Steenrod algebra, as in G. Carlsson [13]. A theorem of Serre [29] then
r(M') is ratio- shows that the variety VG(X) is Fp-rational. Hence V G
hal using Proposition 3.2. The rest of the proof is as in the first
argument and the details are left to the reader, a
42
3.7 Addendum. The examination of the proof shows that in fact the
statement of Theorem 2.1 remains valid, if we replace Z-coefficients
by k-coefficients as well as ~G- and ~C-projective by kG- and kC-free
respectively. Thus one needs that Hi(X;k) = 0 for all sufficiently
large i , instead of the stronger statement with Z-coefficients. a
The above proof also suggests that as in J. Carlson [12], one can
determine the complexity of H*(X,x;k) by the dimension of the varie-
r(~iHi(X,x;k)) = VG(X) for this particular case. This is the ty V G
counterpart of Theorem 2.1 for non-projective modules.
Let p be a fixed prime and let k be a field of characteristic
p , say algebraically closed for convenience sake. We denote by CXG(M)
the complexity of the kG-module M (cf. [2] [23] [12]).
3.8 Theorem. Let X be a connected G-space which satisfies the condi-
tion (DSBC) for each maximal elementary abelian p-subgroup A ~ G and
H*(-;k) . Let M =i~0 Hi(X;k) with the induced kG-module structure.
Suppose CXG(M) = r . Then there exists a p-elementary abelian sub-
group E ~ G of rank r such that cxE(MlkE) = r
Proof: By Alperin-Evens [2], CXG(M) : max{cxA(MlkA) IA c G maximal A
p-elementary abelian} . Thus we may assume that G is elementary abe-
lian. Since V~(M) ~ VG(X) is rational as in the proof of Theorem 2.1
r(M) is the maximum dimension of the rational linear sub- above, dim V G
varieties whose union is V~(M) . Let V 0 be one such linear maximum
dimensional subspace of k n ~ V~(k) , (where we assumed n = rank G )
and let E = G 0 V 0 be the set of rational points of V 0 . Then rank
E = dim V 0 since V 0 is rational. On the other hand, trE,G(V~(MIkE))
= V 0 (cf. [8] and [11] for details) and CXE(MIkE) = dim V0=rank E. •
Let G be a p-elementary abelian group of rank n In [23], Ove
Kroll proves that if CXG(M) = t for a kG-module M , then there
exists a shifted subgroup F c kG of rank n-t such that MIkF is
kF-free. J. Carlson's proof of Kroll's theorem [12] is in essence a
"transversality argument" in the following sense. Since CXG(M) = t ,
dim V~(M) = t , and it is always possible to find an (n-t)-dimensional
linear subspace L of k n ~ V~(k) which is in "transverse position"
to V~(M) , (i.e. it has intersection {0} .) Now restriction to the
shifted subgroup F which is obtained from any k-basis of L yields
r(M)) = 0 which means that MIkF is kF-free. dim Vr(MIkF) = dim(L n V G
43
When V~(M) is rational, one would like to find a subgroup F ~ G
with the above property. But this is not possible in general as it can
be seen from the following simple example:
3.9 Example. Let M = 8E(kG ®kE k) where E runs over all cyclic sub-
groups of G . Then CXG(M) = I and MIkA is not kA-free for any
non-trivial subgroup A c G .
However, the first argument of the proof of Theorem 2.1 above
reveals that we can give a counterpart to Kroll's theorem in a parti-
cular case.
Call a G-space X "k-primary", if the radical of the annihilator
ideal of H{(X,x;k) in H E is prime for all maximal p-elementary
abelian subgroups of G . (Here k is a field of characteristic p
again.) Recall p-rank (G) d~fmax {rank of elementary abelian p-subgroup
E c G} .
3.10 Theorem. Suppose p-rank (G) = n and X is a connected k-primary
G-space which satisfies the condition (DSBC) for all maximal p-elemen-
tary abelian subgroups and H*(-;k)-coefficients. Also, assume that
Hi(X;k) = 0 for all sufficiently large i . Then there exists a p-ele-
mentary abelian subgroup E c G such that rank E = n-max {CXG(H i - i
(X,x;k))} and Hi(X,x;k) is kE-free for all i
Proof: As in the preceding theorems, it suffices to assume that G r
is p-elementary• abelian (Alperin-Evens [2]). By Proposition 3.2 V G
@i Hl(x'x;k)) ~ VG(X) . Since X is k-primary, the first argument (
in the proof of Theorem 2.1 shows that V ~ ( @ i H i [ X , x ; k ) ~ V~( ~ i Hi
(X,x;k)) consists of one rational linear subvariety of k n ~ V~ (k) ,
and its dimension equals to CxG( @ H i ( X , x ; k ) ) = max c x G ( H i ( X ; k ) ) i > 0
H e n c e t h e r e i s a r a t i o n a l l i n e a r s u b s p a c e L t r a n s v e r s e t o V~(H i _
( X ; k ) ) , a n d we may c h o o s e d i m L = n - m a x C X G ( H i ( X ; k ) ) . L e t E b e i > 0
t h e s u b g r o u p o f G whose F p - g e n e r a t o r s g i v e s a n F p - b a s i s f o r L . T h i s
i s t h e d e s i r e d s u b g r o u p , m
3.11 Remark. One can modify the above argument to weaken the hypothesis
that "X is k-primary" or that "X satisfies (DSBC)", etc. But these
hypotheses cannot be removed altogether by the above example 3.9 and
the example in Sections 4 and 5.
44
Section 4. Applications to Steenr0d's proble_mm
In this section we consider the special case of G-actions of Moore
spaces. Suppose M is a finitely generated Z-free ~G-module. Then M
is determined by a homomorphism p : G * GL(n,~) , where n = ran~(M).
Suppose that X is homotopy equivalent to a bouquet of spheres of
dimension k ~ 2 , and Hk(X) ~ Z n . Then E(X) ~ ~0H(X) ~ GL(n,~) by
obstruction theory. Thus p induces a homomorphism ~ : G ~ E(X) such
that the homotopy action (X,~) realizes the ZG-module M . More ge-
if Tor~(M,E 2) = 0 • or if G is of odd order, then an ob- nerally,
struction theory argument (cf. [221) shows that any homomorphism p:G
GL(n,~) (which induces the ZG-module structure of M ) can be lifted
to a homomorphism ~ : G * E(X) . Thus the homotopy action (X,~) rea-
lizes M .
On the other hand, given M , we have the Z-free ZG-module M'
from the exact sequence 0 * M' * F * M * 0 , where F is a free ZG-
module. It is not difficult to see that M is realizable by a Moore
G-space, if and only if M' is realizable by a Moore G-space. Thus,
as far as the question of realizability of ZG-modules is concerned,
one can consider E-free ZG-modules with no loss of generality. There-
fore, the realizability of modules by homotopy actions does not pose a
difficult problem in the contexts where one is primarily interested in
realizability by topological G-actions.
In passing, let us mention that the obstructions for realizability
of a ZG-module by a homotopy action on a Moore space has been studied
by P. Vogel [7] (unpublished). Vogel has shown that for G = ~2 × ~2 '
there is an F2[G]-module which is not realizable by a homotopy action
on a Moore space:
4.1 Example (P. Vogel) [7]. Regard Z 2 × Z 2 as the 2-Syiow subgroup
of GL(2,• 4) , i.e. as 2 × 2 upper triangular matrices of the form I x
(0 1 ) where x belongs to the field with 4 elements. The natural
action of GL(2,F 4) by left multiplication on the column vectors of 2
M = (F 4) makes M into a Z[Z 2 × Z2]-module. Vogel's obstruction
theory shows that this modules is not realizable by a homotopy action
of Z 2 × Z 2 on a Moore space.
4.2 Construction and Examples. Let k be an algebraic closure of Fp ,
= × Z be generated by e I and e 2 . Let I be the and let G Zp P
augmentation ideal and choose the k-vector space L such that I=L@I 2,
45
with {It,12} a k-basis for L , (as in Section 2). Then for almost
all choices of e = (~i,~2) 6 k 2 , the shifted unit u s = I+~iZi+~2£ 2
generates a shifted subgroup S ~ <u > of order p such that $ ~ G
= {I} (cf. Carlson [11] for details on shifted subgroups). More ex-
plicitly, for a (finite) Galois extension K of ~p , choose ~I,~2
6 K such that ue = I+~i(eI-I) + ~2(e2-I) satisfies ue-1 ~ 12 and
a u s ~ g (mod 12) for any g 6 G . The condition l-u~ ~ 12 ensures
that kG is kS-free, and S ~ <u > c kG can be treated like an ordi-
nary subgroup as far as induction and restriction is concerned [11].
In particular, Mackey's formula and Shapiro's Lemma are valid.
Recall that for the local ring kG , projective, injective, co-
homologicaliy trivial, and free modules coincide. First we need the
following:
4.3 Lemma. (i) There exists an indecomposable kG-module M 0 such that
M 0 is kC-projective for all cyclic subgroups C c G , but M 0 is not
kG-projective.
(ii) There exists a finitely generated Z-free gG-module M I which is
ZC-projective for all cyclic subgroups C c G , but M I is not ~G-
projective.
(iii) There exists an indecomposable ZG-module M with the same pro-
perties as in (ii) above.
(iv) In above part (iii), one may choose M such that k ® M ~ M'@ Q,
where M' is an indecomposable kG-module, and Q is kG-free.
Proof: (i) The above discussion, for (almost all) u chosen with S=
<u > , one has S N G = {I} and kG is a free kS-module. Let M 0 =
kG ®kS k be the induced module. Then for each C c G ' AICI = p ' C~ n S
= {1} . Hence H*(C,M 0) = 0 by Mackey's formula. But H(G,M 0) ~ H*
(S;k) ~ 0 by Shapiro's Lemma. Since kG is local, a cohomologically
trivial kG-module is kG-free (= kG-projective). Thus (i) is proved.
such that u = I+~1(ei-I ) + e2(e2-1) , where (ii) One can choose u s
~I and e2 lie in a finite Galois extension of ~p , say k I , and
<u > = S still satisfies the same properties as in (i). Let
M0=kIG® kIskl be the kIG-module which is kIC-free for each C ~ G
but not kIG-free as in (i). Consider the exact sequence 0 ~ M I ~ (ZG) t
M 0 ~ 0 . The long exact sequence of cohomology
...... Hi(C,M I) ~ HI(C, (ZG) t) ~ Hi(C,M 0) ......
46
shows that M I is ~C-projective for all C c G C ~ G , and M I is
not ZG-projective.
I @. .~ r be a decomposition in terms of inde- (iii) Let M I = M I ... M I
composable ZG-modules. Then all M~ are ZC-projective, but at least
1 Then I satisfies (ii) one of them is not ZG-projective, say M I . M I
and it is also indecomposable.
(iv) Tensor the exact sequence of (ii) by k :
t 0 ~ k 8 M I ~ (kG) ~ k ® M 0 ~ 0
Note that we can choose M 0 so that k@M 0 is indecomposable. (Briefly:
dimkk ® M 0 = dimkKG/kS = [G:S] = p , and since k ® M01kC is projec-
tive, the dimension over k of each kC-indecomposable summand, and
hence each kG-indecomposable summand must be divisible by p .) In
the short sequence:
0 ~ M' ~ P ~ k ® M0 ~ 0
where P is the projective cover of k ® M 0 , M' is also indecompo-
sable, since k ® M 0 is indecomposable. Hence Schanuel's Lemma shows
that k ® M I ~ M' @ (projective). •
4.4 Theorem. Suppose G is a finite group such that G D Z × P P
Then:
' which satisfies (i) of Lemma 4.3 (I) there exists a kG-module M 0
(II) There exists a ZG-module M' which satisfies (iv) of Lemma 4.3.
Further, it is not possible to find a Moore G-space X such that H,
(X;k) = M~ as kG-modules.
Similarly, there does not exist a Moore G-space X such that
H.(X;Z) = as ZG-modules.
Proof: Let M 0 be the k[Zp× Zp]-module of Lemma 4.3(i). Let M~
kG@k[Zp×Zp]M 0 Since S n C = {1} , Mackey's formula shows that for
each C c G , ICI = prime, M6/kC is kC-cohomologically trivial, hence
kC-free. But M~ is not kG-free since it is not k[~p x ~p]-free, as
M'IZ_0 p × Zp has M 0 as a direct summand, (or apply Shapiro's lemma).
(If) Let M be as in Lemma 4.3 (iv), and let M' = ZG@z[ZpXZp]M .
The assertion follows as in part (I). Now the non-existence of the
Moore G-spaces realizing these G-modules is a consequence of the pro-
47
jectivity criterion Theorem 2.1. m
(Compare 4.4 with G. Carlsson's theorem [13].)
The case G = Q2n = generalized quaternionic group of order 2 n
is somewhat different, because the maximal elementary abelian subgroup
of Q2n is the subgroup of order two generated by the central element
T 6 Q2n . Therefore kG-projectivity (or ZG-projectivity) of a module
is completely decided by the restriction to k<~> or Z<~> . There-
fore Theorem 2.1 does not help directly in this situation. In the se-
quel, we present first a proof of non-realizability of a kG-module by
Moore G-space (similarly for a ~G-module) in the finite dimensional
case, and we will use the geometric intuition of this case to remove
the finite dimensionality restriction with a different proof.
4.5 Proposition. Let G be the quaternionic group of order 2 n , n~3.
Then there exists a ZG-module M such that M is not ~G-isomorphic
to the (reduced) homology of a finite dimensional Moore G-space X
Similarly, k ® M is not ~G-isomorphic to H,(X;k)
Proof: Let T 6 G be the central element of order 2 and let • gene-
rate T ~ ~2 c G Then G/T ~ = . = D2n- I , the dihedral group of order
2 n-1 . Let M be the module over ~2 × ~2 constructed as in Lemma
4.3 (iv) above and let N = Z[D2n-I]~[Z2 ® ]M . Then M is not × Z 2
Z[D2n-1]-isomorphic to H,(X 0) for any Moore G-space X 0 . In fact,
k I ® M is not k1[D2n-1]-isomorphic to H,(X0;k I) for any field k I
of characteristic 2.
Consider N as a ~Q2n-module, where T acts trivially on N
(To get a G-module on which all elements of G act non-trivially take
ZG @ N , or the group of n-cocycles in a minimal projective resolution
of N over ~G .) . Suppose there exists a finite dimensional Moore G-
space Y such that, yG # ~ and H,(Y) ~ N as ZG-module. Then yT
is a D2n_1-space of finite dimension, and since the Serre spectral se-
quence of Y ~ E T × T Y ~ BT collapses, H*(yT;kl)~H*(T;N)®~,(T;xI)k I as
in the proof of Proposition 1.1. Using this periodicity of H*(T;N) ,
it follows that H*(yT;kl)~(N®kl)~/(I+~) (N®kl)~N®k I . But this means
that N ® k I is realized by the D2n_1-space yT • i.e. H*(yT;kl)~
N ® k I . By Proposition 1.2 or Theorem 4.4 this cannot happen. •
48
Alternatively, the w-stable module M(Y T) up to u-stability is
kiG-isomorphic to N @ Q where N is the indecomposable factor and
Q is kG-free. This is the case because H,(yT;k I) has only one de-
composable kiG-module N as a summand which is not kiG-free. Thus the
construction M(X) and the definition of ~-composite extensions shows
that any ~-composite extension of various Hi(YT;kl) is of the form
N @ Q up to u-stability. Now the Projectivity criterion Theorem 2.1
of Theorem 4.4 shows that N ~ Q of wJ(N) @ Q cannot occur as
H,(L;k I) for any Moore D2n-l-space L . This contradiction shows that
such a Moore G-space cannot exist.
The proof of the above implies the finite dimensional case of the
following corollary. (The details are left to the reader).
4.6 Corollary. If G ~ Q2 n , then there are ~G-modules which are not
ZG-isomorphic to the reduced homology of a Moore G-space. []
Now we proceed to give a different proof which shows that such
Moore G-spaces cannot exist regardless of their dimensions.
Since every quaternionic 2-group contains the quaternionic group
Q8 of order 8, we will prove the theorem for Q8 and deduce the re-
sult for Q2n , n a 3 from it. Suppose that X is any Moore G-space,
where G = Q8 ' such that H*(X,x) ~ M , (x 6 X G ~ ~) . Let M be a
~Q8-module which is Z-free, and T { <T> c Q8 acts trivially on M ,
and let A = Qs/T ~ Z 2 x Z 2 induce a ZA-module structure on M . Con-
sider the Borel construction (W,W0) = E G x T(X,x) which carries a
free A-action. The Serre spectral sequence (X,x) ~ (W,W 0) ~ BT collap-
ses and H*(W,W0;k I) ~ H*(T,M ® k I) . Denote M O k I by M I . Since
T acts trivially on M I , it follows that H*(T,M I) ~ H*(T,k I) ® M
H*(W,W0,k I )
Now consider the Borel construction E A x A(W,W0) ~ BA . In the
spectral sequence of this fibration, E~ '0= 0 for all p and E~ 'I ~
HP(A;HI(w,w0 )) ~ HP(A;M I) . On the other hand, E A × A(W,W0) ~ (W/A,
W0/A) since A acts freely, and (W/A,W0/A) = E G x G(X,x) . Hence
E~,I ~ EL, I ~= HG1(X,x;kl) ~= HI(G;M I) The HA-mOdule structure of E G
XG(X,x) is also related to the HG-structure by the following commu-
tative diagram:
E G × G(X,x) < > E × A(W,W0)
B G ~. B A .
49
At this point, let M I ~ kIA O kiskl , and note that H*(A;M I)
H*(S;k I) ~ k1[g ~] for g~ £ HI(s;k 1) . Let the corresponding generated
be denoted by y 6 HI(A;M I) . Then rad(Ann(y)) in H A is the ideal
j = (~lY+~2x)
On the other hand, let C be the cyclic group of order 4 in k I
[Q8 ] qiven by the extension T ~ C ~ S . If we regard k I as a trivial
module over kS on which T acts trivially also, it follows that
l kiA ® kISkllkiC ~ kiQ 8 0 kick1~kl C .
Thus, H*(Q8;M I) ~ H*(C;k I) , and in the Lyndon-Hochschild-Serre
spectral sequence of T ~ C ~ S , HI(s;k I) ~ HI(C;kl ) while all other
HI(s;k I) map to zero in Hi(C;kl)
Since the diagram
T > C ; S
f
T > Q8 --~ A
commutes, we may identify g~ £ H I (S;k I) with a generator g £
), 1 HI(c;k I) HI (Q8;M I . Under this identifiaction, g 6 H_ (X,x;k~) =
HI(E x A(¢,%~0);kl ) is identified with 7 6 HI(A;MI) ~81(S;ki )I
4.7 Assertion: rad(Ann(g)) = J in H A .
Proof: It suffices to show that f = ~lY+~2x belongs to Ann(g) since
f generates J . But ft Y = 0 since f 6 Ann(y) = J for some t~0
The natura!ity of all the identifications made above shows that ft.y
= 0 ~ ft.g = 0 ~ ft g = 0 ~ (f) = rad(Ann(g))
On the other hand, rad(Ann(g)) must be invariant under Steenrod
algebra, being an associated prime for the module H~8(X,x;k I) over
H A . Hence its variety must be Fp-rational by Serre's theorem [29],
and J is not rational over F by the choice of ~ . This contra- P
diction establishes the theorem.
4.8 Remark. An alternative proof using a complexity argument is briefly
as follows. In the spectral sequence with E~ 'q = HP(A;Hq(W,W0)) which
converges to H*(E G × G(X,x) ~ H*(C;k 1) , for p+q = constant, E~ 'q
0 only for one pair (p,q) . Thus multiplication by ft shifts the
filtration in E . But since there is only one non-zero term, it
50
follows that an appropriate power of ft kills the E -term in this
case. This shows that the radical of the annihilator of the module
contains f . Hence the HA-variety of X is the intersection of the
line £ given by f with possible other lines. If this intersection
does not include I , then it must be zero dimensional, and one argues
that M must be Z 2 × Z 2 -projective accordingly, which is a contra-
diction again.
The above results show the following theorem, due to Carlsson for
G = ~p × Zp [13] and to Vogel for G D Q8 (to appear) using calcula-
tions with the Steenrod algebra. An exposition of Vogel's theorem can
be found in [9].
4.9 Theorem. If all ZG-modules are realizable by Moore G-spaces, then
G is "metacyclic", i.e. all Sylow subgroups of G are cyclic.
4.10 Remark. Jackowski, Vogel and several others have observed that
Carlsson's counterexample for Z × ~ implies that for G m Z × P P P P
the induced module is also a counterexample.
Section 5. Some Examples
We have seen how to construct examples of ZG-modules which are not
realizable by Moore G-spaces. These also give examples of homotopy
actions on Moore spaces which are not equivalent to a topological ac-
tion. The question arises whether these lead to criteria for homotopy
actions on more general spaces to be equivalent to topological actions.
It is helpful to consider the case of spaces which are bouquets of
Moore spaces of different dimensions. We will briefly investigate the
possibility of realizing a given ZG-module M by a topological action
on such a space. This module M arises from a homotopy action (X,~)
and as a consequence our examples reveal some properties of homotopy
actions on such spaces. Note that if a ZG-module M is indecomposable,
then M can be realized only by a Moore G-space. Thus to get new
examples, we will consider decomposable modules.
By means of a simple construction using the modules of Section 4
and the theory of Sections 2 and 3, we will show that for G m ~ ×
the following h01d. P P
(5.1) There is a ZG-module M = M I ~ M 2 , where M i ~ 0 are indecompo-
51
sable, such that neither M nor M i are realizable by ~[oore G-spa-
ces.
(5.2) There is an (n-1)-connected finite G-CW complex X of dimension
n+1 such that @ H.(X) = M as ZG-module. Call this action ~ : G × X i l
~X.
(5.3) X is homotopy equivalent to a bouquet of spheres of dimension
n and n+1 , but (X,~) is not G-homotopy equivalent to a bouquet of
spheres, with a G-action.
(5.4) Let P be the projective cover of M I and O ~ ~(M I) ~ P ~ M I
0 be an exact sequence of ZG-modules. Then an extension of M I and
~(M I) is realizable by a finite dimensional Moore G-space. Similarly
for M . This extension is non-trivial necessarily.
(5.5) We may choose M I = M 2 in the above.
(5.6) Since ~(M I) is not realizable by a Moore space either, we have
also examples of modules M I and M~ = ~(M 1) such that M 1 @ M~ is
not realizable by a topological action on a Moore space, but some non-
trivial extension of M I and M~ is realizable by a Moore G-space.
(5.7) We may construct examples where M I = ~(M I) in the above.
(5.8) There is a homotopy action of G , say ~ , on a finite bouquet
of n-spheres L , such that (L,~) and any suspension of this h-action
(ZiL,zi~) are not equivalent to topological actions. But (LvEL,~vZe)
is equivalent to a topological action.
r(x) , thus the inclusion VG(X ) c V~(X) of Proposi- (5.9) VG(X) ~ V G
tion 3.2 cannot be improved (even for finite dimensional spaces). Here
the varieties are taken over kG . Here VG(X) = 0 while @iHi(X,x;k)
is not kG-free.
(5.10) Radicals of the annihilators in H G of H~(X,x;k) and
H*(G;H*(X,x;k)) are not equal.
(5.11) We may choose M i such that the projectivity criterion 2.1 does
not apply to X . This will follow because we will choose M such 1
that • Hi(X,x) IZC is ZC-projective for all C c G , ICI = prime, but
• iHi(X,x) is not ZG-projective. Thus Theorem 2.1 cannot be extended
to all G-spaces without additional hypotheses (even for finite dimen-
sional G-spaces).
(5.12) For appropriate choices of M I and M 2 , M = MI~ M 2 will not
be realizable by any G-space, M i ~ 0 , i = 1,2
52
5.13 Example. It suffices to consider G = Z × ~ , and the above P P
× ~p or G D Q8 " assertions (whenever applicable) hold fo]~ G m Zp
Consider the ~G-module M I constructed in Theorem 4.4. For some of the
assertions such as (5.5), (5.61}, and (5.7), let p = 2 , otherwise p
is any prime. We may choose M 1 to be Z-free and ZG-indecomposable.
From the exact sequence:
(5.14) 0 ~ M 2 ~ (ZG) r <0 s (ZG) ~ M I ~ 0
it follows that M21ZC is ZC-projective for all C c G , IC! = prime
while M 2 is not ZG-projectiwg, since M I is not ZG-projective.
Therefore M 2 is not realizable by a Moore G-space either. Let M =
M I • M 2 . The same holds for M .
We may take bouquets of s and r free G-orbits of n-spheres,
s r i.e. X I = V (G+^Sn)i and X,) = V (G+AS n)
i=I " j=1 J
There exists a G-map f : X 2 ~ X I such that f,: Hn(X 2) ~ Hn(X I) can
be identified with the ZG-homomorphism ~ : (~G) r ~ (~G)s after appro-
priate identifications Hn(XI) ~ (ZG) s and Hn(X 2) ~= (~G) r . Then the
mapping cone of f is a finite G-space X which satisfies (5.1) and
(5.2) above, in view of the exact sequence (5.14), (5.1) and (5.2)
imply (5.3) .
The projective cover of }41 , namely P satisfies O ~ P ~ F I
F 2 ~ 0 where F I and F 2 are ~G-free (not necessarily finitely
generated). Thus P can be realized via the mapping cone X 0 of the
G-map g : V (G+^S n-l) (G+^S n-l) i i ~ Vj j corresponding to ~ (i.e.
g. = ~ in Hn_ I (-;Z) ). X 0 is also free off the base point. In the
exact sequence :
' ~ P ~ M I ~ 0 (5.15) 0~M I
the homomorphism ~ can be realized by a G-map f': X 0 ~ X which in-
duces f~ : Hn(X0) ~ Hn(X ) f' = , , , ~ by equivariant obstruction theo-
ry (or see [3]). The mapping cone of f' , say Y , is a Moore G-space
and Hn+I(Y ) is the extension in the sequence:
(5.16) 0 ~ M 2 ~ Hn+ I (Y) ~ M~ ~ 0
53
Thus an extension of M~ and M 2 is realizable by the Moore G-space
Y . This proves (5.4). Since M I is a periodic module by construction,
by taking G = ~2 × Z2 we can fulfill (5.5) - (5.7). If we wish to
choose M I ~ M 2 for odd p , just take the exact sequence
(5.17) 0 ~ M I ~ PI ~ P2 ~ MI ~ 0
where Pi are projective covers, and Ker~=M I since M I is chosen
to be indecomposable. (5.17) exists due to periodicity of M I . This
is the analogue of (5.14) and we can use P. instead of F , i=I,2 . 1 l
Since all these modules are realizable by homotopy actions (ob-
struction theory), the assertion (5.8) follows easily from the pre-
vious ones.
r(k O M I) is not ~ -rational by the To see (5.9), note that V G P
construction~ (cf. Section 4). Thus V~(~ ~iHi(X;k)) ~ V~(k ~ ® (MI~ M2))=
V~(M 1)u is not rational over ~p . But VG(X) is rational over r P
r(@iHi(X;k)) Since (see the proof of 2.1). Thus VG(X) # V G VG(M I )
is only one line, in this case it follows that VG(X) = 0 in fact.
Except for (5.12) which will be proved below separately, the other
assertions follow from the above discussion and elementary considera-
tions.
x ~ or Q8 Again, in the following G D Zp P
5.18 Theorem. There exists a decomposable ~G-module M which cannot
be realized by the total reduced homology of any G-space. There are
homotopy actions (X,~) realizing M , and all such (X,~) are not
equivalent to topological actions.
Proof: As before, we may assume G = ~ x Z and the general case P P
follows from this case. Choose ua and u B as in Theorem 4.4, such
that u~ ~ u B (mod 12) and the lines in k 2 given by u~ and u B
are distinct. Corresponding to these choices we get indecomposable
Z-free ~G-modules M~ and M B whose rank varieties are the lines de-
termined by u~ and u B . Neither M~ nor M B is realizable by a
Moore G-space using the projectivity criterion 2.1. For the same rea-
son, ~t(M ) , wS(M B) or any direct sum of them are not realizable
by Moore G-spaces (see Section 3). Any w-composite of M and M 5 is
of the form:
(5.19) 0 ~ ~t(M~) ~ U ~ wS(M B) ~ 0
54
and this extension is determined by a class
By tensoring with k , we get
6 Ex~(et(Ms) , eS(Ms)).
(5.2O) 0 ~ et(M ® k) ~ U ® k ~ ~S(MB® k) ~ 0
and a corresponding class ~'6 EXt~G(et(Ms® k) , eS(M88 k)) . We claim
that this class vanishes, so that (5.20) is split and U ® k ~ e(MB® k)
~t(M ® k) . But this follows from the fact that EXt~G(~t(M ® k),
eS(MB®~k))~HI(G,~t(M 8 k)~® ~S(MB® k)) = 0 , where * means dual with
respect to k . The last assertion is a consequence of J. Carlson ten-
sor product formula ([11] Theorem 5.6) as follows. The rank variety of t
e (Ms® k)* is seen to be the same as V (wt(Ms® k)) = V (Ms8 k) by
the definition of V r , and V~(et(Ms® k)* ® ~S(MB® k)) = V~(M ® k) t s r(M ® k) = 0 by the choice of ~ and ~ . Hence ~ (MR® k)* ® V G
(Ms ® k) is kG-free by (4) of Section 2, and n' = 0 as a consequence.
Now suppose M = Ms8 M B is realizable by a G-space. Then an e-compo-
site of M~8 k and MS® k is realizable by a Moore G-space by Co-
rollary 3.5. By the above discussion, any such u-composite is split
and it cannot be realized by a Moore G-space since it does not satisfy
the projectivity criterion (Theorem 2.1).
Since M is realizable by a homotopy action, the second assertion
follows. •
[I]
[2]
[3]
[4]
[5]
[6]
C7]
[8]
[9]
[10]
[11]
[12]
[13]
55
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Normall[ Linear Poincar6 Complexes
And E~uivariant Splittin~_s
Amir H. ASSADI (*) University of Virginia
Charlottesville, Virginia USA
INTRODUCTION: The study of a number of problems in group actions on
manifolds calls for explicit constructions of actions. Successful appli-
cations of surgery theory in the non-equivariant problems has been a
great motivation for various generalization of surgery to the equivari-
ant set up. However, the variety of problems which may be approached
via surgery in transformation groups is quite rich. The wide range of
phenomena which are to be studied in some of the traditional problems
(such as existence and classification problems) has limited the range
of applicability of the existing equivariant theories. As a result, it
seems appropriate to device specialized surgery theories which aim at
different classes of more specific problems.
In the problems which arise in conjunction with the existence and
classification of actions on manifolds, it is often useful (in agreement
with the general philosophy of surgery) to divide roughly the construc-
tions to two steps. In the first step, one uses methods of algebraic
topology to study the problem in the homotopy category. In the second
step one passes from the homotopy category to manifolds via surgery.
The objects of interest in the first step are Poincar@ complexes. Since
the category of Poincar6 complexes plays an important role in the study
of smooth manifolds, it is natural, thus, to study homotopy related
problems of G-manifolds on the level of equivariant analogues of this
category.
The question arises, then, as to what extent a Poincar~ complex
with G-action should inherit the structure of a G-manifold. In this
paper, we suggest a category of Poincar@ complexes with G-actions, whose
objects are called "normally linear Poincar6 G-complexes" and the
morphisms are "isovariant normally linear maps". This category inherits
(*) The author has been partially supported by an NSF grant, The Center
For Advanced Study of the University of Virginia, and The Max-Planck- Institut Fdr Mathematik whose hospitality and financial support is gratefully acknowledged.
59
all the homotopy aspects of the category of Poincar6 complexes without
G-actions, while it has a certain amount of "manifold information"
(from the category of G-manifolds) built into its objects and morphisms.
This is in the form of a "suitable stratification" and a linearization
of the Spivak normal sphere bundles of strata.
The range of applications and usefulness of this category, of
course, depends on how successfully one is able to translate "the alae-
braic topology" of a problem into the kind of information which would
allow one to construct "homotopy models" in this category. Constructions
of objects in a category of Poincar~ G-complexes becomes difficult if
the candidate Poincar& G-complex is required to have "too much manifold
information" built into it. On the other hand, imposing "insufficient
manifold-like structure" on a Poincar@ G-complex makes it difficult to
construct equivariant surgery problems from such complexes, (mainly due
to lack of equivariant transversality.)
Thus, it appears that the nature of the problem at hand should
determine the extent of manifold-like data required from homotopy models.
We will illustrate this point by studying the problem of equivariant
splittings of closed G-manifolds in our category. Theorem II. I and II. 2
give necessary and sufficient conditions for the existence of splittings
up to homotopy in terms of normally linear Poincar6 G-complexes. Theorems
IV. I , IV. 3 , and IV. 7 illustrate constructions and solutions of the
relevant surgery problems, using the homotopy models of Theorem II. 1 .
To give concrete examples, Theorem IV. 5 considers the problem for
homotopy spheres and yields a generalization of Anderson-Hambleton's
theorem ([I] Theorem A) while Theorem III. I illustrates a shorter and
different proof of their theorem. Further applications of these ideas
will appear in a subsequent paper.
The contents of this paper is as follows. In Section I the category
of normally linear Poincar@ G-complexes is introduced, and some relevant
definitions and background information £s mentioned. Section II contains
the construction of objects of this category which will be used to study
equivariant splittings. Section III illustrates the theory applied to
the special case of homotopy spheres to give another proof of the
Anderson-Hambleton theorem. This section serves to motivate the genera-
lization of this theorem in Section IV. (Theorem IV. 5), and the solution
of the splitting problem up to concordance (Theorem IV. I , IV. 3 and
IV. 7 ) with varying degrees of generality. We conclude the paper by a
brief discussion of the algebraic obstructions which arise in the
general splitting problem.
60
Finally, we would like to point out a few remarks and mention some
features which are implicit in this particular choice of application
for normally linear Poincar6 complexes. First, our methods does not
require "general positionality" or the so called "Gap Hypotheses"
which have been used by most authors. Here, the reader will find a
discussion of the problem of relaxing "general positionality" in the
equivariant surgery problems in Reinhard Schultz' survey article and
collection of problems [20]. Thus, the theories which use general-
position-type assumptions do not apply to our situation. Secondly, we
have considered non-simply-connected manifolds, not only to achieve a
greater degree of generality, but also to illustrate new applications
for the algebraic K-theoretic functor Wh~ of [8] , [9] which is the
relevant functor to capture such obstructions. We have postponed
explicit computations of these obstructions as well as certain other
surgery obstructions to a forthcoming paper. The reader, however, will
find some results in this direction in [9].
The third point concerns the notion of quasisimple actions and
their constructions. The homological hypotheses which are necessary in
the splitting problem and "the extension problem" of [9] use Z z - q coefficients (local coefficients) where Z = Z/q Z . When z is an
q infinite group, one cannot replace ZqZ- coefficients with Z(q)Z -
coefficients, where Z(q) is the integers localized at q . While the
constructions of [9] are given for Z z (in order to provide necessary q
and sufficient conditions for the constructions to exist), they work as
well with Z(q)Z replacing Zq~ everywhere. Thus, in all the homolo-
gical conditions in this paper, one can replace Zq by Z(q) ; but
the sufficient conditions obtained in this form will not be necessary
anymore. S. Weinberger has independently studied "unextended homologi-
cally trivial actions" [23] (which is the analogue of our quasisimple
actions for the case of Z(q)Z- coefficients) using "Zabrodsky mixing".
Weinberger's survey article in [24] contains further ideas and develop-
ments in conjunction with construction of actions. We refer the reader
to [20] for articles of Schultz and Weinberger and their references for
discussions of related results and problems.
Finally, to study G-actions on Poincar@ complexes which are not
quasisimple, one encounters completely new phenomena. The methods of
constructions which assume that G acts trivially on homology do not
apply to non-quasisimple actions. An alternative is to study such pro-
blems via "homotopy actions". This is the point of view of [7] (see
also [4]). Construction of non-quasisimple normally linear Poincar6
61
G-complexes (using homotopy actions) and further applications will be
discussed in a forthcoming paper of the author.
REMARK: It appears to us that the constructions of normally linear
Poincar& complexes, (e.g. as in Section II) may be combined with
Browder-Quinn's paper in Manifolds, Tokyo, 1973, (University of Tokyo
Press 1975) to give a general set up for classification theory of quasi-
simple actions. Moreover, Browder-Quinn theory can be potentially use-
ful to analyze the G-manifold structures on normally linear Poincar&
G-complexes. In this fashion, one may try to refine the results of our
Section IV by analyzing the relevant surgery obstructions in the
Browder-Quinn theory (instead of passing to concordance to bypass
possibly non-zero obstructions).
SECTION I. PRELIMINARY NOTIONS:
Throughout this paper G is a finite group of order q , and we
will work in the category of G-CW complexes, while G-actions on
smooth manifolds are assumed to be smooth. The smoothness assumption
is made only for convenience sake and most of the results, when appro-
priate, are true about more general types of action with some regulariy
conditions, e.g. locally smooth PL actions, etc.
An earlier definition for a Poincar6 G-complex was suggested by
FrankConnolly [11], where all the homotopy analogues of the ingredients
involved in a G-manifold were built into the definition of a so called
"G-Poincar& complex". For our purposes, however, it is appropriate to
introduce G-complexes which have inherited some linear structure on the
regular neighborhoods of various strata. This restriction, in this case,
makes it possible to translate the homotopy problems involving (non-free)
G-manifolds into questions which involve the homotopy structure of the
fixed point sets without losing the linear information naturally given
for their normal bundles. Furthermore, we will discuss methods of
construction for such G-complexes with this richer structure, and obtain
positive answers in a variety of circumstances.
Let C be a category of Poincar@ complexes (pairs). C could be
the category of simple Poincar6 complexes, or the category of finite
Poincar6 complexes, or a more general category, for instance [22]. We
will fix C during the following discussion and suppress any reference
82
to it unless it is necessary. For the applications, the context will
determine the category C .
I.I. DEFINITION: A normally linear Poincar~ G-complex (pair) with
one orbit type is a Poincar@ complex (pair) in C in the ordinary
sense (not necessarily connected). A normally linear Poincar~ G-pair
(X,Y) with (k+1) orbit-types is defined inductively as follows.
Let H be a maximal isotropy subgroup. Then (G • X H , G • yH) is re-
quired to be a Poincar@ G-pair with one orbit type which has an equi-
variant regular neighborhood pair (R,~IR) in (X,Y) such that:
(I) there exists a G-bundle w over G • X H such that (R,91R) is
G-homeomorphic to (D(~) , D(wlG • yH)) ;
(2) there is a normally linear Poincar@ G-pair (C,$C) with k orbit
types and a G-homeomorphism f : S(~) --> ~+Cc ~C such that
X = C U D(~) and Y = ~ C U D(~IG • yH) where ~ C = ~C- ~+C and f - f, -
f, = flS(ml G • yH)
REMARK: Normally linear Poincar@ G-complexes defined above are diffe-
rent from Conolly's [11] G-Poincar@ complexes in at least two different
points. First, the Spivak normal fibre space of one stratum in the next
is already given a linear structure. Second, the Poincar~ embeddings of
our definition are more manifold like in that the complement of one
stratum in the next is also prescribed (subject to the appropriate
identifications coming with the structure). As we shall illustrate in
Section IV, this results in a great simplification of the construction
of surgery problems.
is called the equivariant normal bundle of G" X H . An isova-
riant normally linear map is an equivariant map which preserves the
isotropy types and the normal bundles (after the identification of re-
gular neighborhoods and disk bundles). The G-homeomorphisms f and
f' above are (G-cellular) isovariant normally linear maps of Poincar~
pairs with k orbit types. It is possible to show inductively that for
each subgroup K~G , (xK,y K) is a Poincar~ complex which is Poincar~
embedded in (X,Y) , and its Spivak normal fibre sp~ce has a N(K)-
linear structure. (N(K) = normalizer of K in G ).
Normally linear Poincar@ G-complexes are constructed in [4],[5],
[7],[9] in the semifree case. Smooth G-manifolds are normally linear
63
Poincar~ G-complexes in a natural manner. We drop the prefix G when-
ever the context allows us to de so.
1.2. CONVENTION: All Poincar~ complexes with G-actions are assumed
to be normally linear Poincar~ complexes. If L ~ K ~ G , dimX K - dimX L < 2 .
If X is connected, we assume that X K is connected for all K~G .
All manifolds are compact and all Poincar~ complexes are finite.
We will study first the case of semifree actions which serve as
a model for the inductive proofs of similar results for actions with
several isotropy groups. However, the generalization of the results of
the semifree case is not immediate, even in the case of actions on
spheres (or disks) due to the fact that the fixed point sets of iso-
tropy subgroups of composite order satisfy very little homological
restrictions in general. In fact, Oliver's work [17] shews that in the
case of disks, only certain Euler characteristic relationships are
necessary (and sufficient). Therefore, it is inevitable to consider
some restricted classes of actions where some minimal homological
conditions are imposed on the fixed point sets of various isotropy
subgroups.
A convenient category of G-complexes is the category of quasi-
simple actions.
1.2. DEFINITION: An action ~ : G x X~X is called quasisimple if for
each isotropy subgroup K~G , the action of N(K)/K on the fundamen-
tal group of each component of X K and, subsequently,
H,(X~ ; Zq~I(X~)) are trivial. Note that the triviality of the action
of N(K)/K on ~i (X~) makes it possible to define unambiguously the
action on the homology of X K~ witlh local coefficients ZqZI(X ~)
= (Recall that Zq Z/q Z . One may also use Z(q) systematically).
REMARKS: (I) Quasisimple actions were introduced and studied in [9].
(2) Replacing Zq by Z(q) in the above definition, for a free G-
space X , quasisimplicity means that ~i (X/G) ~ zl (X) x G and G acts
trivially on the homology. This notion has been called "an unextended
action" by S. Weinberger and studied in [23] independently.
64
1.3. DEFINITION: Let X be a connected G-CW complex, where G
is a finite group of order q . X is called a simple G-space (and the
x is fibre homotopy equivalent to action is called simple) if (E G GX)q
(BG x X) . Here X denotes the localization of X which preserves q q
~I (X) and localizes ~i(X) for i > I at Z/q Z . Cf. [10] and [9]
Section II.
Zn dealing with non-simply-connected complexes, it is necessary to
consider simple homotopy types and simple homotopy equivalences. The
equivariant generalizations of the Whitehead torsion are studied in
[18],[15],[14]. To construct a G-action on a simply-connected finite
complex X (up to homotopy type in the category of finite complexes)
the projective class group K0(ZG) and certain subgroups or subquotients
play an important role (cf. [21],[17],[2],[I] etc.). If ~I(X) # I , then
the analogue of K0(ZG) is an abelian group Wh~(~ c F) where ~ = z1(X)
and £ is the extension I --> ~--> ~-->G -->I obtained from the
action of G on zI(X) (whenever defined). Wh~(~ c [') and its alge-
braic properties and topological applications are treated in [9], and
an alternative definition in terms of the fibre of a transfer map bet-
ween Whitehead spaces is given in [8].
We will briefly recall the definition and some properties of
Wh~(~ c ~) when ]~ = ~ × G (the case of quasisimple actions). Let A be
the category whose objects are pairs (M,B) where M is a finitely
generated ZF - projective module which is free over ~ and 8 is a
-basis for M . Two objects are equivalent (M,B) ~ (M',B') if there
is a ~ - simple isomorphism f : (M,8) --> (M',~') . Let A' : A/~
and consider the monoid structure on A' induced by direct sums (and
disjoint union), taking (0,~) as the neutral element. Then (ZF,G)
generates the monoid of trivial elements T , and we define
Wh~(~c£) = A'/T . It is an abelian group which fits into a 5-term i
transfer exact sequence Wh1(£) tr>wh1(~ ) ~>Wh~(~ci~) ~>~0(ZF ) tr>~ (Z~) 0
The homomorphism ~ is induced by the forgetful functor (M,~) --> M .
Furthermore, let Wh~(Z;Zq) = K1(ZqT~)/{±z} . Then one has a conunutative
diagram
Wh1(z) _~B > Wh~(zcF) -~-~> K0(ZF)
can°n'~M /
WhI(~;Z q)
85
where ~ o y is a generalization of the Swan homomorphism (cf. [21])
(Zq)X--> K0(ZG) (when z = I ) . OG:
A topological application of Wh~(zc F) is as follows. Suppose
(X,Y) is a pair, Zl(X) = z , and X is a finite G-complex. Let
: G x y --> y be a free quasisimple action, and let H,(X,Y;Zq~) = 0 .
Then there exists a free finite G-complex X' such that Y is an
invariant subcomplex, and there exists a q-simple homotopy equivalence
f : X' --> X rel Y if and only if y~(X,Y) = 0 in WhT(~ c~ x G) ,
where T(X,Y) is the Reidemeister torsion of the pair (X,Y) (well-
defined in WhI(~;Z q) due to the homological hypothesis). Cf. [9]
Section I for further details.
1.3. LEMMA: Suppose X is a finite semifree simple G-complex. Then
H,(x,xG;Zqn) = 0 and yT(X,X G) 6WhT(zcz × G) vanishes, where
X = zI(X)
PROOF: Cf. [9] Proposition II.3.
We extend the notion of admissible splittings of [I] to non-simply
connected closed manifolds (Poincar6 complexes). Let M n be a closed
manifold and let M n = M~ U M~ be a splitting so that M I N M 2 = ~M 2
It is an "admissible splitting" if z1(~M1) ~ ~1(Mi) ~ z1(M) = z (simi-
larly for Poincar6 complexes).
1.4. LEMMA: Let ~ : G x M n --> M n be semifree and suppose that
M = M I U M 2 is an equivariant admissible decomposition of (M,~)
that M i are simple. Let M G = F and M i N F = F i • Then
H,(Mi,Fi;Zq~) = 0 and yT(Mi,F i) = 0 for i = 1,2,~ = z1(M)
such
PROOF: This follows from 1.3.
In the next section we will show how to construct normally linear
Poincar~ complexes to solve the equivariant splitting problem for closed
G-manifolds on the level of homotopy.
66
SECTION II. SPLITTING UP TO HOMOTOPY:
As before, G is a finite group of order q . Let ~ : G x Z n~ Z n
be a smooth, semifree action on a homotopy sphere. In [I] Anderson and
Hambleton studied criteria for the existance of equivariant homological
symmetry of (Z n ~) i.e zn n D n where each Dn is an invariant , , . ~ = D I U 2 i
disk and Hj ((D<) G)~_ ~ Hj ((D~) G)_ for all j . Roughly speaking, vani-
shing of a semi-characteristic type invariant characterizes (En,@)
which are homologically double in the above sense, provided that
n > 2 dim zG . Anderson and Hambleton call this structure a (strong)
balanced splitting.
Since any homotopy sphere is a twisted double, the results of [I]
may be interpreted as finding obstructions to make a (given) "non-
equivariant symmetry" into an eguivariant one. Besides leading to the
discovery of a new and interesting invariant of such semifree actions,
this equivariant symmetry may be regarded as a homolo~ical regularity
condition (i.e. similarity to the linear actions). From this perspec-
tive, it is natural to ask if such equivariant splittings exist for
more general actions. In this section, we propose to study this ques-
tion for closed manifolds under some homological restrictions which
impose P.A. Smith Theoretic conditions on the fixed point sets of
isotropy subgroups. Our approach is to find invariants which characte-
rize the existence of equivariant splittings on the level of normally
linear Poincar& complexes, thus reducting the problem to an equivariant
surgery problem. Since the fixed-point sets of non-trivial subgroups
are, in general, non-simply connected, we will study the problem with
special attention to the fundamental group. The following theorem gives
necessary and sufficient conditions for the existence of equivariant
splittings in the category of normally linear Poincar& complexes, with
semifree actions. The general case is stated separately and its proof
is an elaboration of the arguments for the semifree case.
II.1. THEOREM: Suppose ~ : G x X --> X is a quasisimple semifree
action such that (X,~) is a normally linear finite Poincar& complex
with (X,~) G = F , m(FcX) = 9 , and a (non-equivariant) admissible
splitting X = X I U X2, F n X i = F i . Suppose (I) H,(Xi,Fi;ZqZ) = 0
and (2) yT (Xi,F i) 6 WhT(~ c ~ x G) vanishes. Then there exists a quasi-
simple semifree normally linear finite Poincar& G-complex X' with the
following properties: (a) (X') G = F, v(FcX') = m ; (b) X' has an
67
' X i n F = F. and X' equivariant admissible splitting X' : X~ U X 2 , 1 i
are simple; (c) there exists a normally linear isovariant map
f : X' --> X which induces a q-simple homotopy equivalence; (d) X! 1
and 8X[ are n-simple homotopy equivalent to X i and 9X i rel F i
and 8F i respectively. Furthermore, the hypotheses (1) and (2) above
are necessary for the existence of such X'
PROOF: Since X is normally linear, there exists a Poincar& pair
(C,~C) with a free G-action, such that 8C = S(~) and X = D(v) U C
(after appropriate identifications.) Let C i = C n x i , and let
9_C i = 9C N X i , 8+C i = C N ~X i , ~0Ci = ~+C i N ~_C i . Note that
~0CI = 80C 2 and 9+C I = 8+C 2 ; denote them by ~0 and 9+ respecti-
vely. Thus we have the following diagram
~_c I ~c 1
0 \ ~~ 9c > c
/ ~_c 2 > c 2
DIAGRAM (n)
in which not all maps are equivariant. If X' exists with the desired
properties, we can write X'= C' U D(~) and obtain a diagram (D') in-
volving C' ,C i' and the analoguous boundary decompositions in which all
maps are equivariant. Furthermore, we will get a map of diagrams
(D') --> (D) with the induced map 96 --> 90 , 9~= --> ~± , 8C' --> 9C ,
and C' --> C being the identity or an equivariant H-simple homotopy
equivalence, as it is clear from the context and the requirements
(a) - (d) above.
Let us use an asterisks to denote orbit spaces (e.g. X* = X/G)
and a bar to denote a covering with the deck transformation group G
68
(e.g. 3" = C in the above situation). Thus we look for a diagram
(D'*) of orbit spaces in which the spaces C[* and 2'* as well as 1 +
the dotted arrows are to be determined:
* > c ' * 3_C~ /--1 ~- / \
/ \ / \
/ \ . . . . > ~'+* \\
\ ~../ /~c,. \ > c,. .. ? % /
\ /
5./ / / ~_c;* -> c;* DIAGRAM (D'*)
The left side and the right side faces of the parallelograms in
(D) , (D') and (D'*) are push-outs with respective push-out maps,
and we denote them by (LD) , (RD), (LD') , etc. Moreover, in (D'*)
we have the following equalities up to homotopy: C'* = C* , ~;* = ~; ,
~_C~* = ~ C~ and ~C'* = ~C* and the appropriate maps are induced - !
by the corresponding maps in (D)
In the terminology of [9] Theorem V.I, we wish "to push forward"
the free action from the push out diagram of free G-spaces (LD) to
the corresponding diagram (RD) after possibly replacing (RD) by
homotopy equivalent complexes. Since the constructions of [9] are
sufficiently functorial, they apply to this situation. Briefly, note
that H,(Ci,~+Ci;Zq~ ) = H,(Ci,~_Ci;Zq~) = H,(Xi,Fi;Zq~) = 0 by
Poincar~ duality, excision and hypothesis (I) of the Theorem. Further,
the quasisimplicity condition ensures that the scheme of [9] applies
to construct the appropriate localizations of the diagrams
~_C I > C i
A A
I ~0 > ~+
69
and then take push-outs. The finiteness obstruction as well as the
Whitehead torsion obstruction for choosing C] to be finite and 1
T-simple homotopy equivalent to C , is the image of the Reidemeister 1
torsion T(Xi,Fi) in Wh~(z cz × G) , and it vanishes by hypothesis
(2). It follows from the duality of the Reidemeister torsion [16] that ! the corresponding obstructions for choosing 2+ to be finite and
(equivariantly) z-simple homotopy equivalent to 2+ vanishes as well
(cf. [9] Theorem 1.13). The existence of an equivariant z-simple homo-
topy equivalence (C',~C') --> (C,~C) and T-simple homotopy equiva-
lence of C. and C~ follows from the constructions and the functo- 1 1
riality of push-outs.
The necessity of conditions (I) and (2) of the Theorem for exis-
tence of X' together with the appropriate equivariant splitting
follows as in [9] Section II.
Equivariant splittings of actions with two isotropy types and
semifree actions can be treated in a similar fashion. This observation
allows one to generalize Theorem II.I to actions with several isotropy
types, provided that the fixed point sets of adjacent strata are rela-
ted to each other in the same manner that the stationary point set of
G and the free stratum are related in the semifree case. The condition
of quasisimplicity as in Definition 1.2 ensures that this is the case.
(The hypotheses of the following theorem may be relaxed at the expense
of introducing more complicated notions and longer statements, but we
will not do this). The proof of this theorem uses an inductive argument
similar to II.1 and we will omit it.
II.2. THEOREM: Let (X,}) be a finite G-Poincar@ complex. Suppose
X = X I U X 2 , and denote F. (K)l = XK NX.I . Assume that the splittings
FI(K) UF2(K) = X K are admissible for all isotropy subgroups K ~G
such that H,(Fi(K) , Fi(L) ; Zq~I(Fi(K))) = 0 and yT(Fi(K) ,
Fi(L)) 6WhT(zI(FI(K)) c~1(Fi(K)) × G) vanish. Then there exists a
finite G-Poincar~ complex (X' ,~) with (X',~) G = (X,}) G and an
equivariant admissible splitting X' : X~ U X½ such that (a) there
exists a normally linear isovariant simple homotopy equivalence
f : X' --> X extending the inclusion X 'G = X G c X ; (B) X i and
X' and X i' nx 'K are simple homotopy equivalent to Xi and Xi NX K
repectively and X 'G nx~ = X G NX. l 1
70
SECTION III: A SPECIAL CASE
In the special case where M n is a homotopy sphere, an equiva-
riant splitting is obtained as an application of II.1, or by a direct
argument. This yields another proof for a Theorem of Anderson-
Hambleton [I]. We will mention this special case separately to illus-
trate the theory in a concrete case.
111.1. THEOREM: Let zn be a homotopy sphere, ~ : G × Z n --> zn
a semifree action and F k = EG where v = v(F c E) and dim v > k
Given a splitting F = F I U F 2 , there exists a corresponding equiva-
riant splitting zn = Din U D 2n into disks such that D01 N F = Fi if and
only if H,(Fi;Z q). = 0 and 0(F i) = 0 in ~0(ZG) , where
0(F i) = [ (-I)IOG(H j (Fi)) and CG is the Swan map of Section I. j>0
n En PROOF: Choose X I c to be diffeomorphic to D n and X I N F = F I
and SX I ~ F = ~F I This follows easily from handle body theory and
general positionality, since n > 2k , and we are working non-equiva-
riantly. By Theorem II.1 we have a normally linear finite Poincar6
complex X~ such that (X~,~X~) G = (FI,~F I) , and v(F I cX4) = vlFI ,
' --> Z which extends the and there is an isovariant map fl : Xl
inclusion on D(vlF I) . Since Hi(FI) : 0 for i ~ k - I , it follows
that X~ is obtained from D(v) by adding free G-cells of dimension
at most k . Thus, fl can be deformed into an isovariant embedding
extending the inclusion of D(vlFI) . Let R be an equivariant regular
neighborhood of f1(X~) U D(~) in Z n . Then closure (R- D(~IF2)) is n zn n n
diffeomorphic to D I and is equivariantly split as D I U D 2
n = zn _ int(D~) The necessity of these conditions follows where D 2
easily as in [1] or [2] Section II.
III.2. REMARK: The existence of X~ follows from a direct argument,
by attaching free G-cells of dimension ~ k to S(~IF I) as in [2] II.V ! or [3] Section II. Then the equivariant map fl : Xl --> zn extending
the inclusion D(~IF I) --> E n is a direct consequence of obstruction
theory, and it can be deformed rel D(vIF I) to an isovariant map using
general positionality of F .
71
SECTION IV. SPLITTING UP TO CONCORDANCE:
in this Section we use the existence of equivariant splittings of
Section II to find equivariant splittings of a G-manifold (M,~) based
on a given non-equivariant splitting. This illustrates the construction
of surgery problems from a given normally linear Poincar@ G-complex.
When the appropriate obstructions for the existence of an equiva-
riant splitting in the category of normally linear Poincar@ complexes
vanish, we obtain (X',~') which is isovariantly z-simple homotopy
equivalent to (M,}) . Next, we return to the category of G-manifolds
by smoothing (X',~') equivariantly, while preserving the splitting up
to equivariant homotopy. The result will be (M',~) which is isovari-
antly z-simple homotopy equivalent to (M,~) (relative to an equivari-
ant regular neighborhood of M G = M 'G ). Rather than a detailed analy-
sis of the relevant surgery exact sequence (leading to the surgery
obstructions in order to arrange (M',~) to be G-diffeomorphic to
(M,~) and inherit the desired splitting from (X',}')), we pass to a
restricted concordance in order to get a positive answer. Namely, we
change the action on the free part of (M,~) tel S(m(MG)) to get
(M,9) concordant to (M,~) (rel M G) such that (M,~) is equivariant-
ly split as desired.
If ~i (M) = I , then this change in action is merely taking the
equivariant connected sum of (M,~) and an "almost linear" sphere
(sn,0) . Thus in this case, the G-homeomorphism type of (M,~) is not
changed in order to be equivariantly split. Again we give the proof in
the case of semifree actions and only state the general case.
IV.1. THEOREM: Let # : G x M n --> M n be a quasisimple smooth semi-
free action with (M,~) G = F k, m(F cM) = ~ and a (non-equivariant)
admissible splitting of the closed manifold M = M I UM 2 , M i N F = F i
Assume that (I) H,(Mi,Fi;ZqZ) = 0 , and (2) yT(Mi,F i) 6 Wh~(z cz x G)
vanishes. Then there exists a quasisimple semifree G-action on M ,
say ~ : G × M --> M , such that: (a) (M,~) is concordant to (M,~) !
relative to F ; (b) (M,~) has an equivariant splitting M = M~ U M 2
where M i' are simple, M[1 N F = Fi and M[I and $M~l are q-simple
homotopy equivalent to M i and ~M i respectively. Furthermore, con-
ditions (I) and (2) are necessary for the existence of (M,9)
72
PROOF: Since by Theorem II.1 the conditions (I) and (2) are necessary
for the existence of equivariant splittings in the category of normally
linear Poincar& complexes, (cf. If.l) we need to show only their
sufficiency.
First, we construct the concordance on the level of normally
linear Poincar6 complexes. Thus we have an equivariantly split G-com-
plex X' which satisfies all the stated properties if we replace X
by M in Theorem II.1.
IV.2. PROPOSITION: Under the hypotheses of IV.I, there exists a
normally linear G-Poincar@ pair (Y,~Y) such that: (I) yG = F × [0,1]
and ~(yG cY) = ~ x [0,1] ; (2) ~Y = M U X' where the induced action
on M is ~ and on X' is the action given by Theorem II.1.
PROOF: Let f : X --> M be the isovariant map of II.1, and let Y
be the mapping cylinder of f .
We continue the proof of IV.1 by finding a normal invariant for
(Y,~Y) which restricts to the natural one given on Mc ~y . Using the
normal linearity, let Y = D(~ x [0,1]) U Y' where
~Y' = C U S(~ x [0,1]) U C' using the notation of II.1, and Y' has a
free quasisimple G-action.
Let BG be Stasheff's classifying space for stable spherical
fibrations. As before, we denote the orbit space by an asterisk:
X* mX/G . Let ~ : Y'* ~> BG be the classifying map for the Spivak
spherical fibration of Y'* . Then e I C* lifts to BO since C* is
a manifold. Also this lift extends over S(v x [0,1])* . The obstruc-
tion to extending this to a lift of ~ to BO is an element
X 6 h*(Y'*,C* U S(~ x [0,1])*) , where h* = generalized cohomology theory
of G/O . Since H*(Y',C U S(~ x [0,1]);Zq) = 0 by excision, the Cartan-
Leray spectral sequence for the covering pair (Y',C) --> (¥'*,C*)
collapses and H°(B ,h*(Y',C)) ~ h*(Y'*,C*) . From the hypothesis of
quasisimplicity, it follows that G acts trivially on h*(Y',C)
(cf. [9] II.6 and Le~ma II.10) and h*(Y'*,C*) = h*(Y',C) . Thus X is
q-divisible. On the other hand the transfer tr(X) 6 h*(Y',C) vanishes,
since Y' is (non-equivariantly) homotopy equivalent to C x [0,1]
78
Therefore X = 0 and e lifts to BO
This yields the desired normal invariant, say
f : (wn+I,$w) --> (Y'*,~Y'*) such that SW = C* U S(~ x [0,1])* U V n
and f I C* U S(~ x [0,1])* is the inclusion. The splitting
C'* = C~* U C½* (as given in II°1) induces an equivariant decomposi-
tion V = V I UV 2 , V 1 n V 2 = V 0 = ~V 1 = ~V 2 . Let fi I V i , i = 0,1,2
The surgery obstruction to making fl : (VI,~V I) --> (C{*,~C~*) into
a homotopy equivalence rel S(~ x I)* such that ~I : (VI'~VI) -> (C{,~C~)
is a z-simple homotopy equivalence ,el S(v x 1) vanishes by [22]
.n+1 be this normal cobordism Theorem 3.3 (cf. [9] Theorem II.7) . Let ml
.n+1 to V n÷1 along V I to obtain a new normal map (after and add N I
smoothing corners, etc.). Then f' : W' --> Y'* with
~W' = C* u s(v x [0 I])* UV' v' = , , v~ UV~ , and
f' I V I : (V~,~V~) --> (C~*,8C~*) is a homotopy equivalence
rel S(m × I)* N ~C~ (and the induced map on the G-coverings is z-simple).
Next, we can do surgery on f' ,el C* U S(m × [0,1])* UV~ to make it
into a homotopy equivalence of pairs, applying again Wall's Theorem
([22] Theorem 3.3) since ~1(C½) ~ z1(Y') ~ ~ . Call the new map
f" : W" --> V' where ~W" = C* U S(~ x [0,1])* UV" and V" = V~ U V~
V~' = V~ and f" I V~ is also a homotopy equivalence (and f" : V"--> C'
and f" I ~ are z-simple equivalences). Adding D(~ x [0,1]) back
to W" along S(V x [0,1]) yields the desired concordance. (The reader
can easily verify that W" is an s-cobordism with a free G-action, and
M~ = V~ U D(~ x I I FI) and M~ = V~ U D(~ x 11 F2) yield the equivariant
splitting required by the Theorem).
IV.3.3 THEOREM: Suppose x1(M) = I in IV.I. Then there exists an
almost linear sphere (sn,o) such that the equivariant connected sum
(M,~) = (M,~) # (sn,o) admits an equivariant splitting as in the con-
clusion of Theorem IV.I.
PROOF: Let T 6 Whl(G) be the torsion of the relative h-cobordism W"
with respect to C/G . ChOose X 6 M G and the linear sphere
S(TxM@R) = S n where the tangent space TxM has the linear represen-
tation induced by ~ . Let K n+1 be the concordance S n x [0,1] ob-
tained by adding free 2-handles and 3-handles to the free stratum of
S n so that the resulting G-h-cobordism has torsion -T . The new
74
equivariant concordance M x [0,1] # S n x [0,1] (where the connected
sum is along an arc {X} x [0,1] in the stationary point sets) is
actually an equivariant s-cobordism, and hence a product. But
$(M x [0,1] # S n x [0,1] with the induced action is G-diffeomorphic to
(M,~) U (M,% # o) where ~ is the "alomost linear" action induced on
S n x {I] in the concordance S n x [0,1] (See [5]).
IV.4. COROLLARY: Given (M,~) as in IV.I, and so that ~I(M) = I
there exists a smooth action ~ : G x M --> M such that (M,~) has
an equivariant splitting as in the conclusion of Theorem IV.I, and
(M,#) is G-homeomorphic to (M,~)
If M n is a homotopy sphere, then we get an equivariant decom-
position into disks, thus generalizing the Anderson-Hambleton Theorem
[1] Theorem A. Note that the methods of [I] which are based on general
position arguments do not apply here, since codimensions could be quite
small.
~n IV.5. THEOREM: Let } : G x E n --> be a semifree action with
(En,~)G = F and v(F c E) = v , dim 9 > 2 . Assume that En = Din U D 2n
with Dn N F = F is a non-equivariant splitting. Then there exists l l
a smooth semifree G-sphere (E,9) such that (Z,~) is G-homeomorphic
to (Z,9) , (E,~)G = F , ~ I G x ~ : } i G x ~ , and (Z,9) has an equi-
variant splitting into disks Z = D n U D n with (Dn) G = F~ if and 1
only if j>0Z (-1)J °G (Hj (F i) = 0 in K0(ZG) , i = I or 2 .
IV.6. REMARK: Suppose F k as a mod q homology sphere. Then Anderson
and Hambleton prove that the necessary and sufficient conditions for
existence of a "balanced splitting" of F k (i.e. F is homologically
a double) is that a certain semicharacteristic type invariant vanishes
(cf. [I] Theorem B). Thus Theorem IV.5 can be applied to generalize
this result of Anderson-Hambleton and improve their dimension hypothe-
sis in Theorem B of [I] from dim v ~ k + 2 to dim ~ > 2 .
As in Section II, we can generalize the above results to actions
with many isotropy subgroups. The proofs of the semifree cases can be
75
adapted to serve as the inductive step of the following theorem. The
normally linear Poincar~ G-complex which is the homotopy model in this
case is provided by Theorem II.2. We omit the details.
IV.7. THEOREM: Let (xn,¢) be a smooth closed G-manifold with an
admissible splitting X = X I U X 2 , satisfying all the hypotheses of
Theorem II.2. Then there exists a smooth G-action ~ : G × X --> X such
that (X,~) is concordant to (X,¢) rel X G and (X,~) has an equi-
variant splitting X = X~ U X~ which satisfies the conclusions (a) and
(b) of Theorem Ii.2.
SECTION V. REALIZATION OF OBSTRUCTIONS:
One may use normally linear Poincar& complexes to construct actions
with admissible splittings which do not admit necessarily equivariant
splittings. Again, the results of this section may be specialized to
the situation considered by Anderson-Hambleton [I] to give an alterna-
tive proof of their Theorem C. The important algebraic calculations of
the hyperbolic map in the Rothenberg-Ranicki exact sequence for the
quaternionic groups are due to Anderson-Hambleton ([I] Proposition 5.2
and [13] Lemma 6.1) who applied it in their examples of actions on
spheres without balanced splittings. These calculations are used to take
care of the case where the 2-Sylow subgroup is the quaternion group of
order 8 , denoted by Q8 "
V.I. THEOREM: Let M n be a simply-connected closed manifold, and
M n = M1n UM 2n be an admissible splitting. Suppose that F k cM is a
closed submanifold with normal bundle ~ which admits a G-bundle
structure with a free representation on each fibre, where G is a sub-
group of SU(2) whose 2-Sylow subgroup is either (i) cyclic or (ii) , . n + l k + l
Q8 and K ~ 1 mod 4 . Assume that ~M 0 ,F 0 ) is a manifold pair
such that ~(M0,F0) = (M,F), F i = M i n F satisfying the hypotheses:
(1) for i = 1,2 z1(Mi) = I and H,(Mi,Fi;Z q) = 0, where i = 0,1,2, .
(2) The G-bundle structure of ~ extends to the normal bundle of
F 0 in M 0 . Then there exists a quasisimple semifree action
¢ : G × M' --> M' such that M 'G = F, where M' is homotopy equiva-
lent to M. Further, (M',¢) has an equivalent splitting
M I' U M 2' , M~I A F = F.l if and only if Z(-I)JoGH(MI,FI) = 0 in K0(ZG).
76
The idea of the proof of this theorem is the following. Using the
hypotheses (I) and (2) in this context, we construct a normally linear
Poincar~ pair (X,~X) with semifree G-action such that X G = F 0 and
(X,~X) ~ (M0,3M0). This pair is not necessarily finite, however, one
shows that the finiteness obstruction for the boundary vanishes, so
that 3X is a finite Poincar~ G-complex. Then a surgery problem is
set us as in [9] and in the spirit of section IV of the present paper.
To realize the obstructions for equivariant splittings, one may choose
(M0,F 0) such that for any choice of an admissible splitting, the
cohomology class in. ~(Z2;K0(ZG) represented by the finiteness A ~
obstruction Z(-I)3OG(Hj (MI,FI)) be non-zero. E.g. when G = Q8 K0(ZQs) ~ Z2, and there are such pairs (M,F) with non-zero obstructions.
One instance of this is Anderson-Hambleton's example using
thickerings of Moore spaces with appropriate homology. The crucial
algebraic fact is that this non-zero element contributes non-trivally
only to the surgery obstructions which arise in the process of
equivariant splittings i.e. (MI,FI). This contribution is zero when
the surgery problem is considered over all of ~X. This is reflected
in the algebraic calculations of Anderson-Hambleton [I] of the hyper-
bolic map in the Ranicki-Rothenberg exact sequence. In fact, the
approach of constructing the normally linear Poincar~ model of this
problem simplifies and shortens only the geometric part of the proof
of Theorem C of [I]. The more delicate algebraic computations are
already treated in [I], and we use them almost in the same way as in
L1] (only at the last stage to complete the surgery and produce M'
which is G-homotopy equivalent to 3X tel F.) We comment that the
cobounding surgery problem (X,F) is only auxiliary and simplifies
the study of the surgery obstruction on 3X.
This theorem may be generalized to actions with several
isotropy subgroups. The full proof of this theorem and further
applications of normally linear Poincar~ complexes will appear
elsewhere.
77
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
D. ~ A n d e r s o n a n d I . H a m b l e t o n : "Balanced splittings of semi- free actions of finite groups on homotopy spheres", Com. Math. Helv. 55 (1980) 130-158.
A. Assadi: "Finite Group Actions on Simply-connected Manifolds and CW Complexes", Memoirs AMS, No. 257 (]982).
A. Assadi: "Extensions of finite group actions from submani- folds of a disk", Proc. of London Top. Conf. (Current Trends in Algebraic Topology) ~MS (1982)o
A. Assadi: "Extensions Libres des Actions des Groupes Finis dans les Vari&t&s Simplement Connexes", (Proc. Aarhus Top. Conf. Aug. ]982) Springer-Verlag LNM 105].
A. Assadi: "Concordance of group actions on spheres", Proc. AMS Conf. Transformation Groups, Boulder, Colorado (June 1983) Editor, R. Schultz, AMS Pub. (1985).
A. Assadi a~d W. Browder: "On the existence and classification of extensions of actions of finite groups on submanifolds of disks and spheres", (to appear in Trans. AMS).
A. Assadi and W. Browder: "Construction of free finite group Actions on Simply-connected Bounded Manifolds", (in preparation).
A. Assadi and 9. Burghelea:"Remarks on transfer in Whitehead- Theory",Max-Planck-Institut, Preprint 86-6 (1986).
A. A@sadi and P. Vog£1: "Finite group actions on compact mani- folds", (Preprint). A shorter version has been published in Proceedings of Rutgers conference on surgery and L-Theory, 7983, Springer-Verlag LNM 1126 (1985).
A.K. Bousfield and D.M. Kan: Springer-Verlag LNM, No. 304 (1972).
F.Con~olly: (Talk in Oberwolfach meeting in Transformation groups, August 1982).
A. FrShlich, M. K¢ating and S. Wilson: "The class groups of quaternion and dihedral 2-groups", Mathematika 21 (1974) 64-71.
I. Hambleton and I. Milgram: "The surgery obstruction groups of finite 2-groups", Inv. Math. 61 (1980) 33-52.
H. Hauschild: "Aquivariante Whitehead torsion", Manus. Math. 26 (1978) 63-82.
S. Illman: "Whitehead torsion and group actions", Ann. Acad. Sci. Fenn. Ser. AI. 588 (1974) 1-44.
I. M i l n o r : "Whitehead torsion", Bull. AMS. 72 (1966) 358-426.
R. Oliver: "Fixed-point sets of group actions on finite acyclic complexes", Comm. Math. Helv. 50 (1975) 155-177.
78
[18]
[19]
[20]
[21]
[22]
[23]
[24]
M. Rothenb£rg: "Torsion invariants and finite transformation groups", Proc. Symp. Put Math. vo!. 32, Part I, AMS. (1978).
A. Ranicki: "Algebraic L-theory I: Foundations", Proc. Lon. Math. Soc. (3) 27 (1973) 101-125.
R. Schultz, Editor, Proceeding of AMS summer conference in transformation groups, Boulder Colorado 1982, AMS. R.I. (1985).
R. Swan: "Periodic resolutions and projective modules", Ann. Math. 72 (1960) 552-578.
C.T.C. Wall: "Surgery on compact manifolds", Academic Press, New York 1970.
S. Weinberger: "Homologically trivial actions i" and "II", (preprint), Princeton University (7983).
S. Weinberger: "Constructions of group actions", Proceedings of AMS summer conference in transformation groups, Boulder Colorado 1982, AMS. R.I. (1985).
FREE (2Z/2)k-AcTIONS AND A PROBLEM
IN COMMUTATIVE ALGEBRA
Gunnar Carlsson
(I) INTRODUCTION. In ~1,2J, the following theorem is proved.
THEOREM I.I. Suppose G = ~/p~)k acts freely on a finite complex
X , where X is homotopy equivalent to (sn) ~ , and suppose that G
acts trivially on n-dimensional mod-p homology• Then ~ ~ k .
The analogous theorem for G = ($1) k is proved in E6J. In fact,
for this case, the theorem is proved for S nl x...x S n~ , where the
ni's may be distinct. The proofs of these theorems rely heavily on
the special homological properties of the spaces involved, in parti-
cular on the non-vanlshing of cup-products in H (X; ~/p ~) or .
H (X;Q) . One's initial reaction is to attempt to remove the hypo-
thesis of trivial action on homology in Theorem 1.1, to extend the n I n~
result to S x...x S . However, in attempting this, one is still
utilizing the special properties of the spaces involved; a more
appealing approach is to try to find a priori homological properties
which__ must be satisfied by spaces which admit free (2Z/p ~)k or
(s1)k-actlons, and which apply in a wide family of examples.
Such general properties are hard to come by; an example is:
THEOREM 1.2 C3]. Let X be a finite free G-complex, G = ~Z/2~)k
or ($1) k , and suppose G acts trivially on H (X; ~/2 ~) , if
= ~Z/2~)k . Then X has at least k non-trlvial homology groups. G
We now propose as a conjecture the following much more striking
a priori restriction.
I.~. Suppose G = ~Z/p~) k or ($1) k , and suppose CONJECTURE X
• ~ rk~/p~ Hi(X,~/p ~) or is a finite free G-complex Then
2 k ~ rkGHi(X,Q) , respectively, is
REMARK: The rational version of this conjecture has also been pro-
posed by S. Halperin.
. The author is an Alfred P. Sloan Fellow, and is supported in part by N.S.F. Grant 82-01125.
80
So far, this conjecture can be proved for ~/2~ )k and ($I) k, with
k~3 (see ~4], where the case of (2g/2 ~ )k is handled. The proof
for ($1) k is entirely similar.) The case k=4 can probably also be
carried through with these techniques.
In this paper, we'll formulate the algebraic analogue of the con-
jecture for G = ~Z/2~ )k , and prove its equivalence with a question
concerning differential graded modules over polynomial rings.
We'll also briefly discuss its relationship with commutative al-
gebraic conjectures of Horrocks, related to the study of algebraic
vector bundles on projective spaces.
The author wishes to thank L- Avramov, S. Halperin, and J.E. Roos
for stimulating discussions concerning this subject.
(II) THE ALGEBRAIC FORMULATION. We consider ~2CG~ , G = ~Z/2~ )k,
and let A k = ~2~G~ . As an algebra, A k is isomorphic to the ex-
terior algebra E(y I .... 'Yk ) ' Yi = Ti+1 ' where {TI, .... Tkl is a
basis for ~Z/2 ~ )k . We view A k as a graded ring by assigning the
grading 0 to all elements of A k •
Let A. be a graded ring.
DEFINITION 11.1.
graded A.-module
of degree (-I)
A DG (Differential Graded) A.-module is a free,
M with a graded A.-module homomorphism d : M * M
so that d 2 = 0
A DG A.-module is said to be finitely generated, bounded above,
or bounded below if its underlying graded module is. The homology of
M , H.M is defined in the usual way; H,M is itself a graded A.-
module. The notions of homomorphism, chain homotopies, and chain
equivalences of DG A.-modules are the evident ones.
Now, for a graded ring A. , we let ~(A.) denote the category
of finitely generated DG A.-modules, and if A. is bounded above,
we let ~(A.) denote the category of bounded above DG A.-modules.
~(A.) is of course a subcategory of ~(A.)
The algebraic formulation of our Conjecture 1.3 is the following.
CONJECTURE II.2. Let M E ~(Ak) . Then rk~2H.M ~_ 2 k
We observe that Conjecture II.2 implies Conjecture 1.3. For if
~ X X is any finite G-complex, then the cellular chains C.( ;~F 2) are
a finitely generated chain complex of free ~2~G~ = Ak-mOdules,
which is the same as an object of ~(A k) , and H (X;~)=H.(~.(X;~2) ) * 2
81
Suppose that the ring A. is an augmented algebra over a field
k , so that k is a module over A.. If M E ~ (M) , we denote by
H.(M,k) the homology of the DG k-module k ®A M . A k is of course
an augmented ring over ~2 via the augmentation A k ~ F 2, T i - I
Now, let Pk denote the polynomial ring ~2[Xl,...,Xk] , which we
grade by assigning each variable the grading (-I) Pk is also
augmented over ~2 ; the augmentation is determined by the require-
ment that x i - 0 for all i . Recall from C3] that there is a
functor ~ : ~ (Ak) ~ ~(Pk ) defined as follows. For a DG Ak-mOdule
(M,b) the underlying module of ~(M,~) is M ®F2 Pk ~ and the
differential 6 on B(M) is defined by 6(m®f)=~m@f+i~lYim= ®xif •
The Pk action is on the right hand factor. We also have
PROPOSITION II.~. [3; Propositions II.1 and II.2]. There are natu-
ral isomorphisms H.M ~ H.(~M;~ 2) and H.(M;~ 2) - H.~M .
An immediate consequence is
COROLLARY II.4. [3; Corollary II.3]. For any M E ob~(Ak) , H.BM
is finitely generated as an F2-vector space.
For any graded ring A. , bounded above, we let h~(A.) and
h~(A.) denote the "homotopy categories" of ~(A.) and ~(A.)
These are obtained from ~(A.) and ~(A.) by inverting all chain
equivalences. Let ~°(Pk) and ~(Pk ) denote the full subcategorles
of ~(Pk ) and ~ (Pk) , respectively, whose objects are the DG-P k-
modules (M,b) for which H.M is a finitedimensional F2-vector
space. We also let h~°(Pk ) and h~(Pk) denote the corresponding
homotopy categories. Finally, let ~(A k) denote the full subcate-
gory of ~ (A k) whose objects are chain requivalent to objects in
~(A k) and let h ~(Ak) denote the corresponding homotopy category.
Let h~ : h~(A k) ~ h~ (Pk) be the induced map on homotopy categories.
Then Corollary II.4 shows that h~ factors through h~°(Pk). More-
over, it is easy to check that it extends to a functor H :h~(A~-h~.
DEFINITION II.5. Let (M,~) E ob ~(A.) , where A. = 0 for
. > 0 , and where A. is augmented over a field k . We say that
(M,~) is minimal if the map ~ ® id : M ® A k ~ M ®A k is the zero
map.
PROPOSITION 11.6. For every (M,b) E ob(~(A.)) , there exists
(M,~) E ob(~ (A.)) , where (M,~) is minimal and is chain equivalent
to (M,~)
82
PROOF. This is Proposition 1.7 of [4].
We now prove our main theorem.
THEOREM II. 7.
gories.
H : h~(Ak) - h~(Pk) is an equivalence of care-
PROOF. We first construct a functor G :~°(P k) ~°(A k) as
follows. Given a DG Pk-mOdule (M,~) , the underlying module is of
G(M,~) is M ®F2 A k , and the differential 5 on G(M,~) is de-
k fined by 5(m ® ~) = ~m ® ~ + ~ xim ® yi ~ . One proves, by argu-
i=I
ments identical to those in the proofs of Propositions II.1 and II.2,
that H.(G(M)) = H.(M,I~2) and H.M = H.(G(M);I~2) . To see that
G(M) £ obo~°(Ak) , we note that since M £ ~ o ~(Pk ), dim~H.M < +~.
Therefore dim~2H.(G(M);F 2) < +~. Let ~ be any minimal DG A k-
module, chain equivalent to G(M) . Then ~ ®Akl~2 ~ H.(G-~-M~; =
H.(G(M);I~2) so ~ is finitely generated which was to be shown.
We now construct a natural transformation N : G oH ~ Id as follows.
The underlying module of G oH(M) is M ®F2 Pk ®IF 2 Ak ; let
¢ : Pk " F2 be the augmentation, and let ~ : M ® A k - M be the
structure map for M as a Ak-mOdule. Then we define N(M) to be
id ® ¢ ® id the map M ®F2 Pk ®~2 Ak .... ~ M ®F2 A k ~-M ; it is easily
checked to be a chain map, and a chain equivalence. Similarly, we
define N'(M) : H oG(M) ~ M to be the composite M ®i~ 2 A k @F2 Pk
M ®F2 Pk - M . This is also easily checked to be a chain equivalence,
which proves the theorem.
This equivalence of categories leads us to propose the following:
CONJECTURE 11.8. Let M E ob~D°(Pk ) • Then rkpkM > 2 k .
Finally, we prove
PROPOSITION II.~. Conjecture II.8 is equivalent to Conjecture II.2.
PROOF. By Theorem II.7 and Proposition II.3, Conjecture II.2 is
equivalent to the conjecture that for all M E ob~°(Pk) ,
rk~2H.(M;F 2) ~ 2 k . But Proposition II.6 shows that M is equivalent
to a minimal DG Pk-mOdule M " rkPkM > -- rkPk~= rk~2~ ®Pk F2 =
rk~2H*(~;~ 2) ~ 2 k .
83
(III) THE RELATION WITH HORROCKS' CONJECTURE. G. Horrocks' has
conjectured the following (see [5] for discussion of related materiaL)
CONJECTURE III.1. Let M be an Artinian graded module over the
polynomial ring R = F[Xl,...,x k] , where F is a field. Then
rkFTOrRi(M,F) >_ (k) .
We may weaken this slightly to
CONJECTURE III.2. Let M be an Artinian graded module over the
polynomial ring R = F[Xl,...,x k] , where F is a field. Then
~ rkFTOr~(M,F)- ~ 2 k . i
The relationship between our conjectures and this one is now
given by the following.
PROPOSITION III.~. Conjecture II.8 implies Conjecture IIi.2 for
F=F 2
PROOF. Let M be any Artinian graded module over Pk ' and let
R(M) denote a minimal graded resolution of M . Then Pk
rkPkR(M) = ~ rk~2Tor i (M;F2) , and R(M) may certainly be viewed
as an object of ~(Pk ) . Since H.R(M) ~ M , and M is ~2-finite
dimensional (since it is Artinian), R(M) is in fact an object of
°(P k) . Thus, if Conjecture II.8 holds, then
i rk~2Tor~k(M,~2) = rkPkR(M) >- 2k
REFERENCES
[I] CARLSSON, G.: On the non-existence of free actions of elementary abelian groups on products of spheres, Am. Journal of Math., 102, No. 6, (1980), pp. 1147-1157.
[23 CARLSSON, G.: On the rank of abelian groups acting freely on
(sn) k, Inventiones Math., 69, (1982), pp. 393-400.
[33 CARLSSON, G.: On the homology of finite free @Z/2~-complexes, Inventiones Math., 74, (1983), pp. 139-147.
[4] CARLSSON, G.: Free (~/2 ~-actions on finite complexes, to appear, Proceedings of a Conference in honor of John Moore.
[5] HARTSHORNE, R.: Algebraic vector bundles on projective spaces: a problem list. Topology, 18 (1979), pp. 117-128.
[6] HSIANG, W. y.: Cohomology Theory of Topological Transformation Groups, Springer Verlag, 1975.
Department of Mathematics, University of California, San Diego
La Jolla, CA 92093
V e r s c h l i n K u n g s z a h l e n yon
Fixpunktmen~;~en in D a r s t e l lurlssformen-
I3[
Tammo tom Dieck und Peter L~ffler
Abstract: Let G = H 0 X H I be a product of two cyclic groups of odd order.Let
ji:S n(i)---~ S n(0)+n(|)+l , i=0,1, be any two imbeddings of standard spheres
into the standard sphere. Suppose
a)The integers n(0) and n(1) are both odd and greater or equal to 5.
b)The normal bundles ~i of the imbeddings Ji,i=0,1, are both trivial.
c)The linking number k of J0(S n(0)) and JI(S n(1)) is a unit in Z/ IG I and
lies in the kernel of the Swan homomorphism s G : ZlIGI* .... , K(ZG).
Then there is a smooth action of G on X = S n(0)+n(1)+l such that
1)the isotropy groups are l, H0, HI,
H. 2)the fixed point sets X i are the spheres ji(sn(1)), i=0,1.
Ziel dieser Note ist es, den folgenden Satz zu beweisen:
Satz I: Sei G = H 0 X H I ein Produkt yon zwei zyklischen Gruppen ungerader
Ordnung. Fur im0,l seien Ji:Sn(i)---~ S n(0)+n(|)+l disjunkte Einbettungen yon
Standardsph~ren in die Standardsph~re. Es gelte
a)Die Zahlen n(0) und n(1) sind beide ungerade und grSBer oder gleich f~nf.
b)Die Normalenb~ndel ~i der Einbettungen Ji' i=0,1, sind trivial.
e)Die Verschlingungszahl k yon J0(S n(0)) und von jI(S n(1)) in sn(0)+n( | )+I
sei eine Einheit in Z/IG[ und liege im Kern des Swan Homomorphismus' s G :
ZIIGI* ~ K(ZG).
Dann gibt es eine glatte Operation yon G auf X S n(0)+n(|)+l = ~ so dab
i) die Isotropiegruppen I, H0, H I sind,
H i 2) die Fixpunktmengen X die Sph~ren Ji(S n(i)) sind, i=0,1.
Dieser Satz verallgemeinert den Hauptsatz aus [tDL], wo die Existenz einer
85
solchen Verschlingunskonfiguration der Fixpunktmengen bewiesen worden war.
In [Le] wird allerdings gezeigt, dab man mehrere solcher Konfigurationen bei
festem n(O) und n(1) vorgeben kann. Der hier angegebene Beweis differiert
auch erheblich yon dem aus [tDL] und kann vermutlich - mit gewissen
Einschr~nkungen - auf einfach zusammenh~ngende rationale Homologiesph~ren
mit komplexen Strukturen auf den NormalenbUndeln der E~nbettungen erweitert
werden.( F~r die EinschrEnkungen vergleiche etwa [Sch] ) Die hier benutzten
Methode der Erweiterung yon Gruppenoperationen, die auf dem Rand einer
Mannigfaltigkeit vorgegeben sind, haben sich schon an anderer Stelle
bew~hrt.
Wir setzen n = n(O)+n(1)+l. Ohne Beschr~nkung der Allgemeinheit d~rfen wir
n(1) ) n(O) ) 5 voraussetzen. Bekanntlich gilt dann, dab Jo:S n(O) S n his
auf Isotopie die Standardeinbettung ist([Le]).
Nach Voraussetzung b) k~nnen die Einbettungen Ji zu disjunkten Einbettungen
~i : S n(i) X B n-n(i)''~ S n
verdickt werden.
Wit setzen
X = S n - ~o(S n(O) X ~n-n(O)) _ 71(sn(l) X ~n-n(ll).
X ist eine Mannigfaltigkeit mit Rand 5X.
Es gilt 8X = BoX V 51X
mit 5iX ~ S n(0) X S n(1) , i=O,l,
wobei die Diffeomorphismen dutch die 3i induziert werden.
Man errechnet
Hi(X'51X) ~ { Z/ko sonst.i = n(1)
Bekanntlich gibt es Inkluslonen
io: S n(O) ~ S n _ JI(S n(1))
und
if: S n(1) ~ S n _ J0(sn(O)),
die HomotopieEquivalenzen sind [M].Seien
$o:S n - Ji(Sn(1)) ---~ sn(O)
¢i:S n - jO(S n(O)) ~ S n(1)
Homotopieinverse zu diesen Inklusionen.
Wir betrachten nun das folgende Diagramm
86
50X , X , ~l x
- 1(sn(1)) sn- i (sn(0)) (~) sn ~ i~O ~I ~ 0
sn(O) sn(1)
wobei i i = k i o ~i ' i = 0,I , gesetzt ist
und k i die offensichtliche Inklusion ist. Wir definieren schlieBlich
~': X ~ S n(0) X S n(1)
dutch a" = (#ooko)X(~iOkl).
Ist r E ~, so bezeichne [r]: S a , S a elne Abbildung vom Grad r.
Lemma I: Wit haben ein homotopiekommutatives Diagramm
80X ,X : SIX
II I II S n(O) X S n(1) ~" S n(O) X S n(1)
[k] )< [i] ~ ! //~i] X [k]
S n(0) X S n(1)
Beweis: Dies folgt leicht aus dem Diagramm (*) und der Definition und den
Eigenschaften der Verschlingungszahl.
Wir benotigen nun den folgenden Satz:
Satz 2: Sei G eine endliche Gruppe der Ordnung g. Sei X n eine kompakte
Mannigfa]tigkeit, n ~ 6, mit 8X = ~0 x Q ~i x und 50X ~ $i x = ~. Es gelte
l) ~(X) = ~i(50 x) = ~i(51 X) = 0 i = 0,1.
2) H.(X,~0X) ® Z(g) = 0
3) G operiere frei auf 50X und die auf H,(50X) @ Zig -|] induzierte
Operation sei trivial.
4) Es bezeichne h i d~e Ordnung yon Hi(X,SoX). Setze
P(X,SoX) = ~ h2i/~ h2i+l . Wegen 2) definiert P(X,SoX) dutch Reduktion ein
87
Element in (Z/g)*- Es sei SG(P(X,8oX)) = 0 (s G = Swan Homomorphismus yon
G).
Unter dlesen Voraussetzungen gibt es auf X eine freie G-Operation, die die
auf ~0 x gegebene erweitert. Diese induziert auf H,(X) ~ ~[g-l] wieder die
triviale G-Operation.
Dieser Satz wurde yon mehreren Autoren unabh~ngig voneinander bewiesen [AB],
[W], um nut zwei Quellen zu nennen.
Der Satz wurde so zitiert, dab er genau auf unseren Sachverhalt paBt. Jede
G-Operation auf SoX (die der Bedingung 3) genUgt) kann auf X erweitert
werden und induziert eine auf 51X.
Wir w~hlen nun fur i = O, | freie Hi-Darstellungen V i mit
dim ~ V i = n(i)+l. Seien S(Vi) die zugehSrigen Einheitssph~ren mit
induzierter freier G-Operation. Wie in [tDL] 2.2 zeigt man, dab die
Normalenabbildung
k.id : k,S(Vi) ~ S(Vi) vom Grad k dutch Umhenkeln
(n(i)-l)-zusammenh~ngend gemacht werden kann. So erh~It man eine Sphere
~(k,Vi) mit freier G-Operation. Wit nennen ~(k,Vi) (mit der Grad-k
Normalenabbildung) ein k-laches yon S(Vi). Sicher ist E(k,Vi) nicht
eindeutig. Aber da L~(G) verschwindet [B], werden je zwei Vertreter yon
~(k,V O) X S(V I) bzw. S(Vo) X E(k,VI) h-kobordant. Nun gilt:
Satz 3: Versehen wir ~0 X mit der G-Operation E(k,V0) X S(VI) , und versehen
wir X mit der durch Satz 2 garantierten freien G-Operation, so wird auf ~IX
gerade S(Vo) X ~(k,V I) induziert.
Folgerung: Satz Iist richtig.
Beweis: Man betrachte
E(k,V 0) X B(V i) V X V B(V 0) × E(k,Vl)-
DaB man naeh Vergessen der G-Operation w~eder das Objekt erh~It, mit dem man
anfing, liegt daran, dab der Diffeomorphlsmus
~|X ~ S(Vo) X ~(k,V I )nach Vergessen der G-Operation die Identit~t ist
( vergleiche den Bewels nach Lemma 3 ).
Es bleibt Satz 3 zu zelgen:
88
Dazu versehen wir X mit der durch Satz 2 garantierten freien G-Operation.
Man betrachte nun dam folgende Diagramm:
80 X = ~ ( k , V O) X S(V 1) ~ x , 8i X i o
s(v 0) X s(v I )
Lemma 2: Die G-~quivariante Normalenabbildung 5 0 kann (bis auf Homotopie)
eindeutig zu einer G-~quivariante Normalenabbildung ~ erweitert werden.
Beweis:a) Existenz einer G-Abbildung.
Wegen Lemma I gibt es ~', eine nicht ~quivariante Abbildung. Invertieren wit
die Gruppenordnung g, so hat man ein homotopiekommutatives Diagramm
(5oX/G)(I/g) '(X/G)(I/g )
12 II
180Xl(l/g)X BG , I X l ( l / g ) X BG
(18oXI bzw IXI bezeichnet den Raum 80X bzw X mit trivialer G-Operation) und
wir setzen ~[I/g] = ~'[I/g] X idBG- (F~r die Bezeichnungen und den
Hintergrund uber Lokalisierungen vergleiche man etwa [ELP]). Lokalisieren
wit an der Gruppenordnung, so wlrd i0(g ) eine Homotopie~quivalenz. Wir
setzen a(g) = ~(0)o10(g). Die Abbildungen all/g] und a(g) passen rational
zusammen und definieren ~.
b)Behauptung: Ist T(X/G) das Tangentialb~ndel yon X/G, so gilt als Gleichung
in KO(X/G)
(V 0 • V I) X G X ~ T(X/G).
Beweis: B e t r a c h t e h i e r z u
89
KO(X, BoX)
"t KO(X/G, BoX/G)
, KO(X)
"1 ~2
, KO(X/G)
, KO(50X) "l ~3
.... , KO(BoX/G)
(hierbei mind die N durch Projektionen induziert).
Aum der Atiyah-Hirzebruch Spektralfolge ergibt sich:
I) NI ist ein Isomorphismus;
2) ~2 ~ Z[I/g] ist ein Isomorphismus.
Setze a = T(X/G) - (VoeVl)XG x .
Man hat nun
~2(A) = T(X) = 0
sowie i0(A) = 0,
weil die geforderte Gleichheit micher ~ber 50X/G gilt. Wegen 2) ist demhalb
nut g-Torsion. Wegen I) und der Struktur yon H,(X,~0X) besteht
KO(X/G,~0X/G) abet nut aus k-Torsion.
c)Behauptung: Die Erweiterung g yon g0 kann eindeutig (bis auf Homotopie)
als G-~quivariante Normalenabbildung gewEhlt werden.
Beweis: W~hle einen Isomorphismus
¢: T(X/G) ~ g N ~ g (T(S(V 0) X S(VI))/G) • EN+I
den es wegen b) gibt. Man betrachte
KO -I(X,~OX) ~ KO -I(X) p KO -I(~0 X) ,KO(X,5oX)
I I I T ~I ~2 ~3 ~4
KO -I(X/G,~0X/G ) --~ KO-I(X/G) ~ KO-I(50X/G) --* KO(X/G,~oX/G )
Wir mUmsen zeigen, dab es einen Automorphismum ~ des stabilen Bffndelm T(X/G)
gibt, der die folgenden Eigenschaften hat:
i) ~o~IB0 X ist die gegebene Normalenabbildung.
2) ~ ist eindeutig bestimmt.
Nun entsprechen stabile Automorphismen yon T(X/G) gerade KO-I(x/G).
Schr~nken wit ~ auf BoX/G ein, so gibt es einen Automorphismus ~I ~ber BoX/G
mit den geforderten Eigenschaften. Eine Diagramm~agd zeigt, dab man ein
mit den geforderten Eigenschaften linden kann. Torsionsbetrachtungen wie
unter b) zeigen die Eindeutigkeit.
90
Bemerkung:Eigentlich ist Lemma 2 ein Tell eines ausf~hrlichen Beweises yon
Satz 2.
Wie in [.M]
Normaleninvarianten vom Grad k.
Element.
Seien Wi, i = 0,I, freie
Normalenabbildung vom Grad k
E(k,Vi) , S(Wi) vom Grad
Beweis yon [tDL] 2.2).
Man betrachte jetzt
bezeichne Nk(((S(V0) X S(VI))/G) die Menge der
Wir w~hlen (80X/G,~0) als ausgezeichnetes
Hi-Darstellungen , so dab die gegebenen
~(k,Vi) J S(Vi)zu Normalenabbildungen
1 hochgehoben werden kann (vergleiche den
NI(((S(W 0) X S(VI))/G) ' Nk(((S(V O) X S(VI))/G) ' [k] x [i]
' NI(((S(V 0) X S(WI))/G)- [i] x [k]
Es definiert (50X , C0) das ausgezeichnete Element auf der linken Seite und
in der Mitte. Offenbar besagt Lemma 2
[ 50X, 60] = [ 51X, ~i ] 6 N k-
Andererseits gilt (siehe [tDL] 2.3)
[~(k,V O) X S(VI) ] = [S(V O) X E(k,VI)] 6 N k-
Lemma 3: Die ~quivariante Normalenabbildung
a I : 81 x , S(V O) X S(V I)
vom Grad k kann zu einer ~quivariante Normalenabbildung
~I : 51X ~ S(V O) X S(W I)
vom Grad i hochgehoben werden.
Beweis: Mit Lokalisierungen beweist man dies analog zu Teil a) aus Lemma 2.
Damit haben wir die Gleichheit
([I 3 X [k])[51X/G,~I] = ([I] X [k])[(S(V O) X ~(k,VI))/G] E N k
91
Aus IBM] Proposition 4.6 folgt) dab der Kern yon [I] X [k] nut aus k-Torsion
besteht. Beachten wir andererseits) dab die Projektion
: S n(O) X S n(1) ) (S(V O) X S(Vl))/G
einen k-lokalen Isomorphismus
~(k):[(S(Vo) X S(VI))/G, OS0/Cat](k ) ) IS n(O) X S n(1) ) QsO/cat](k)
induziert (gist jetzt invertierbar) und folgern daraus
~(k):(51X/G' ~I ) = ~(k) ((S(Vo) X ~(k,VI))/G))
so ergibt sich) dab die Normaleninvarianten von (51X) ~i ) und
(S(V 0) X ~(k,VI))/G ubereinstimmen m~ssen. Da L~(G) verschwindet [B], mussen
beide h-kobordant sein.
Literatur
[AB] A. Assadi-W. Browder: In preparation.
[B] A. Bak: Odd dimension surgery groups of odd torsion groups
vanish. Topology 14(1975)) 367-374.
IBM] G. Brumfiel-I. Madsen: Evaluation of the transfer
and the universal surgery class. Inv. math. 32(1976), 133-169.
[tDL] T. tom Dieck-P. L~ffler: Verschlingungen yon Fixpunktmengen
in Darstellungsformen. I, Math. Gottingensis 1 (1985) und Alg.
Top. G~tt. 1984, Proc. LNM 1172(1985), 167 - 187.
[ELP] J. Ewing-P.L~ffler-E. Pedersen: A Local Approach to the
Finiteness Obstruction) Math. Gott. 40(1985).
[HM] I. Hambleton-I. Madsen: Local surgery obstructions and
space forms, preprint 1984.
[Le] J. Levine: A classification of differentiable knots. Ann. of
Math. 82(1965), 15-50.
[M] W. Massey: On the normal bundle of a sphere imbedded in
Euclidean space. Proc. AMS I0,(1959)) 959-964.
[Sch] R. Schultz: Differentiability and the P. A. Smith theorems for
spheres: I. Actions of prime order groups. Conf. on Alg. Top.,
London, Ont., 1981, Can. Math. Soc. Conf. Proc. Vol. 2) Pt. 2
( 1 9 8 2 ) , 235-273.
[W] S. Weinberger: Homologically trivial group acLions,
preprint 1983.
An algebraic approach to the generalized Whitehead group
Karl Heinz Dovermann
Department of Mathematics
Purdue University and
Unversity of Hawaii at Manoa
and Melvin Rothenberg
Department of Mathematics
University of Chicago
Abstract: The notions of simple homotopy theory and Whitehead torsion
have generalizations in the theory of transformation groups. One does
not have to consider free actions. A geometric description of a
generalized Whitehead group was given by Illman. The approach
resembles that of Cohen. An algebraic approach was pursued by
Rothenberg. This approach has been developed only under certain
assumptions. In this paper we generalize the approach to give an
algebraic description of the generalized Whitehead group for a finite
group. In particular we put no restrictions on the component structure
of the action and we do not assume that H fixed point components are
1-connected. We prove that our and Ii!man's approach lead to the same
group.
Partially supported by NSF Grant MCS 8100751 and 8514551
Partially supported by NSF Grant MCS 7701623
93
0. Introduction
Simple homotopy theory was introduced by J.H.C. Whitehead [13]
attempting to find a computational approach to homotopy theory. This
notion turned out to be different from homotopy theory. Two standard
references for simple homotopy theory and the related notion of White-
head torsion are Milnor [ii] and Cohen [3]. These references also
include many geometric applications. Applications to the theory of
free transformation groups are obtained by passing to quotient spaces.
The notions of simple homotopy theory and Whitehead torsion have
generalizations in the theory of transformation groups. One does not
have to consider free actions. A geometric description of a general-
ized Whitehead group was given by Illman [7]. The approach resembles
that of Cohen. An algebraic approach was pursued by Rothenberg [12].
Which approach is preferable depends on the particular application one
has in mind. Rothenbergs approach has been developed only under certain
assumptions. In this paper we generalize this approach to give an
algebraic description% of the generalized Whitehead group for a finite
group G. In particular we put no restrictions on the component struc-
ture of the action and we do not assume that H fixed point components
are 1-connected. As one may expect, the groups defined by Illman and
by us are related to each other. We prove that our and Illman's approach
lead to the same group.
The paper is organized as follows:
In the first nine sections we introduce the basic categorical
notation, K0, KI, and the Whitehead group Wh. In sections 10-14 we
define the generalized torsion of a G-homology equivalence. The con-
94
cepts required are strictly algebraic. Theorem A states that our
algebraically defined group coincides with Illman's geometrically
defined one. This result is based on Theorem B which describes the
generalized Whitehead group as a sum of classical Whitehead groups.
Finally we state the basic geometric properties of the generalized
Whitehead torsion as well as the most important geometric conclusions.
The generalized Whitehead torsion has been considered in several
other articles by Illman, Hauschild, Anderson, and ourselves [9,10,6,1,5]
but the formalism and generality of our present approach is new. For
some more recent articles by Araki, Araki-Kawakubo, and Steinberger-West
see also [14,15,16].
i. Basic categories
A generic category will be denoted by M. All categories con-
sidered in the next seven sections will be assumed to have unique
initial-terminal objects ~, and all functors will be assumed to
preserve them. For such M, any two objects are connected by a
uniquely defined trivial map, denoted by 0. A morphism a is an
epimorphism if ba = 0 implies b = 0. Projective objects are defined
through the common universal property. The category C(M) of finite
chain complexes over M is defined in the obvious manner. An object
of C(M) will be denoted by (Cj,dj), where j E ~- All categories
we consider will be small, so that the usual set theoretic operations
can be performed. We will systematically surpress mentioning that fact.
2. Exact sequences
An ES structure (ES = exact sequence) on M is a collection
ES(~0 = {Cp,i)} of pairs of morphisms, where domain p = range i,
95
such that for isomorphisms ~,y,¢ of M, (p,i) E ES(M) if and only
if (~py-i,yi¢-l) ~ ES(M). We further assume that for the initial-
terminal object ~ the pairs (Ol,Id) and (Id,O 2) are in ES(M),
where O1: ~ + A and 02: A ~ ~ Subcategories always inherit ES
structures, as does C(M), if M has one. For abelian categories we
always use the usual ES structure.
3. K 0 (M)
For the category M with an ES structure K0(M) is well
defined. It is the free abelian group generated by isomorphism
classes of objects M subject to the relations ~ = 0, and if
(p,i) ~ ES(M) then domain p = range i = domain i + range p.
If d: M I + M Z is an exact functor, i.e., d preserves ES structures,
then d induces a homomorphism d,: K0(M I) ~ K0(M2). The inclusion
of a subcategory is an example of such an exact functor.
4. Category of F chain complexes
If F: M I -~ M 2 is a functor we define C(F), the category of
finite F chain complexes, as follows. An object of C(F) is a
sequence (Cj,dj) with Cj ~ M I, dj E M2(F(Cj), F(Cj_I)), and with
dj_id j = 0. We assume all but a finite number of the Cj's are ~.
A map ~: (Cj,dj) -~ (~j,~j) in C(F) is a sequence ~j: Cj -~ %,
where ~j ~ MI(Cj,~j) and ~jF(~j) = F(~j_l)d j. When M 1 has an
ES structure, C(F) inherits one and the natural functor
Jl: C(MI) + C(F) is exact. If M 1 ÷ M 2 is exact then the natural
functor J2: C(F) ÷ C(M 2) is also exact.
96
5. Categories of acyclic complexes
Let F: M I ~ M 2 be as in 4. If M 2 is an Abelian category,
we denote by C [M2) c C(M2) the full subcategory of acyclic complexes,
Ca(F ) c C(F) the full subcategory whose objects are in j21(Ca(M2)),
and Ca(MI,F) the full subcategory of C(M I) whose objects are in
J lljil (C~(M2)) .
6. K 1 IF) and ~1 [F).
Let M I be an ES category, M 2 an Abelian category and
F: M I ~ M 2 a functor. We define
K 1 (F) = K o (c a ( ~ ) ) / j 1 . (K0 (Ca (MI' F) ) .
Consider elements in Ca(F ) of the form
. . . . . . . . . . A "Id>A .......
These sequences generate a subgroup I in KI(F). We define
~I(F) = K l ( F ) / I .
7. K I of a ring
We now specialize to the categories we are interested in. R will
be a ring with identity and R will be the category of left R-modules.
Let S be the category of base pointed sets and base point preserving
maps. TO assure that S satisfies the assumptions of (i), we assume
base point of A = base point of B = #, for A,B in S and that
= {#}. Let f: ~ + S be the forgetful functor, and F:S + [ the
97
left adjoint of f. That is, F(A) is the f r e e R-module on A - {#}.
For A, B in S there exists a coproduct unique up to isomorphism,
any of whose representatives will be denoted by AvB. If A n B = {#}
we can take AvB = A U B. The ES structure on S is given by pairs
(p,i), i: A ~ AvB and p: AvB ~ B, the injection and projection of
the coproduct. The category R is an Abelian category and we take
the natural ES structure from exact sequences. The functor F, but
not f, is exact. KI(F ) is not interesting since we have not yet
imposed a finiteness condition. However, if we let F 0 = FIS0, where
S O is the subcategory of S consisting of finite sets, then KI(F0)
is the usual KI(R). This motivates the notation of (6).
8. Categories of functors
To get the Whitehead groups we proceed as follows. For
categories M 1 and M Z we consider the functor category C(MI,M 2)
whose objects are functors from M 1 to M 2 and whose morphisms are
natural transformations. Note that C(MI,M 2) will have an initial-
terminal object if M 2 does. We need no such assumption on M I. If
M2 has an ES structure then C(MI,M2) does by setting
i _p__ al-->a2 >~3 to be in ES of C(MI,M 2) if and only if for each
A E M I, el(A) i(A) >a2(A) p(A) >e3(A) is in ES of M 2. If
G: M 2 ~ M 3 is a functor, the composite yields
G,: C(MI,M 2) + C(MI,M3). If G is exact so is G,. With the
notation from (7) we now set M 2 = S, M 3 = R, and G = F. Again,
KI(F,) is not yet interesting since we have not yet imposed finiteness
or projectivity conditions.
98
9. The Whitehead group of a category
A functor ~: M ~ S is of finite type if for each A ~ M,
~(A)/(Iso(A)) is finite. Here Iso(A) denotes the invertible
elements of M(A,A) and Iso(A) acts on ~(A) via the functor
We let C0(M,S ) c C(M,S) be the full subcategory consisting of
projective functors of finite type. From (7) and (8) we have
F,: C(M,S) ~ C(M,R). We let F 0 be the restriction of F, to
C0(M,S ). Finally, we set
Wh (M,R) = K1 (Fo).
We repeat that for this M need not have an initial or terminal
object. This definition is related to the classical one in the
following example. Let G be a group, and G the category with one
object whose morphisms are the elements of G. Then Wh(G,R) is the
classical Whitehead group of G with coefficients in R.
i0. The generalized Whitehead group.
Consider the following category O(G), crucial in transformation
groups. The objects of ~(G) are the subgroups of G. The morphisms
O(G)(HI,H2) are G maps from G/H 2 to G/H I . Alternatively,
= {g E GIH 2 c gHlg-l}/H I. H I acts by right multiplication O(G)(HI,H 2)
on {g E GIH 2 c gHlg-l}. This category is sometimes described as the
orbit category, but this is deceptive since G/H and G/gHg "I are
indistinguishable as orbits but represent different, although
isomorphic, objects of @(G).
99
This category is central because for a G CW complex X the
map which assigns to each H c G the n cells of X H determines a
functor X: @(G) ~ S which encodes the G cell structure of X. To
continue our examples, we have Wh(~(G),R) = Wh(G;R), the generalized
Whitehead group of G defined in [12].
II. Partially ordered G sets
To continue our setup we need to digress and consider G posets.
This notion is helpful in the study of the combinatorial structure of
a G action, and it has been discussed in much detail in [4]. Suppose
is a partially ordered set and G acts on ~ preserving the
partial order on ~. Then we call ~ a partially ordered G set. As
example consider S[G) = {H c GIH is a subgroup of G}. A partial
ordering is given by H ~ K if and only if H m K. The G action
is given by conjugation. Suppose p: ~ ~ S(G) is an order preserving
equivariant map. For any a E H we set
ii.i ~ = {B ( ~IB ~ ~},
[(a) = {B E ~IgB { e for some g E G},
G = {g E Glg~ = ~}.
Throughout we assume that (an assumption satisfied in 11.4).
S(G) is injective. II.2 p(~) ~ G and p: ~ P(~)
= c = NG(P (~)) Note that S(G)p(~) S(p(a)) c S(G ) and G Gp(~)
the normalizer of p(~) in G. A pair (~,p) as we just discussed
it is called G poset. As example of a G poset consider (S(G),Id).
If H is a subgroup of G, then S(G) H = {K E S(G) IK c H} and
100
S(G)(H ) = {K ( S(G) IgKg "I c H for some g E G}. In general,
H a is a G ° poset and H(a) is a G poser.
A G poset (H,p) is called complete if
11.3 P: Ea ~ S(G)p(a) is bijective for all ~ E H.
To any G space X we associate a G poset (H(X),PX). Set
11.4 ~(x) -- ~ ~o(X H)
Here ,Jj denotes the disjoint union. The action of G on X
provides an action of G on H(X). If e E ,0(X H) we set px(e) = H.
If ~ E ~0(X H) then a is the name of a path component of X H. We
denote this subspace of X by either X or l~I. The partial
< c X B and p ( a ) ~ p ( 8 ) . ordering on H(X) is given by: ~ _ ~ if X _
Often we. abbreviate (H(X),Px) by (H(X),p). Notice that (H(X),p)
is always complete.
Definition Ii.5 Let (H,p) be a G poset. A (H,p) space is a G
space X with a collection of distinguished subspaces {X~I~ ( K},
X could be empty, such that
(i) Xg~ = gX for all g ( G and a ~
(ii) X ~ X8 if a, 8 E K and ~ ~ 8
Ciii) xH : If x
If X is a G CW complex and the Xa's are subcomplexes we call
(X,{X }e E H}) also a (E,p) complex. We say that X is a
101
(~,p) space or complex if the X's are understood.
The obvious example is as follows. Let X be a G CW complex
and ~ = ~(X) as in 11.4. The natural choice for the subcomplex in
11.5 are the spaces X distinguished in the paragraph before 11.S.
Let (E,p) be a G poser. To each ~ E K we associate
I1.6 W(~) = G /p(e).
Suppose (~,p) and (~',p') are G posers. A G poset map
a: (~,p) + (~',p') is an equivariant order preserving map a: ~ ~ ~'
such that p(~) = p'(a(~)). Suppose f: X ~ Y is an equivariant
map of G spaces. This map f induces a map f: (~(X),p x) + (~(Y),py)
by setting f(~) = 8 where B is defined by PX(a) = py(8) and
f(X ) ~ Y~. By restriction f induces the map
11.7 f: X ~)
12. The Whitehead group for 1-connected fixed point components
Let (~,p) be a G poset. We define an associated category
-~-~/~-. The objects are the elements of E. For ~,y ( E, define
N(e,y) = {g E Glg~ < y}. Then p(~) acts on N(e,y) by right
multiplication. The morphisms of the category are
"~-~-y(~,y) = N(~,y)/p(e). Composition of morphisms is defined by
multiplication in G. The Whitehead group Wh(~-~;R) is a
generalization of the Whitehead group Wh(O(G);R) from (I0). It is
appropriate for the study of actions of G where, for subgroups H
of G, the H fixed point set need not be connected but each component
is simply connected. The case of an empty H fixed point set is
included.
102
13. The Whitehead group for non 1-connected fixed point components.
Next, we wish to describe algebraically the Whitehead groups for
a G complex X where the fixed point sets need not be simply
connected. We define a category #(X). The objects will be the
components of X H, as H runs over the subgroups of G with X H ~ ¢.
So, the objects are the elements of H(X). For each component
E ~(X), we select a base point x(~) in X (11.4). A morphism
from ~ to y will be an element of N(a,y)/p(~) where N(~,y)
consists of pairs (g,k),g E G, g~ ~ y, and X is a homotopy class
of paths in X joining gx(~) to x(y). The subgroup p(~) of Y
G acts on the pair by acting on the first factor on the right.
Notice that if each component of X H is simply connected we are
exactly back in the category of (12). The product of elements of G,
along with the composition of paths, describes a composition law for
morphisms in #(X). Strictly speaking, the category depends on the
choices of base points. However, choosing paths connecting two
different sets of base points, defines an isomorphism from the category
with one set of base points to the category with another set. This
isomorphism is not canonical but it depends on the choices of paths.
We now define the group Wh(X;R) = Wh(@(X);R) and see that, at least
as abstract group, its isomorphism class is independent of the choices
of base points. We claim that this is the algebraic description of
Illman's group Wh(X~ when R = ~, see Theorem A below.
14. The Whitehead torsion of a G homology equivalence mod R.
Suppose we are given two finite G CW complexes X and Y and
a G map f: X + Y such that f maps components bijectively
103
(3: ~(X) ÷ ~(Y) is a bijection) and on each component f is a
mod R homology isomorphism ((f),: H,(X ,R) + H,(Y~(~),R) is an
isomorphism). Naturally, we suppose that we selected base points for
X and Y~(~) and that f preserves them. In addition, we suppose
that f~ induces an isomorphism from ~I(X ) __t° ~I(Y~(~)). For
notation see (II). So, our notion of an R homology equivalence is
tied to the category and makes stronger assumptions than usual. We
will see how to get an invariant x(f) in Wh(X;R) from f.
The two functors, which assign the cellular chains of ~ and £Z
of Y~(~) to ~ are finite chain complex functors on O(Y). They
can be checked to be projective. If f is a G cellular map, it
induces a transformation of chain complexes which induces an iso-
morphism on homology mod R by assumption. The mapping cone of this
transformation is then an R valued acyclic functor from O(Y) to
finite R complexes, which is projective and thus defines an element
of Wh(Y,R). The argument of [12, p. 285] shows that this element
depends only on the G homotopy class of f and thus the invariant
is well-defined.
Let IWh(X) denote Illman's Whitehead group [7]. The con-
struction of this paragraph defines a homomorphism
~: Iwh(x) + WN(X, 2Z).
We then have
Theorem A. ~ is an isomorphism.
The proof will be carried out in the next few sections.
particular it will follow from Theorem B of the next section.
In
104
IS. Computation of Wh(X,R).
The proof of Theorem A follows from a calculation, lllman
calculated his group; we shall calculate ours and see that one gets
the same result. For each u ~ ~(X) we defined
G = {g ~ Gjg~ = ~} = {g E GIgX a = X } 0 NGP(~ ) and
w(oO -- GoJP(cO.
Let X be the universal covering space of X .
selected base point so that this canonical. Let ~(~)
of homeomorphisms of X which cover the action of G °
We then have the exact sequence
Recall that we have
be the group
o n X •
in each G orbit of K(X) pick one representative. Call the
set of components so constructed A. Then
Theorem B .
WN(X,R) = ~EA Wh(~(~),R).
In the theorem Wh(~(~),R) is the classical Nhitehead group
with coefficients in R.
Theorem A follows from Theorem B ~ince it is easily seen
that a, composed with the isomorphism of Theorem B, is just
Illman's isomorphism. We shall prove Theorem B in the next few
sections.
105
16. Extending functors.
proposition. Let V c W be a full suhcategory. Given any functor
T: V ~ S there exists a unique (up to natural isomorphisms pre-
serving T) minimal extension T: W + S satisfying the following
property. Given a natural transformation a: T ~ FIV , where F is
any functor from W to S, there exists a unique natural trans-
formation a: T ~ F extending ~.
It follows easily from the universal property that if T is
projective so is ~. In the cases that we are interested in, we can
check that, if T is of finite type so is T. This is in particular
true for each pair of objects A and B of W the morphism set
W(A,B) is finite.
P!oof 0f the Proposition. We construct T as follows. For A E W
set:
T(A) = {(f,x) I f E W(C,A) for some C E V and x E T(C)}/~U~.
The relation ~ is the smallest equivalence relation suoh that
(fl,Xl) ~ (f2,x2] if there exists f3,C3,Jl,J2 and a commutative
diagram
Jl J2 C I -> C 3 < C 2
~ v A
106
and T(Jl)(Xl) = T(J2)(x2). If I E W(A,B) set T(1)(f,x) = (lf,x).
For A ( V we have a natural indentification of T(A) and T(A).
Futhermore, if a is a natural transformation T + FIV, we define
~: T(A) + F(A) by ~(f,x) = F(f)(~(x)). The properties of T
asserted above follow immediately from the construction.
17. Restricting projective functors
propositign ,. Let W be any category, V c W a full subcategory such
that for A in ob(W) ob(V) and B E ob(V) then W(A,B) = ~.
If J: W ~ S is projective, then Jl = JIV is projective.
Proof. Let ~: F ~ Jl be a surjective natural transformation where
F is any functor form V to S. We may extend ~ to ~: F ÷ J
by (16), however ~ may not be epi. Let %: W + S be a functor,
with ~(A) = ~ if A E V and %(A) = J(A) if A ( W - V. By the
assumptions of the proposition % is a functor and ~ extends
naturally to ~: F v % -~+ J. Since J is projective there exists
y: J + F v ~ with ~y = Id. By construction Y1 = YIJI factors
through F and ~YI = Id. Hence Jl is projective.
18. Quotients of natural transformation
Let TI,T2: W ~ S be functors as above and let ~: T 1 ~ T 2
be a natural transformation. We can form the functor T2/a(Ti),
where T2/~(TI)(A) = T2(A)/~, and ~ is the equivalence relation
which identifies points of ~(TI(A)) with #. The functor T2/~(T I)
applied to a morphism has the obvious meaning, namely, it is the
induced morphism on quotients. With this notation we have
107
Propositon. Suppose for each A EW such that T2(A ) # ~(TI(A))
f E W(A,B) we have that f has a two sided inverse. If T 2 is
projective so is T2/a(TI).
and
Proof. There exists a natural transformation n: T 2 ~ T2/e(TI).
The functor T2/a(TI) is projective if and only if ~ splits, i.e.,
there is j: T2/~(TI) + T 2 with ~j = Id. There is an obvious unique
base pointed splitting of n(A), j(A): T2/a(TI)(A ) ~ T2(A), and we
must show that this is functorial in A. This is true if and only if
x ~ T2(A), x # a(y) implies that for all f E W(A,B), T2(f)(x ) # a(z),
z ~ =. By assumption, if there exists an x in T2(A) and x # a(y)
then f in W(A,B) is invertible. But if T2(f)(x) = a(z), then
x = T2(f-l)T2(f) (x) = T2(f-l)~(z) = ~(Tl(f'l)(z)),
which is a contradiction.
lg. Consequence of (17) and (18)
Corollary of (17) and (18). Let V be a full subcategory of W such
that if A E W - V, then W(A,B) = ~ for B E V and a E W(A,C) is
invertible for C E W - V. If T: W + S is projective, then
T 1 = TIV is projective and T/J(TI) is projective, where j: T 1 + T
is the natural transformation of 16 extending the identity trans-
formation.
20. Decomposition of the Whitehead group
Let V be a full subcategory of W as in 19. Then
Wh(W;R) = Wh(V;R) @ Wh[W,V;R)
108
where Wh(W,V;R) = K-(F00), F00 = F01C00(W,V), and C00(W,V) is the
full subcategory of C(W,S) of projective functors of finite type, y,
such that 7(A) = = for all A in V.
Proof. To see this we need only show that if for the functor Tn,d n
d .... T n(A) n >T n.l(A) ....
is acyclic for each A in V, then
. . . Tn(A ) dn Tn- 1 (A)
is acyclic for all A E W. Here %'~n is a minimal extension of
Tn,d n. This follows by an easy Meyer-Victoris argument. So we have a
splitting of the natural map p: Wh(W,R) ÷ Wh(V,R). It follows also
from 19 that the kernel of p is generated by complexes whose terms
are of the form T/j(TIV), that is, by elements which come from
Wh(W,V;R).
21, Proof of Theorem B
Let X be a finite G CW complex.
of the subgroup H of G let
X [ ~ = U KE[H]
X [HI = {x E X [HI -1 s IG x ; gHg
By r e p e a t e d a p p l i c a t i o n o f (20) we have
For the conjugacy class [H]
X K and
for some g E G}.
109
~ ( O ( X ) ; R ) = X Wh(@(x[H]), e (x~H]) ; a) [HI
where the summation runs over conjugacy c l a s s e s of subgroups of
Now, an easy c a l c u l a t i o n as in [12, p. 274] shows t h a t
Wh(O(x[H]), e(X~ HI) ;a) = [Wh(W(a);R).
G.
Here a runs over elements in H(X) such that 0(a) = H and we have
to pick one such a in each G orbit. This proves Theorem B which
was stated in (15).
22. Geometric properties
As a first step we discuss induced maps between generalized
Whitehead groups. Let s: (~,~) ÷ (~',p') be a poser map such that
W(a) = W(s(a)) for all ~ E ~. Taking direct sums of the chain
complex functors we obtain an induced map s,: Wh(~,p) + Wh(~',p').
More generally, suppose g: B ÷ Y is an equivariant map. Pick
collections of base points in B and Y (see 13) and suppose that
g preserves them. Using an appropriate definition of universal
coverings we have base points ~(a) in 181. Now suppose that for all
E E(B) (f~)#: ~l(Ba) ÷ ~l(Y~(a)) is an isomorphism and that
W(~) = W(f(a)). Then we have the induced maps W(~) ÷ W(f(~)) (compare
(compare [2, p. 65]) which are isomorphisms by the Five Lemma. We
can continue as above and take direct sums of chain complex functors
to obtain an induced map g,: Wh(B,R) ÷ Wh(Y,R). Obviously, g~
can be naturally defined if g is a G homotopy equivalence.
We give the geometric interpretation of the process we just
described. Suppose s: (g,p) ÷ (H',p') is a G poset map and X is
110
a ( ~ , p ) s p a c e . Then X c a n be u n d e r s t o o d a s ( ~ ' , p ' ) s p a c e by
setting X B = U_ 1 X . Suppose f: A ÷ B is a G homotopy
equivalence of finite G CW complexes, or, more generally, f is
just a mod R G homology equivalence as in (143. So T(f) ~ Wh(B,R)
is defined. Setting H(B) = 9, and assuming s or g as above, we
obtain an element s,(~(f)) ~ Wh(]~x-~) or g,(~(f)) ~ Wh(Y,R).
We generalize the definition given in (14). Suppose
fl A ........ > B
Igl f2 Ig2 X - - > Y
is a square of equivariant maps of finite G CW complexes. Suppose
the square is G homotopy commutative and h: g2fl m f2gl is a
G homotopy. This data determines an induced map
F: (MgI,A) ÷ (Mgz,B)
which defines
f3: Mgl/A ÷ Mg2/B
Suppose, g2 and p: y incl.>~ proj,>M /B induce maps on the level g2 g2
of Whitehead groups:
Wh (B,R) ( g 2 ) * P* • > Wh(Y,R) >Wh(Mg2/B,R).
111
If F is a G- R homology equivalence we set
T(F) = T(f3) E Wh(Mg2/B,R).
If T(fl) E Wh(B,R) and T[f2) E Wh(Y,R) are defined
(22.1) p,T(f 2) = T(f 3) + p,(g2),~(f I) E Wh(Mg2/B,R).
It follows from this formula that z(f3 ) will not depend on the
choice of the particular homotopy h in this case.
There is an interesting special case. Suppose the H fixed
point set of B and Y are l-connected (possibly empty) for each
H c G. In this case we call B and Y G simply connected. In a
natural way Wh(B,R) and Wh(Y,R) are subgroups of Wh(point,R),
and so is Wh(Mg2/B,R ). Remember that Wh(point,R) is Wh(G,R) from
[12]. This was pointed out in (i0). With this understood, 22.1
simplifies to
2 2 . 2 ~(f2 ) = T(f3) + ~(fl ).
If i: Z ÷ W is an inclusion we set ~(i) = T(W,Z). If fi and gi
are inclusions 22.1 can be reformulated as
22.3 T(Y,X) = T(B,A) * T(Y,X U B) = T(B,A) + 7(Y/B,X/A)
Let f: X + Y and g: Y ÷ Z be G-R homology equivalences.
A standard proof [12, 1.32] based on the exact sequence of the
appropriate chain complexes of mapping cones implies
112
22.4 T(gof) = g,T(f) + ~(g)
Here are some properties of the generalized Whitehead torsion.
A generalization of [12, 2.5] is
Proposition 22.5 The generalized Whitehead torsion of an equivariant
subdivision is zero.
The existence and uniqueness of a smooth equivariant triangulation
of a smooth G manifold has been shown in [8]. From this follows
Proposition 22.6 The torsion of a G homotopy equivalence (mod R
G homology equivalence) between smooth compact G manifolds is well
defined and vanishes for diffeomorphisms.
Independently, this has also been shown by Illman [9, Theorem 3.1
and Corollary 3.2]. We use the notation of a G h-cobordism as it has
been defined in [! ? , 3.1]. If X is a G space and H c G we set
X [H] = {x E XIG x is conjugate to H.}. Let i: X ÷ ~Y be a G
imbedding of compact G manifolds with dim X = dim Y - I. We call
i a G cobordism of X to ~Y - X. We call i a G h-cobordism if
for each H c G the induced maps i[H]: x[H] ~ y[H] and
i,[H]: ~y _ x[H] + y[H] are homotopy equivalences. Here i' is the
inclusion ~Y - X ~ ~Y.
We restate some results from [12, section 3] in our language.
The results are more general but the proofs are similar.
Eq__uivariant s-Cobordism Theorem: Let i: X + Y be a G h-cobordism
such that Y = X × I if dimX ~ 4. The pair (Y,X) is G
diffeomorphic to (X × I,X x 0) if and only if T(i) = 0 in Wh(Y, ~).
Let z represent a class in Wh(Y,R) for some finite
113
G CW complex Y. The direct sum decomposition of Theorem B
determines components ~ 6 Wh(W(e),R).
Realization Theorem: Let X be a compact G manifold. Let z be
any element of Wh(X, ~) such that ~ vanishes if dim X ~ 4. Then
there exists a G h-cobordism i: X ~ Y such that ~(i) = T and
Y = X x I whenever dimX ~ 4.
Classification of h-cobordism Theorem: Let i.: X + Y. be a G 3 3
= (Yj) = ×I if dimX < 4 h-cobordism, j 1,2. Suppose that ~ X a ~ _ ,
E ~(X), and ~(il) = T(i2). Then there exists a diffeomorphism
l: Y1 ÷ Y2 with i 2 = li I.
The notions of elementary expansions and collapses (briefly
deformatlons) discussed in [7] and [12, p. 288] have natural general-
izations in the category of (~,~) complexes. Such deformations
have vanishing torsion. We use the symbol A/~B to denote that A
and B are connected through a sequence of equivariant elementary de-
formations. More precisely, we are given a sequence of spaces
Ci, i ~ i ~ £, and maps ki: C i ÷ Ci+ I, 1 ~ i ~ £-i such that
A = CI, B = C£, and k i is either an equivariant collapse, or k i is
an inclusion and the G homotopy inverse of an equivariant collapse.
So it makes sense to consider maps which are G homotopy equivalent
to a sequence of elementary deformations. Finally, we have
Proposition 22.7 Suppose f: A + B is a G homotopy equivalence of
finite G CW complexes. Then f is G homotopic to a sequence
of elementary deformations if and only if T(f) vanishes in W~(B, ~).
The proof is standard based on [7, Theorem 3.6'] and our Theorem A.
For a special case this was observed in [12, 2.3].
114
References
i. D. Anderson, Torsion invariants and actions of finite groups. Michigan Math. J. 29(1982), 27-42.
2. G. Bredon, Introduction to compact transformation groups, Academic Press, 1972.
3. M.M. Cohen, A course in simple homotopy theory. Springer Verlag, Berlin-Heidelberg-New York, 1970.
4. K.H. Dovermann and T. Petrie, G surgery II. Memoirs of the AMS, Vol. 260, (1982).
5. K.H. Dovermann and M. Rothenberg, The equivariant Whitehead torsion of a G fibre homotopy equivalence, preprint (1984).
6. H. Hauschild, Aquivariante Whitehead torsion. Manuscripta Math. 26(1978), 63-82.
7. S. Illman, Whitehead torsion and group actions. Annales Academiae Scientiarum Fennicae, Vol. 588(1974).
8. , Smooth equivariant triangulations of G manifolds for finite group. Math. Ann. 233(1978), 199-220.
9. , Equivariant Whitehead torsion and actions of compact tie groups, Group action on manifolds, Contemporary Mathematics Vol. 36(1985), 91-106.
, A product formula for equivariant Whitehead torsion, preprint (1985), ETH Zurich.
J. Milnor, Whitehead torsion. Bull AMS 72(1966), 358-426.
M. Rothenberg, Torsion invariants and finite transformation groups. Proc. of Symp. in Pure Math., AMS, Vol. XXXII, (1978), 267-311.
J.H.C. Whitehead, Simplicial spaces, nuclei and m-groups, Proc. London Math. Soc. (2), 45(1939), 243-327~
S. Araki, Equivariant Whitehead groups and G expansion categories, preprint.
S. Araki and K. Kawakubo, Equivariant s-cobordism theorem, preprint.
M. Steinberger and J. West, Approximation by equivariant homeo- morphisms, preprint.
i0.
ii.
12.
13.
14.
15.
16.
Almost complex sl-actions on c ohomoloqy comD!ex projective spaces
To the memory of A. Jankowski and W. Pulikowski
Akio Hattori
i. Introduction.
Let: X be a closed C manifold which has the same cohomology
ring as the complex projective space CP n. Such a manifold will be
called cohomology complex projective space or, briefly, cohomology CP~
There is a conjecture due to T. Petrie [P] to the effect that if the
group S 1 acts non-trivially on X then X has the same total
Pontrjagin class p(X) as CP n. The conjecture was partially solved
in various special cases; cf. [D], [HI], [M], [P], [YI ] and [Y2 ] . In
[P] Petrie presented an interesting example of exotic sl-action on a
cohomology complex projective space X whose normal representations at
fixed points are different from the linear sl-actions on CP n but,
nevertheless, the Pontrjagin class of X is the same as p(cpn).
On the other hand the author investigated certain almost complex
sl-actions in [H2]. The results there suggest the following conjec-
ture.
Conjecture A. Let X be an almost complex manifold which is a
cohomology CP n such that Cl(X)n[x] > 0 and T[X] ~ 0 where T[X]
denotes the Todd genus of X. If X admits an almost complex S l-
action with only isolated fixed points then the normal representations
of S 1 at fixed points are the same as those of a linear action on
fpn (the precise statement will be given in the statement of
conjecture B). In particular, the total Chern class c(X) is the same
as c(cpn), i.e.
c(X) = (l+x) n+l
where x is a generator of H2(X; Z).
In the present paper we shall present a proof of the above conjec-
ture for n ~ 3. In fact we shall formulate a more general conjecture
concerning certain almost complex sl-actions and prove it affirmatively
when the complex dimension of the manifold is less than 4.
2. Main results.
First we recall some of the results in [H2]. Let X be a compact 1 connected almost complex manifold with an S -action which preserves the
almost complex structure. Our basic assumptions in the sequel are the
116
following:
2.1 The fixed points are all isolated.
2.2 The Euler number × of X is equal to n + i.
with (2.1) implies that there are exactly n + 1
P0' PI''''' Pn"
2.3
This together
fixed points
Cl(X)n[x] > 0 where Cl(X) is the first Chern class of X and
IX] (H2n(X;Z) is the fundamental class of X. Moreover there
exists x 6 H2(X;~) such that
xn[x] = I and Cl(X) : kx with k > 0.
2.4 T[X] ~ 0.
Let < be a complex vector bundle such that Cl({) = x.
Lemma 2.5. The action of S 1 on X can be lifted to <.
This follows from [H 2, C o r o l l a r y 3 . 3 ] s i n c e ~k = AnT(x)
a lifting where ~(X) denotes the complex tangent bundle of X.
If we restrict ~ to each fixed point P a 1 1
<IP, which is of the form t where t 1
1 - d i m e n s i o n a l s l - m o d u l e and a, ( ~. 1
admits
we get an sl-module
is the standard
Lemma 2.6. The integers {a i} are all distinct.
This follows from [H 2, Corollary 3.8] in view of (2.2) and (2.3).
On the other hand the normal representation at each Pi takes
the form m..
~(X) IP i : It ±3
Lemma 2.7. The integers {mij} are related to
(2.8) [ m ~ = ka + d j i] 1
where d is a fixed integer.
This is an easy consequence of the identity
[H 2, Corollary 3.15] for some related results.
The problem is to determine the possible values of
Theorem 5.1] it is proved that
a. by the formula 1
A n ~(X) = <k; cf.
k. In [H 2 ,
(2.9) k < n + 1
under more general assumption than (2.1), (2.2), (2.3) and (2.4).
Conjecture B. Under the assumptions (2.1), (2.2), (2.3) and (2.4) the
117
only possible value of k is n+l.
In [H2, Corollaries 5.8 and 5.9]
implies the following
it is proved Conjecture B
Consequence 2.10. Under the assumptions (2.1), (2.2), (2.3 and (2.4)
the weights {m i} at each fixed point Pi are given by
{mi } = {a i - aj}j~ i,
that is, the weights are the same as those of a linear actzon on £pn.
If moreover X is a cohomology CP n then
n+l c(x) = (l+x)
Conjecture B combined with Consequence 2.10 reduces to Conjecture
A when X is a cohomology ~pn.
Theorem 2.11. Conjecture B and hence Conjecture A is true for n ! 3.
Remark. Conjecture B is also true for n = 4. We can give a proof
similar to the one which will be given in the next section. However it
is too cumbersome to be reproduced here.
Cl(X)n[x] > 0 can be relaxed to Cl(X)n[x] Also the condition
0 when n is even. But we do not insist on this point here.
3. Proof of Theorem 2.11.
As in [H2] we set
Pi : number of ~ such that miv > 0
and
pq : number of i such that Pi = q"
By [H2, Proposition 2.6 and Remark 2.10] we have
(3.1) Pn-q : Pq for all q and P0 = Pn : T[X].
For each i we set
a -a (l-t z 3)
~i(t ) = j~i m ' 0 _< i _< n.
H(l-t )
The gi(t) is a Laurent polynomial of
3.7] we know that
t, and by [H 2, Proposition
118
(a.-a)
(3.2) 5oi(i) : j~i I 3 = i.
~m,
From (3.2) it follows easily that
(3.3) Pi ~ i mod 2 for i = 0, l, .... n.
Moreover [H 2, Theorem 4.2] implies the following
Proposition 3.4. If we set z = n + 1 - k then there are Laurent
polynomials r0(t),..., rz(t) such that
a ha ~i(t ) = r0(t ) + rl(t)t l +...+ rz(t)t z for all i,
r0(t) = T[X] = P0 = Pn
and
rz_s(1) = rs(1), 0 i s ! Z.
As a consequence of Proposition 3.4 we deduce the following
Lemma 3.5. £ must be even, i.e.
k ~ n+l mod 2.
In fact if £ = 2s + 1 then
~i(1) = 2(r0(1) +,..+ rs(1)).
But this contradicts (3.2).
|
Proposition 3.6. Let p be a prime. For each i let xj and x v
be the exponents of p in the prime factor decomposition of a i - aj,
j ~ i, and miv respectively. Then {xj}j~ 1 and {x~} v coincide up
to permutations.
Proof. The following argument is essentially due to [P]. ~i(t can
be expressed as a product of cyclotomic polynomials ~ d(t) (up to
multiplication by some unit ±t N) where d ranges over those integers
such that the number of { j; jgi, d divides a i - aj} is strictly
larger than the number of {v; d divides miv}.
On the other hand it is well known that ~d(1) = q if d is a
power of prime q and ~d(1) : 1 otherwise. Since ~i(1) = 1 the
conclusion follows easily.
Corollary 3.7. Let m > 1 be an integer. Let Y be a component of
the fixed point set of the restricted ~/m-action on X. If Pi and
Pj both belong to Y then m divides a i - aj. Conversely if Pi
119
belongs to Y and m divides a. - a~ then P. also belongs to Y 3 3
provided m is a power of a prime p. In this case the Euler number
x(Y) of Y is equal to dimcY + i.
Proof. The first statement is easy; see e.g. [H2]. The remaining
part follows from Proposition 3.6 and the observation that × (Y)
equals the number of j such that Pj belongs to Y and dimly is
equal to the number of v such that m divides m once we choose a
fixed point P in Y. ±
With these preliminaries we can now proceed to the proof of
Theorem 2.11.
First we consider the case n = I. Since 0 < k < 2 and k is
even by (2.9) and Lemma 3.5, k must equal 2.
Remark 3.8. It is known that the only Riemann surface a~mitting an S l-
action with non-empty fixed point set consisting of only isolated fixed
points is CP 1 and thus the first Chern class c I evaluated on the
fundamental class is equal to 2. It follows that the conclusion k = 2
holds without the assumption (2.3).
In the sequel we shall assume n ~ 2. We also suppose that the
fixed points {Pi } are indexed so that a 0 < al<. • .<a n-
Suppose n = 2. Let p be a prime number dividing a 2 - a 0. Let
Y be the component of the fixed point set of the restricted
~/p-action containing P2" By Corollary 3.7, Y also contains P0"
We may assume the action on X is effective. Then P1 can not be
contained in Y. Therefore dimcY = i, and by Remark 3.8 and
Consequence 2.10 applied to the case n = 1 we see that a 2 - a 0 is
the weight of Y at P2" Hence from (3.2) it follows that the
remaining weight of X at P2 must be equal to a 2 - a I. Similarly
the weights at P0 are precisely a 0 - a 2 and a 0 - a I. Then from
(2.8) for i = 0 and i = 2 we deduce that k must be equal to 3.
Remark 3.9. We have proved Theorem 2.11 and Consequence 2.10 under the
following milder condition (2.3)' instead of (2.3). Let ~ be a
complex line bundle on which the action can be lifted and such that n
x IX] = 1 where x = Cl(~). We define the integers {a i} as before.
Then under (2.1) and (2.2), the integers {a i} are mutually distinct
and the equality (2.8) holds for some integers k and d as was
proved in [H 2, Corollary 3.8, Corollary 3.15 and (3.17)]. Now we state
the condition (2.3)'
(2.3)' There exists a complex line bundle { as above with xn[x] = 1
and k > 0.
Note. It was shown in [H2, Theorem 4.2] that if we also assume (2.4),
120
i.e. T[X] @ 0, then k > 0 in (2.3)'
We now proceed to the case n : 3.
The possibilities are k : 2 or k = 4. Assuming k = 2 we
shall deduce a contradiction. It is known that the numbers {a i} are
altered to {a i + a } for some a if we take another lifting of the
action to ~ ; see e.g. [H2]. Therefore we may assume that 0 < d _< 1
in the equality when k = 2. We divide into three subcases.
Subcase i: a 0 < a I < a 2 < 0 < a 3. Evidently we have P3 = 3, i.e.
all the m3v are positive. Therefore
H ) 3
~3(i ) j~3 (a3-aj a 3 = .... > > i. 2a3+i =
~m3vv (___7)3
This contradicts the assumption ~3(I) = 1.
Subcase 2: a 0 < 0 ~ a I < a 2 < a 3. Similarly to Subcase 1 we deduce
~i(I) > 1 which is a contradiction.
Subease 3: a 0 < a I < 0 < a 2 < a 3. First assume a 3 - a 2 > 1 and let
p be a prime integer dividing a 3 - a 2. Let Y be the component of
the fixed point set of the restricted ~/p action containing P3" We
may assume the given sl-action is effective. Thus dimcY is equal to
2 or i.
Assertion. If dimcY is equal to 2 then k must equal 4.
Proof of Assertion. We first show that K : (xly)2[Y] = i. In fact
assume k > i. There is a unique Pi not contained in Y where
i = 0 or I. If m is the weight at ]?3 normal to Y then from
(3.2) we get
(3.10) m = K(a 3 - ai).
Let Y' be the component of the fixed point set of the restricted
Z/m action containing P3" There exists Pj 6 Y', j ~ 3. Then, from
the equality (cf. [H2, (3.20)])
= -n+ld = -2d [ as k
and the assumption 0 ~ d ~ 1 it follows that
(3.11) a 3 - aj ! 3a 3
On the other hand we see that
lajl £ 2a 3 and hence
(3.12) m = K(a 3 - a i) h 2a 3
121
since i = 0 or 1 and we assumed K > 1.
Now m divides a 3 - aj by Corollary 3.7.
(3.11) and (3.12), m must equal a 3 - aj, i.e.
Hence by virtue of
a 3 - aj : K(a 3 - a i) = m
where i = 0 or 1 and j / 3. If j = i then K = i. If j ~ i
then Pj 6 Y so that p divides a 3 - aj : m; but this can not
happen since we have assumed the action is effective from the first.
In any case we have proved K = i.
If K = 1 then the weight of X at P3 normal to Y is equal
to a 3 - a i as above, and similarly the weight at P0 normal to Y is
a 0 - a i. Putting this in (2.8) we get
2
= + (d- a i ) [ ms~ a s ~=i
for s = 0 and s = 3 where msl and ms2 are the weights of Y
at P . But s 2
[ m = k'a + d' v=l s~ s
for all s, and k' must equal i. This contradicts Remark 3.9 and
completes the proof of Assertion.
If a 3 a 0 has a common prime divisor with a 3 - a I or a 3 a 2
then we can apply the above procedure and we have k = 4 by Assertion.
Similarly if a 0 - a I has a common divisor with a 0 a 2 or
a 0 a 3 then k = 4.
Thus we are left with the case where a 3 - a 2 is prime to both
a 3 - a I and a 3 - a 0 and a 0 a I is prime to both a 0 - a 2 and
a 0 a 3. Then a 3 - a2, a 3 - al, a 3 - a 0 are prime to each other.
Let q0 and ql be prime integers dividing a 3 - a 0 and a 3 - a 1
respectively and let Yi be the component of the fixed point set of
the restricted ~/qi action containing P3 for i = 0, i. We see
easily that dimcY i = I. Hence by Remark 3.8 and Consequence 2.10
that the weight of Yi at P3 is a 3 - a i, i = 0, i. Thus the
weights of X at P3 are precisely {a 3 - aj}j~ 3. A similar argument
shows that the weights of X at P0 are precisely {a 0 - aj} j~0"
Putting these in (2.8) for i = 3 and i = 0 we get k = 4.
This completes the proof of Theorem 2.11 for the case n = 3.
[D ]
References
I.J. De]ter, Smooth sl-manifolds in the homotopy type of CP 3,
Michigan Math. J. 23(1976), 83-95.
122
[HI] A. Hattori, SpinC-structures and sl-actions, Invent. math.
48(1978), 7-13.
[H2] A. Hattori, S 1-actions on unitary manifolds and quasi-ample line
bundles, J. Fac. Sci. Univ. Tokyo, Sect IA, 31(1985), 433-486.
[M] M. Masuda, On smooth sl-actions on cohomology complex proj.ective
spaces. The case where the fixed point set consists of four
connected components, J. Fac. Sci. Univ. Tokyo, Sect IA, 28(1981),
127-167.
[PI ] T. Petrie, Smooth sl-actions on homotopy complex projective spaces
and related topics, Bull. Amer. Math. Soc. 78(1972), 105-153.
[YI ] T. Yoshida, O__nn smooth s_emi-f______[ree sl-ac_____~tions on cohomolog_yy complex
projective spaces, Publ. Res. Inst. Math. Sci. 11(1976), 483-496.
[Y2 ] T. Yoshida, sl-actions on cohomology complex p__[rojective spaces,
Sugaku 29(1977), 154-164(in Japanese).
Department of Mathematics
University of Tokyo
A PRODUCT FORMULA FOR EQUIVARIANT WHITEHEAD TORSION
AND GEOMETRIC APPLICATIONS
By S~ren Illman
Dedicated to the memory of
Andrzej Jankowski and Wojtek Pulikowski
In the following, G and P denote arbitrary compact Lie groups, unless
otherwise is specifically stated. Let f: X + X' be a G-homotopy equivalence
between finite G - CW complexes and let h: Y + Y' be a P-homotopy equivalence
between finite P - CW complexes. In this paper we shall give a formula which
determines the equivariant Whitehead torsion
(G x P)-homotopy equivalence
t(f x h) ~ WhGxp(X x y) of the
f x h: X x y ~ X' x y' (I)
in terms of the equivariant Whitehead torsions of f and h, and various Euler
characteristics derived from the G-space X and the P-space Y. We are here
concerned with equivariant simple-homotopy theory and the corresponding notion of
equivariant Whitehead torsion as defined in [7]. We wish to point out that even
in the case when G = P our formula for the equivariant Whitehead torsion of (I)
deals with the situation where (i) is considered as a (G x G)-homotopy equivalence
between finite (G x G)-complexes. Nevertheless we are able to give in Corollary B,
for G a finite group, a geometric application in which we are dealing with the
diagonal G-action on X x y and X' x y', see also Corollaries D and G.
In the case when P is a finite group we obtain as a corollary of the product
formula the geometric result given in Theorem A. Specializing further we obtain
in the case when G = P, a finite group, the application given in Corollary B.
THEOREM A. Let G be a compact Lie group and let f: X + Y be a G-homotopy
equivalence between finite G - CW complexes. Assume that P is a finite
group and that B is a finite P - CW complex, such that X(B~) 0 for each
BQ of any fixed point set B Q. Then component P
f x id: X x B ----+ Y x B
is a simple (G x P)-homotopy equivalence.
124
COROLLARY B. Let G be a finite group and f: X + Y a G-homotopy equivalence
between finite G - CW complexes. Assume that V is a unitary complex repre-
sentation of G. Then
f × id: X x S(V) > Y × S(V)
is a simple G-homotopy equivalence, where G acts diagonally on X × S(V) and
Y × S(V).
Theorem A does not hold in general if P is a non-finite compact Lie group
and Corollary B does not either hold for a non-finite compact Lie group G, see
section 8.
Recall that in the case of ordinary simple-homotopy theory we have the follow-
ing. Let f: X + X' and h: Y ~ Y' be homotopy equivalences between finite
f × h: X × Y ~ X' × yt connected CW complexes. Then the Whitehead torsion of
is given by
• (f × h) = x(Y)i,T(f) + x(X)j,~(h) , (2)
Here i: X + X × Y and j: Y + X × Y denote inclusions given by i(x) = (x,y O)
and j(y) = (Xo,Y) , for some fixed Yo ~ Y and x ° ~ X, and X denotes the Euler
characteristic. (See e.g., 23.2 in [i].) In particular we have that the map
f × id: X × S 2n-I ~ X' × S 2n-I (3)
has zero Whitehead torsion and hence is a simple-homotopy equivalence for each
n > i. The fact that (3) is a simple-homotopy equivalence is an important result
in geometric topology. Our Corollary B establishes, for any finite group G, the
corresponding result in equivariant simple-homotopy theory. Our formula for the
equivariant Whitehead torsion of (i), valid for arbitrary compact Lie groups G
and P, is a generalization of the classical formula (2).
This paper also contains some other results than those already mentioned and
a quick survey of the contents of the paper is as follows. Section 1 contains
a review of the algebraic description of the equivariant Whitehead group WhG(X),
where G denotes an arbitrary compact Lie group and X is a finite G - CW
complex. In Section 2 we define the Euler characteristics that we will use.
The statement of the product formula for equivariant Whitehead torsion is given in
Section 3 and the proof of the product formula is given in Section 4. In Section
5 we prove Theorem A and Corollary B. Section 6 gives a formula for the equi-
variant Whitehead torsion of the join of two equivariant homotopy equivalences,
and corresponding formulae in the case of the smash product and reduced join are
given in section 7. In Section 8 we give an example which shows that equivariant
Whitehead torsion in the case of a compact Lie group G is no___~t determined by the
125
restrictions to all finite subgroups of G. This example also shows that Theorem
A does not hold when P is a non-finite compact Lie group and that Corollary B
does not hold for G a non-finite compact Lie group.
In the case of a finite group G and with the additional assumption that each
component of any fixed point set X H and yK is simply connected, a product for-
mula is given in Dovermann and Rothenberg [4], see the Corollary on p. 3 of [4].
They consider the product spaces X × Y and X' × Y' as G-spaces through the
diagonal action of G. In fact they are mainly concerned with the more general
situation of a G fiber homotopy equivalence. They work with the generalized
Whitehead torsion as defined in Rothenberg [14], and they also establish formulae
for the generalized Whitehead torsion of joins and smash products. There is also A A
some unpublished work by Shoro Araki on product formulae for equivariant Whitehead
torsion. For product formulae for equivariant finiteness obstructions see
tom Dieck [2], tom Dieck and Petrie [3] and Liick [ii], [12].
i. Review of the componentwise formula for WhG(X)
We will need to recall the algebraic determination of WhG(X)
see also [9]. (The first algebraic determination of WhG(X), for
Lie group, is due to H. Hauschild [5].) We have an isomorphism
as given in [8],
G a compact
9: WhG(X) = + Z Wh(~o(WK)~). (i) c(x)
The direct sum is over the set [(X) of equivalence classes of connected (non-
empty) components X K of arbitrary fixed point sets X K, for all closed subgroups
K of G. Two components X K and L of the fixed point sets X K and X L,
respectively, are in relation, denoted X K L ~ X$, if there exists n e G such
L Given a component X K of X K that nKn -I = L and n(X ) = X B. a we define
(WK)a = {w ~ WKIwX K = xK}.
Here WK = NK/K. There is a short exact sequence of topological groups
e - - A ~ (WK)[ ~ (WE) ................ e
where A denotes the group of deck transformations of X~, and hence
A ~ ~,(xK). The group (WK)* is a Lie group (not necessarily compact)
which acts on the universal covering X of ~ by an action which
X K" covers the action of (WK) on For the details of the construc-
tion of (WK)* we refer to Section 5 of [8]. Observe that the groups (WK)
126
and (WK)~ in fact depend on the actual geometry of the G-space. When
we find it necessary to emphasize this fact we will use the following
more complete notation:
(WK)~ = W(X~), and
= W*(X).
Using the more complete notation we may write the above exact sequence as
By ~o(WK)~ we denote the group of components of
Whitehead group of the discrete group ~o(WK)~.
We may also think of the direct sum over C(X)
(WK)~ and Wh(~o(WK) ~)
as a double direct sum
is the
E E Wh(~o(WK)~) (2) (K)
where the first direct sum is over all conjugacy classes (K), of closed subgroups
of G, for which X K + ~, and the second direct sum is, for a fixed K represent-
ing the conjugacy class (K), over the set of NK-components of X K, with one
connected component X K representing the NK-component (NK)X K.
The isomorphism ~ is defined as follows. Let s(V,X) ~ WhG(X) be an
arbitrary element in WhG(X). Thus (V,X) is a finite G - CW pair with
i: X ~ V a G-homotopy equivalence. Let K be a closed subgroup of G and X K
a connected component of X K, and let V K be the corresponding component of
V K. We denote
V >K = {v ~ vKIK ~ Gv } ¢~
-(vK,x K U V~ K) ~ is a finite (WK)~ - CW pair, such that (WK) Then
on vKo; - (X~ U V >K)a , and t h e i n c l u s i o n
acts freely
j: X K U V >K ----+ V K
is a (WK) -homotopy equivalence, see [8] Corollary 4.5 and Corollary 8.5b. Let
V K be a universal covering space of V K and let U ~ V >K be the induced
universal c o v e r i n g space of X K U V >K. Now (vK,x K U V >K) i s a f i n i t e
(WK)* CW pair, where (WK)* acts freely on - (X a U V ), and the inclusion
I: "'~-X K U V >K + V WK
127
is a (WK)~-homotopy equivalence, see [8], Theorem 6.6 and Corollary 8.6.
We now consider the chain complex
c(vK,x K U V >K) (3)
where Cn(A,B) = Hn(An U B,A n-I O B;Z) and A n denotes the equivariant n-skeleton
of A, (here, the (WK)~-equivariant n-skeleton), and Hn( ;Z) is ordinary singular
homology with integer coefficients. We have that (3) is a finite acyclic chain
complex of finitely generated free based Z[~ro(WK)~]-modules, see [8], Section 9.
Hence (3) determines an element in the Whitehead group of ~o(WK)*, which we denote
by
• (v,x)~ =
The isomorphism
"c(c(vK'xKe~ o~ U v>K))~ ~ Wh(~o(WK)~).
is given by
~(s(V,X))K, ~ = ~(V,X)~.
Here we think of the right hand side of (i) as given in the form (2), and
#(s(V,X))K, ~ denotes the (K,~)-coordinate of ~(s(V,X)). We shall also denote
~(s(V,X)) = ~(v,x).
Observe that we have ~(V,X)~ = 0 unless
V K~ - (X~ U V >K)~ + #.
2. Euler characteristics
Let X be a finite G - CW complex and let
Then we have
X (K) = {x ~ Xl(K) ~ (Gx) } = GX K
and
X >(K) = {x ~ XI(K) ~ (Gx) } = GX >K
where X >K = {x ~ XIK ~ Gx}. Now let
then define
K be a closed subgroup of G.
X K be a connected component of X K. We
128
X (K) = {x e x(K) IGx n X K + ~}.
Then we have
X (K) = GX K.
Furthermore we define
X >(K) = X (K) n X >(K)
and it then follows that
X>(K) = GX >K
where X >K = X K n X >K.
For any n ~ 0 we set
Vn,K,~(X) =~{G-n-cells of type G/K in x(K)}.
Another way to express this is that Vn,K, (X)
n-cells in X (K)~ - X >(K)~ , and hence Vn,K, (X)
cells in
equals the number of G-equivariant
equals the number of ordinary n-
(X~ K) - x>(K))/G = x(K)/G _ x>(K)/G.
We also have that Vn,K, (X) equals the number of (WK) -equivariant n-cells in
X K - X >K, i.e. the number of ordinary n-cells in
(X~ - x>K)/(WK) = xK/(wK) - x>K/(wK)~ a"
We now define
s
X~(X) : I (-l)nvn,K, ~(X)
n=O
where s = dim X. It follows from the above discussion that we in fact have
x~(X) : x(x(K)/G,x>(K)/G) : x(X[/(WK) ,x~K/(wK) ). ( i )
It is immediate that the following holds.
129
LEMMA. Let f: X + Y be a G-homotopy equivalence.
all (K,~). (Here ~Y~(~) denotes the component of
-K -K Then xa(X) = Xf(a)(Y) for K f X K Y that contains ()-)
3. Statement of the product formula
Let G and P be compact Lie groups. Let f: X + X' be a G-homotopy
equivalence between finite G - CW complexes and let h: Y + Y' be a P-homotopy
equivalence between finite P - CW complexes. Then the equivariant Whitehead
torsion ~(f x h) of the (G × P)-homotopy equivalence
f x h: X x Y > X' x y'
is given as follows. Given a connected component A ~K of a fixed point set X K
and a connected component Y~ of YQ, where K and Q are closed subgroups of
G and P, respectively, we have the connected component X K~ × Y~ = (X x y)KXQ~x~
of (X x y)KXQ. The (KxQ,axB)-coordinate of ~(f x h) is given by
(f x h)~x SKxQ = x~(Y)i...~(f)K + . . xK(x)j,~(h)Q. (i)
Here i: ~o(WK)~ + ~o(WK)~ × Wo(WQ)~ and j: ~o(WQ)~ + ~o(WK)~ × Wo(WQ)~ denote
natural inclusions. Furthermore any coordinate ~(f x h)$ of ~(f x h), the
where (S,~) is not of a product form as above, equals zero.
4. Proof of the product formula
We shall begin by proving the following fact. Given any element s(V,X)
WhG(X) and any finite P - CW complex B the equivariant Whitehead torsion
~(V x B,X x B) of the (G x P)-pair (V x B,X x B) is given as follows: If X K
and B~ are connected components of X K and B Q, respectively, we have that
KxQ = x~(B)i,~(V,x)K • (V x B,X x B)~x~ (i)
and T(V x B,X x B)$ = 0 whenever (S,7) is not of a product form.
The very last statement is easily seen to be true for the following reason.
Every isotropy subgroup occurring in V x B is of the product form Gv x Pb'
and therefore (V × B) S - ((X x B) S U (V x B) >S) = ~ and consequently
T(V x B,X x B)$- = 0, for each component (X x B)$- of (X x B) S, if S is not
a product of a closed subgroup of G and a closed subgroup of P.
Now consider a subgroup of G × P of the form K x Q, where K and Q are
closed subgroups of G and P, respectively. Then we have
(X x B) Kxq = X K x B q.
Moreover any connected component
niKx Q X K ~ x B = (X x ~'~x$ '
130
(X x B)~KxQ of (X x B) KxQ is of the form
where X K~ and B~ are connected components of X K and B Q, respectively, It now
remains to prove that (i) holds.
By definition
~(V x B,X x B)KXQ~x$ ~ Wh(~o(W(K x Q))~×~) (2)
is the torsion of the chain complex
C((V x B)ax~,( X x KxQ ..... °J~x$~>(K×C)~ B)ex ~ U ~V ^ j (3)
which is a finite acyclie complex of finitely generated free based Z[~o(W(K × Q))~× ]-
modules. Observe that we have
((V B) K×Q (X B) K×Q ">(KxQ)l -~x~ ' ~×~ u (V × x x B)~x ~ ,
= (V~ × BQ,x K ~ V >K ~ V K ~Q) x B U ~ x B U ~ x B
Q >Q = (V~,X~ U v~K)x (B$,B~).
It follows that the chain complex (3) equals the chain complex
C((vK,x Ka ~ U V >K)~ x (B~,B~)) (4)
which is isomorphic to the chain complex
c(vK,x K U V> K) ®Z Q >Q )' (5) ~ ~ C(B~,B~
It is easy to see that (W(K x Q))~xB = (WK)~ x (WQ)~,~ and moreover we have a
canonical isomorphism of rings
Z[~o(WK)~ x ~o(WQ)~] m Z[~o(WK)~] ®Z Z[~o(WQ)~]" (6)
Taking into account the canonical ring isomorphism (6) we have that the chain
complexes (4) and (5) are isomorphic as based chain complexes over the ring (6).
131
All in all it follows that the torsion of the chain complex (3) equals the torsion
of the chain complex (5).
For simplicity we denote
C = c(vK,x K U V >K)
C' Q >Q = C(B~,B~ )
and set ~ = no(WE)* , ~' = ~o(WQ)~, R = Z[~] and R' = Z[~']. It follows by the
Product Theorem in [i0] that the torsion of the chain complex (5), i.e., the
R ®Z R' complex C ®Z C', is given by
• (C ®Z C') = XR,(C')i,~(C) (7)
where i...: Wh(~) -~ Wh(~ × ~') is induced by the natural inclusion i: ~ + ~ x ~'
and XR,(C') denotes the Euler characteristic of C' as an R'-complex. But we
have that
×R'(C') = XR,(C(B~,B~Q) ) n ~
= x(BQ/(WQ) g, B~Q/(WQ) g)
= ~(B)
where the last equality is given by (2.1). Since T(C) = ~(V,X) K we have that
(7) shows that the formula (i) holds as claimed.
Now let f: X + X' be a G-homotopy equivalence between finite G - CW com-
plexes. By the equivariant skeletal approximation theorem (see Theorem 4.4 in
[13] or Proposition 2.4 in [6]) we may assume that f is skeletal. The geometric
equivariant Whitehead torsion of f is then by definition
t(f) = s(Mf,X) ~ WhG(X) ,
where Mf denotes the mapping cylinder of f. (In [7], Section 3 the element
t(f) is denoted by ~g(f).) On the algebraic side we use the notation
• (f) = ~(Mf,X) = ~(t(f))
132
for the equivariant Whitehead torsion of f. Let B be any finite P - CW
complex and consider the (G x P)-homotopy equivalence f x idB: X x B ~ X' x B.
The mapping cylinder of f x id B equals Mf x B and hence
~(f x idB) = ~(Mf x B,X x B).
Therefore we obtain from (i) that
~(f x id~)Kx~b ~x# = X~(B)i"t(f)~,, . (8)
We are now ready to complete the proof of the general product formula.
the (G x P)-homotopy equivalence f x h: X x y + X' x y' as a composite
f x h = (id X, x h) o (f x idy)
and use the formula for the geometric equivariant Whitehead torsion of a composite
([7], Proposition 3.8) to obtain
t(f x h) = t(f x idy) + (f x idy)ilt(idx,~ x h). (9)
Applying the isomorphism
of {(t(f x h)) = ~(f x h)
to (9) and considering the (K x Q,~ x $)-coordinate
we obtain
~(f x h)K~ ~n = ~(f × id)K~ + (~(f x id),it(id- × h)) KxQ (I0) ~ ~ ~ ~ x ~ "
Using a naturality property of the isomorphism ~ we now obtain
"~f(~)×~"
Here (fK id),: Wh(~oW*(xKf~") "-~ ~ "'~') is × × ~oW*(YQ)) + Wh(~oW*((X')f(~)) x ~oW~(y~)
induced by the map fK x id: X K x Y~ ~ (X')K Q )f(~) ~ f(~) x y , where (X' K denotes
the component of (X') K that contains f(xK). By (8) we have that
T(id x h) KxQ = -K ')ji~(h)~ f(=)×S xf(~)(x
where j ~oW*(yr-~) ' x ~oW*(y~) : ÷ ~oW*((x')f(~))
., -K ' ~(X) (f~ × id), o j, = 3,, and Xf(~)(X ) =
obtain that
denotes the natural inclusion.
by the Lemma in Section 2 we now
We write
Since
(fix id 1 idx (12
133
Applying the basic formula (8) to the first term on the right hand side of (Ii)
and using (12) we see that the formula (Ii) establishes the product formula.
5. Proof of Theorem A and Corollary B
Assume that P is a finite group and let B be a finite P - CW complex
such that x(B~) = 0, for each component B~ of any fixed point set B Q, It Q >Q
then follows that also x(B Q) = 0 and~hence\ x(B$,B~ ) = 0. Using (2.1) and
the fact that acts freely on we now obtain
1 y(B Q n>Q~ = ~ , , . ~,~ ~ = 0
for each component B~ of any fixed point set B Q.
Now let f: X + Y be a G-homotopy equivalence between finite G - CW com-
plexes, where G is a compact Lie group. It then follows by the product formula
(3.1) (or in fact by the simpler formula (4.8)) that the (G x P)-homotopy equiv-
alence f x id: X x B ~ Y × B has algebraic equivariant Whitehead torsion equal
to zero, i.e., T(f x id) = 0. Since ~ in (i.i) is an isomorphism we also have
that t(f x id) = 0 e WhG×p(X × B), and therefore f × id: X × B ÷ Y × B is
a simple (G x P)-homotopy equivalence, by TheoremS.3.6' in [7]. This completes
the proof of Theorem A.
In the case when G = P, a finite group, we thus have that f × id: X x B
Y × B is a simple (G × G)-homotopy equivalence. It is an easily established
geometric fact that if one restricts the transformation group to any subgroup
H of G x G one still has that the H-map f x id: X x B ~ Y × B is a simple
H-homotopy equivalence. In particular this applies to the case when H is the
diagonal subgroup of G x G, i.e., in the case when we are considering X × B and
Y × B as G-spaces through the diagonal G-action on them. Taking B to be the
unit sphere S(V) in a complex unitary representation space V of G we see
that Corollary B holds. D
6. Equivariant Whitehead t prsion of the join of two equivariant homotopy
equivalences
In this section we denote I = [-i,i]. The join of X and Y is by defini-
tion
X * Y = (X x y × I)/~
134
where stands for the identifications (x,y,-l) ~ (x,y',-l) for any x e X
and all y,y' ~ Y and (x,y,l) ~ (x',y,l) for any y ~ Y and all x,x' ~ X.
The join X * Y has the quotient topology induced from the natural projection
p: X x y x I + X * Y, and we denote p(x,y,t) = [x,y,t]. If X is a finite G - CW
complex and Y is a finite P - CW complex, where G and P are compact Lie
groups, then X * Y is a finite (G × P) - CW complex. We have the natural
imbeddings
i : X ~X*Y
i+: Y ----~ X * Y
Jo: X x Y ----+ X * y
_ ~ ~ X defined by i (x) = [X,Yo,-i ] and i+(y) = [Xo,Y,l] where Yo ~ Y and x °
are arbitrary, and Jo(x,y) = Ix,y,0], for all x ~ X and y ~ Y. Let ~i:
G x p + G denote the projection onto the first factor• Then i is a skeletal
co-~l-ma p from the G - CW complex X into the (G x p) - CW complex X "~ Y, i.e.,
we have i_(~l(g,p)x) = (g,p)i_(x) for all (g,p) ~ G × P and x ~ X. Hence i_
induces a homomorphism
i_,: WhG(X)÷ WhG×p(X * y)•
This homomorphism is defined as follows. By changing
through ~I: G × P + G we obtain a homomorphism
!
~i: WNG(X) + WNGxp(X).
X into a (G × P)-space
!
( I t is not difficult to see that ~ is a monomorphism.) Let us denote by
1 : X ~ X * Y the inclusion i when X is considered as a (G x P)-space
through ~i • Then i is a (G × P)-map and induces a homomorphism 1 ,:
WhGxp(X) ~ WhGxp(X * Y)• We now define
!
i_, = I_, o ~i"
Similarly i+ is a co-z2-ma p from the P-space Y into the (G x P)-spaee X * Y,
where ~2: G × P ÷ P is the projection onto the second factor, and the induced
homomorphism
i+,: Whp(Y) + WhGxp(X * y)
is defined in complete analogy with the above definition of i_,. Finally the
t35
(G × P)-imbedding Jo induces a homomorphism
Jo*: WhG×p(X x y) + WHGxp(X * y).
Now let f: X + X' be a G-homotopy equivalence and h: Y ÷ Y' a P-homotopy
equivalence, where X' and Y' also denote finite G and P, respectively,
CW complexes. Then we have.
PROPOSITION C. The equivariant Whitehead torsion t(f * h) ~ WGxp(X * Y) of the
(G x P)-homotopy equivalence f * h: X * Y + X' * Y' is given by
t(f * h) = i ,t(f) + i+,t(h) - Jo,t(f x h). (i)
Proof. Let us denote Z = X*Y and Z' = X'*Y' and define
Z = {[x,y,t] * ZI-I < t < 0}
z+ = {[x,y,t] ~ zi0 < t < i}.
' defined similarly. Then we have Z = Z U Z+ and The spaces Zi and Z+ are
Z_ n Z+ = X x y x {0} = X x y, and Z' has an analogous decomposition. By the
sum theorem for equivariant Whitehead torsion (see [7], Theorem II.3.12) we have
t(f * h) = j_,t((f * h)_) + j+,t((f * h)+) - Jo,t(f x h). (2)
I Here (f * h)_: Z + Z' and (f * h)+: Z+ ÷ Z+ are the (G x P)-maps induced by
f * h: Z + Z', and j_: Z ÷ Z and j+: Z+ + Z denote the inclusions. Now let
k : X + Z_ be the natural (G x P)-inclusion defined by k (x) = [X,Yo,-l], where
Yo ~ Y is any element in Y, and define a (G x P)-retraction rl: ZJ ~ X' by
r'[x',y',t] = x', where -i < t < 0 and x' ( X', y' ~ Y'. Then we have
f = r' o (f * h) o k : X > X' (3)
where f is considered as a (Gxp)-homotopy equivalence. (The factor P of
G x p acts trivially on X and X'.) The torsion of f: X + X' when considered !
as a (G x P)-map equals ~i(t(f)) ~ WhGxp(X), where t(f) ~ WhG(X) is the torsion
of the G-homotopy equivalence f. Applying the formula for the torsion of a
composite map (see [7], Proposition 3.8) to (3) we obtain
~t(f) = t(k_) + k[~t<(f * h) ) + ((f * h) o k_),it(rl). (4)
But Z collapses (G x P)-equivariantly to X. (This follows for example from
136
Corollary II.l.9 in [7].) Thus k : X + Z is a simple (G × P)-homotopy equiva-
lence and hence t(k_) = 0. Similarly t(r') = 0. Since j o k = 1 : X + Z
we now obtain from (4) that
!
* = = l_,~it(f) = i ,t(f). j_,t((f h)_) j_,k_,~t(f)
Similarly we also obtain j+,t((f * h)+) = i+,t(f). Making these substitutions in
(2) we see that we have proved (i). u
COROLLARY D. Let the assumptions be the same as in Theorem A. Then the equivariant
Whitehead torsion of the (G x P)-homotopy equivalence f*id: X * B + X' * B is given
by
t(f * id) = i,t(f) (5)
where i,: WhG(X) + WhGxp(X * B) is induced by the natural inclusion i: X + X * B.
In case G = P, a finite group, and we consider f*id: X * B + X' * B as a G-homotopy
equivalence, where X * B and X' * B have the diagonal G-action, the same formula
(5) holds but now as an equality in WhG(X * B).
7. Torsion of smash products and reduced joins
In this section we give simple explicit formulae for the equivariant Whitehead
torsion of the smash product and the reduced join of two equivariant homotopy
equivalences. First we need the following geometric result.
LEMMA E. Let (X,X) be a finite G - CW pair such that X collapses equiva- O O
to {Xo}, where x ° e (X~) G.-_ Then the natural projection p: X ÷ X/X ° riantly
is a simple G-homotopy equivalence.
Proof. Clearly p: X + X/X is a skeletal G-map. In order to prove that p o
is a simple G-homotopy equivalence we need to show that S(Mp,X) = 0 e WhG(X).
We shall show that Mp in fact collapses equivariantly to X. Let Cl,...,c k be
all the equivariant cells of X - X ordered in such a way that dim c. < dim c. o i J
implies i < j. Let us denote
X i = X ° U c I U...D ci, 1 < i < k,
and Pi = PlXi: Xi + Xi/Xo' 0 < i < k. By a direct use of the definition of an
equivariant elementary collapse we now have that for each i, 1 < i < k, there
is an equivariant elementary collapse
Mpi U X ~Mpi_l U X.
137
Thus M collapses equivariantly to M U X = CX U X. But since X o col- P Po o
lapses equivariantly to {Xo} it follows that CX ° U X collapses equivariantly
to C{Xo} U X, see Lemma 3.1 in [7]. Since C{Xo} U X collapses equivariantly
to X we have completed the proof, o
Using the above Lemma and the sum theorem for equivariant Whitehead torsion we
can now prove the following.
PROPOSITION F. Let f: (X,A) ÷ (Y,B) be a G-map between finite G - CW pairs
such that f: X + Y and flA: A + B are G-homotopy equivalences. Then the
equivariant Whitehead torsion of the induced G-homotopy equivalence f: X/A + Y/B
is given by
t([) = p,t(f) - q,t(flA)
where p: X * X/A denotes the natural projection and q = plA: A + X/A is the
constant map q(a) = {A} ~ X/A for every a e A.
Proof. Let X U CA be the union along A of X and the cone CA on A,
and define Y U CB analogously. Then f induces a G-homotopy equivalence
f: X U CA ÷ Y U CB. Since both CA and CB collapse equivariantly to their
respective vertices v A and v B it follows that C(flA): CA + CB is a simple
G-homotopy equivalence and hence t(C(flA)) = O. Thus we have by the sum theorem
for equivariant Whitehead torsion (see [7], Theorem 11.3.12) that
t(f) = il,t(f) - io,t(fIA) ( 1 )
where il: X ~ X U CA and io: A ÷ X U CA
(X U CA)/CA = X/A and (Y U CB)/CB = Y/B
X U CA ~ Y U CB
I X/A , Y/B
denote the inclusions. Since
we now have the commutative diagram
where p and p' denote the natural projections collapsing CA and CB, respec-
tively, to a point. The maps p and p' are simple G-homotopy equivalences
by Lemma E and hence t(p) = 0 and t(p') = 0. Therefore the above commutative
diagram and the formula for the equivariant Whitehead torsion of a composite map
(see [7], Proposition II.3.8) together with (i) give us that
where
t(f) = p,t(f) = P, il,t(f) - P, io,t(flk) = p,t(f) - q,t(flA)
p = p o if: X + X/A is the natural projection and q = p o i ° = plA:
138
A + X/A equals the constant map q(a) = (A} ~ X/A for every a ~ A. u
Now assume that X = (X,x o) and X' = (X',x~) are two finite pointed G - CW
complexes and let f: X + X' be a G-homotopy equivalence such that f(x o) = X'.o
Similarly let Y = (Y,yo) and Y' = (Y',y~) be finite pointed P - CW complexes
and let h: Y + Y' be a P-homotopy equivalence such that h(y o) = y~. It then
follows that f and h are equivariant homotopy equivalences between pointed
equivariant CW complexes. Therefore we have that the smash product f A h:
X A Y + X' A Y' is a (G x P)-homotopy equivalence between two finite (G × P) - CW
complexes. Since X A Y = (X x Y)/(X V Y) we have by Proposition F that
t(f A h) = p,t(f x h) - q,t(f V h)
where p: X × Y ~ X A Y denotes the natural projection and q: X V Y ÷ X A Y is
the constant map onto the point [Xo,Yo] ~ X A Y. By the sum theorem the torsion
of the (G x P)-homotopy equivalence f V h: X V Y ~ X' V Y' equals t(f V h) =
il,t(f) + i2,t(h). Here X V Y = X × {yo} U {Xo} x y, and il: X + X V Y and
i2: Y + X V Y denote the natural inclusions. Thus the equivariant Whitehead
torsion of the smash product f A h: X A Y ~ X' A Y' is given by the formula
t(f A h) = p,t(f x h) - ql,t(f) - q2,t(h). (2)
Here ql: X + X A Y and q2: Y + X A Y denote the constant maps onto the point
[Xo,Yo] ~ X A Y, and p: X x y + X A Y is the natural projection as above.
In particular we have the following.
I COROLLARY G. Let f: (X,x o) ~ (X',x o) be a G-homotopy equivalence between finite
pointed G - CW complexes, where G is a compact Lie group. Assume that P is
a finite group and that B = (B,b o) is a finite pointed P - CW complex such
that ×(B~) = 0 for each component B~ of any fixed point set B Q. Then the
equivariant torsion of the (G x P)-homotopy equivalence f Aid: X A B ~ X' A B
is given by
t(f Aid) = -q,t(f) (3)
where q: X + X A B is the constant map q(x) = [Xo,bo], for every x ~ X. In
case G = P, a finite group, and we consider f Aid: X A B ÷ X' A B as a G-
homotopy equivalence the same formula (3) holds, now as an equality in WhG(X A B).
Proof. By Theorem A we have that t(f x id) = 0 ~ WhGxp(X × B). Therefore
we obtain from (2) that
t(f Aid) = -q,t(f) ~ WhG×p(X A P).
139
Now the second assertion follows directly when one recalls that q,: A ! A
WhG(X) + WhGxG(X A B) is defined by q, = q, o ~i' where q: X + X A B equals I
the map q considered as a (G x G)-map, and observes that res o ~ = id. Here
res: WhG×G(X) + WhG(X) denotes the map induced by restriction to the diagonal
subgroup of G x G, and G x G acts on X by having the second factor G act
trivially, o
Let us now return to the general case where G and P are compact Lie groups.
We shall consider the reduced join of f and h. The reduced join of X and Y
is by definition
X * Y = (X * Y)/(X * {yo} U {Xo} * Y).
We claim that the (G × P)-subcomplex X * {yo } U {Xo} * Y collapses equvariantly
to {xo}. This is seen as follows. Since X * {yo} equals the cone on X we
have that X * {yo} collapses (G × P)-equivariantly to {Xo} * {yo}, by Lerama II.
1.8 in [7]. Likewise {Xo} * Y collapses equivariantly to {Xo} * {yo}, which
then collapses equivariantly to {Xo}. Thus we have a commutative diagram
f*h X * Y ~ X' * Y'
X ~ Y f * h ~ X' * Y'
where the natural projections ~ and ~' are simple (G x P)-homotopy equivalences
by Lemma E. Hence we obtain, in the same way as in the proof of Proposition F,
that
t(f * h) = ~,t(f ~'~ h).
Applying Proposition C we now obtain
t(f * h) = ql,t(f) + q2,t(f) - Po,t(f × h) (4)
where ql: X + X ~ Y and q2: Y + X ~ Y are constant maps to the base point
{*} c X ~ Y and Po: X × Y + X ~ Y is given by Po(X,y) = [x,y,0] ~ X ~ Y. But
observe that Po is (G x P)-homotopic to the map sending (x,y) to [x,y,l] =
[Xo,Y,l] = {*}, i.e., to the constant map qo: X × Y ~ X * Y onto the base point
{*} ~ X ~ Y. By Lemmall.2.1 in [7] we have that Po* = qo* and hence we may write
(4) in the form
t(f ~ h) = ~±,t(f) + q2,t(f) - qo,t(f x h). (5)
140
Having established this formula (5) for the equivariant Whitehead torsion of
the reduced join of two equivariant based homotopy equivalences let us point out
that (5) is in fact already contained in the formula (2) for the equivariant
Whitehead torsion of the smash product of two equivariant based homotopy equiva-
lences. Namely, there is a natural (G x P)-homeomorphism X ~ Y ~ X A Y A S I, and
a double application of the formula (2) gives us exactly the formula (5). Thus we
have that the basic formulae are: the product formula, the formula (6.1) for the
(unreduced) join, and in the based case the formula (7.2) for the smash product.
8. Restricting equivariant Whitehead torsion to finite subgroups~ an example
We shall give an example which shows that equivariant Whitehead torsion in the
case of a compact Lie group G is not determined by the restrictions to all finite
subgroups of G. Examples of this kind are suggested by our product formula for
equivariant Whitehead torsion, but we are in fact able to give a very simple
example which only involves the use of the product formula for ordinary Whitehead
torsion.
Let f: X + X' be an ordinary homotopy equivalence between ordinary finite
CW complexes such that t(f) ~ 0 ~ Wh(X).
and consider the sl-homotopy equivalence
f x id: X x S 1 + X' × S I.
Since the action of S 1 on X x S 1
~,: WhsI(X
see [7], Theorem 2.7.
Let S 1 act on S 1 by multiplication
(i)
is free there is a natural isomorphism
x S i ) ~ ~ Wh((X × S l ) / S l ) = Wh(X)
We have $ , ( t ( f × i d ) ) = t ( f ) ~ 0 and h e n c e t ( f x i d ) + 0
WhsI(X x SI), and consequently (I) is not a simple sl-homotopy equivalence.
Before we continue let us observe that this fact already shows that Theorem A does
not hold when P is a non-finite compact Lie group and that Corollary B does not
hold when G is not finite.
We now claim that for any finite subgroup K of S 1 the K-homotopy equiva-
lence f x id: X x S 1 ÷ X' x S 1 has equivariant Whitehead torsion
t(f x id) = 0 ~ WhK(X x SI). (2)
In order to prove this claim we note that in this case we have $,(t(f × id)) =
t(f ' ), where × l d s 1 / K
141
%: WhK(x × s I) ~ ~Wh(X × (sl/K)) (3)
is an isomorphism. Since SI/K is a circle we have x(SI/K) = 0, and hence it
follows by the product formula for ordinary Whitehead torsion (see, e.g. [I],
Theorem 23.2) that f × id: X × (SI/K) ÷ Y × (SI/K) has Whitehead torsion
t(f x idsl/K ) = 0 ~ Wh(X x (SI/K)). Since ¢, in (3) is an isomorphism we also have
that t(f × id) = 0 ~ WhK(X × SI), which proves our claim that (2) holds. Thus we
have that (i) is a non-simple sl-homotopy equivalence which is such that when the
action is restricted to any finite subgroup K of S 1 becomes a simple K-homotopy
equivalence.
References
[i] M. Cohen, A course in simple-homotopy theory, Graduate Texts in Math. i0,
Springer-Verlag, 1973.
[2] T. tom Dieck, Uber projektive Moduln und Endlichkeitshindernisse bei
Transformationsgruppen, Manuscripta Math. 34 (1981), 135-155.
[3] T. tom Dieck and T. Petrie, Homotopy representations of finite groups, Inst.
Hautes Etudes Sei. Publ. Math. No. 56 (1983), 337-377.
[4] K.H. Dovermann and M. Rothenberg, The generalized Whitehead torsion of a G
fibre homotopy equivalence. (Preprint, 1984).
[5] H. Hauschild, Aquivariante Whiteheadtorsion, Manuscripta Math. 26 (1978),
63-82.
[6] S. lllman, Equivariant singular homology and cohomology for actions of
compact Lie groups, in Proceedings of the Second Conference on Compact
Transformation Groups (Univ. of Massachusetts, Amherst, 1971), Lecture
Notes in Math., Vol. 298, Springer-Verlag, 1972, pp. 403-415.
[7] S. lllman, Whitehead torsion and group actions, Ann. Acad. Sci. Fenn. Ser.
A 1 588 (1974), 1-44.
[8] S. lllman, Actions of compact Lie groups and the equivariant Whitehead group,
to appear in Osaka J. Math. (Almost identical with the preprint: Actions of
compact Lie groups and equivariant Whitehead torsion~ Purdue University 1983.)
[9] S. lllman, Equivariant Whitehead torsion and actions of compact Lie groups,
in Group Actions on Manifolds, Contemp. Math. Amer. Math. Soc. 36 (1985),
pp. 91-106.
[i0] K.W. Kwun and R.H, Szczarba, Product and sum theorems for Whitehead torsion,
Ann. of Math. 82 (1965), 183-190.
[ii] W. LHck, Seminarbericht "Transformationsgruppen und algebraische K-Theorie",
GSttingen, 1982/83.
[12] W. LHck, The Geometric Finiteness Obstruction, Mathematica G~ttingensis,
Heft 25 (1985).
142
[13] To Matumoto, On G - CW complexes and a theorem of J.H.C. Whitehead,
J. Fac. Sei. Univ. Tokyo Sect. I A Math. Vol. 18 (1971), 363-374.
[14] M. Rothenberg, Torsion invariants and finite transformation groups, in
Proc. Symp. Pure Math., Vol. 32, Part 1 (Algebraic and Geometric Topology),
Amer. Math. Soc., 1978, pp. 267-311.
Department of Mathematics University of Helsinki Hallituskatu 15 00100 Helsinki Finland
Balanced orbits for fibre preserving maps
of S I and S 3 actions
Jan Jaworowski
Abstract. Let G = S I or G = S 3 , and let p : Z + X be a bundle
with a fibre preserving action of G . Let q : V + Y be a vector
space bundle with a fibre preserving action of G . Let f : Z ÷ V be
a fibre preserving map. The paper studies the size of the subset Af
made up of the orbits over which the average of f is zero. The size
of Af depends on the cohomology index of the action on Z and on the
type of the action on V which can be described in terms of a Euler
number. The result can be viewed as an extension of a continuous version
of the Borsuk-Ulam theorem.
~. The average of a map.
Let G be a compact Lie group, let Z be a G-space, let V be a
finite dimensional representation space for G and let f : Z ÷ V be
a map. The average of f is the map Av f : Z ÷ V defined by
(Av f)x := / g-1(fgx)dg ,
where f denotes the Haar integral on G . The classical version of the
Borsuk-Ulam theorem says that for any map f : S n + ~n there is an
orbit {x,-x} over which the average of f (with respect to the anti-
podal ~2-actions) is zero. In [8], Liulevicius proved an extension
of the Borsuk-Ulam theorem for arbitrary free compact Lie group actions
on the sphere using the averaging construction. In [7] we studied the
set of points where the average of a map f : Z ÷ V from a single
G-space Z to a representation space V is zero. In this note we stu-
dy such a set in a fibre bundle setting : a single map f is replaced
by a fibre preserving map of a bundle p : Z + X over X whose fibre
is a sphere (or, more generally, a suitable manifolc~) with a free,
144
fibre preserving action of G ; and the vector space V is also re-
placed by a vector ~pace bundle. An alalogous extension of the Borsuk-
Ulam theorem for ~2-actions was done in [5], [6] and [9].
The average can be defined in the same way for a fibre preserving
map f : Z + V of a bundle p : Z ÷ X with an action of G to a
bundle q : V ÷ Y of representations of G . It has the following
properties:
(1.1) For any map f , Av f is an equivariant map.
(1.2) If f is equivariant then Av f = f
We say that f is balanced at x if (Av f)x = 0 . If f is
balanced at x then it is balanced over the entire orbit of x . Let
Af denone the set of points where f is balanced. It is an invariant
closed subset of Z and we have
-I (1.3) Af = A(A v f) = (Av f) 0 ,
where 0 is the zero section of q : V + Y .
If f : S n + R k , then Af = {x E S n fx = f(-x) } (with res-
pect to the antipodal involution).
~. The index and the characteristic homomorphism
A free action of the groups G = S ° , G = S I or G = S 3 is well
described by its characteristic class, or by the index of the action.
Let d = I , d = 2 or d = 4 according to whether G is the unit
spherein the field E of real numbers, • , complex numbers, • or
quaternions, ~ . The universal space EG for these groups is the
infinite dimensional sphere and the classifying space EG/G = BG is
the infinite projective space P ~ . The cohomology of BG is a poly-
nomial algebra on one generator. In the case • = R , it is ~2[cE]
generated by c R ~ HI(p ~; Z 2) ; for F = ~ or ~ = ~ , the
cohomology with integer coefficients is ~[c E] generated by c~ 6 Hdp • .
For simplicity, we will drop the subscript ~ and write c := c~
If G acts freely on a space Z then the characteristic class of Z d
is the image cE(Z) = c(Z) := (~/G)*c ~ H (Z/G) of c under a classi-
fying map ~ : Z + EG . The case of • = R , from our point of view,
was studied in [6] ; in this note we will deal with the cases E =
and F = E
We will write Z := Z/G for the orbit space of the action; and
if f : Z + X is a map, then ~ : Z + X will denote the induced map
of the orbit spaces.
The index for G = ~2 was defined by Yang [10] and Conner an~
Floyd [I]. Fadell, Husseini and Rabinowitz [2, 3] defined and studied
145
the index for compact Lie groups other than ~2 ' including non-free
actions. For G = S I or G = S 3 we will define it as follows: If Z
is a free G-space, then IndF(Z) = Ind (Z) is the largest n such that
cn(z) is an element of infinite order in Hdnz . The following pro-
perty of the index is often used:
(2.1) Proposition. If Z and Z" are two free G-spaces and f : Z ÷ Z"
is an equivariant map then Ind(Z) ~ Ind(Z')
To study fibre preserving actions, the characteristic class, or
the index alone, are not sufficient: one must take into account the
action of the cohomology of the base space on the cohomology of the
total space of the bundle. Just as it was done in [6] for G = ~2 '
one can define a "characteristic homomorphism" associated with the
action.
(2.2) Definition. Let G = S 1 or S = S 3 ; let Z be a free G-spa-
ce; and let p : Z X be a map such that the action of G on Z is
fibre preserving with repect to p . The characteristic homomor~hism
for p is the map
A A • '
pj = pj (Z) : HIX ~ HI+3Z
A
defi~ed by pjx := (~*x) oc j (Z)
!- Equivariant cohomology
We will be using the Alexander-Spanier cohomology with integer
coefficients and the Borel equivariant cohomology. If Z is a G-space
then Z G := EGXGZ , where EG is the universal space for G , G
acts on EGxZ by g(e,z) := (ge,gz) and EGXGZ := EGxZ The map
Z G ~ EG = BG induced by the first projection EGxZ ~ EG is a bundle
with fibre Z If G act trivially on Z then Z G ~ BGxZ.
The equivariant cohomology of Z is H~Z := H*Z G . If G acts
freely on Z then the map Z G ~ Z induced by the second projection
EGxZ ~ Z is a bundle with a contractible fibre EG ; hence
H~z ~ ~*~
If - denotes a one-point space then the constant map EG + •
induces an isomorphism H~(.) ~ H*BG . We will be identifying the
groups H*GEG = H*BG and H~(.) under this isomorphism. In our case
of G = S I and G = S 3 ,this ring is a polynomial algebra on the
generator c ~ Hdp F , the universal characteristic class.
Suppose that W is a representation space for G with dim~ = m
and let W o := W - (0) The map ~ : W G + BG induced by the first
~rojection is an orientable bundle with fibre W and it has its Thom
class U(W) ~ Hm(WG,WoG) C ~G(W,Wo)- The restriction U' (W) of U(W)
146
to W , U' (W) e H~W , corresponds to the Euler class of z under
the isomorphism 7" : Hm(.) ~ ~ . The Euler class of z will also
be called the Euler class of W and denoted by e(W) Of course,
e(W) = 0 unless k is a multiple of d = dim r . In our case of
G = S d-1 if m = dk Hm(-) is freely generated by c k and e(W) t t
can be characterized by an integer x(W) such that U' (W) = X(W)-cm(w).
This integer, X(W) , will be called the Euler number of W . Here c
is viewed as a class in ~G (.) = ~GEG .
(3.1) Lemma. Let Z be a free G-space (G = S I or S 3) and let W
be a representation space for G with dimrW = k and with the Euler
number X(W) Let f : X ÷ W be an equivariant map. then
f~U' (W) = X(W)'ck(z)
Proof. Let ~ : Z + EG be a classifying map for Z . Let
c ~ H~EG = H~(.) be the universal characteristic class. Because of the
isomorphism H*[ ~ H~Z , we can consider c(Z) = ~c . Let y : W + •
be the constant map. Since W is equivariantly contractible, the
diagram
W is commutative, ~ = fG7 G . Hence f~U' (W) = f~(x(W).ck(w))
= X(W).f~(y~ck(.)) = X(W)-(f~y~c) k = x(W)'(~c) k = X(W).ck(z)
Suppose now that q : V + Y is an orientable vector space bundle
with a fibre preserving linear action of G ; i.e., a bundle of rep-
resentations of G . If y ~ Y , let vy :=q-ly be the fibre of q
Let dimlY y = m . The map qG : VG ÷ Y induced by q is over Y 4
a bundle whose fibre over y E Y is q~ly = V~ and qY : V~ ÷ BG is
a bundle with fibre V y.
Let V be the complement of the zero section in V and let o
~G(V,V o) be the equivariant Thom class of qG " It is charac- UG(V)
terized by the fact that for each y e Y it restricts to the Thom
class U(V y) £ ~G(VY,v~) of the bundle qY : V~ + BG which, in turn,
restricts to the orientation class in Hm(VY,v~) in every fibre V y .
The restriction U~(V) of UG(V) to V will be called the equivari-
ant Euler class of V . For each fibre V y it restricts to the
Euler class U' (V y) . The Euler class is locally constant on Y . If
Y is connected, it is constant and, as for a single representation
space, is characterized by an integer, the Euler number.
147
(3.2) Proposition. If W is a representation space for G ( = S I
or S 3 ) free outside the origin the the Euler class of W is zero.
This proposition was proved in [7; (5.2) and (5.3)] . For the
standard (scalar multiplication) representation W = F k , the rest-
riction homomorphism Hdk'FkG ~ ,~k'o; ~ HdkFk is an isomorphism, hence
e(~ k) = ck(r k) Now if W is any representation for G free outside
the origin, with dim~W = dk , then there is an equivariant map
: r k + W and ~*U' (W) = U' (r k) ; therefore U' (W) is non-zero.
(3.2) Remark. If p : Z + X is a bundle with a fibre preserving
action of G = S 1 or S 3 , then Ind(Z x) is a locally constant
function of x ~ X ; and Ind(Z) ~ Ind(Z x) because a fibre inclusion
is an equivariant map. If X is connected, Ind(Z x) is constant; it
will be called the fibre index of Z
4. Main Result.
As usual, G = S I or G = S 3 , and F = • or r =E , respec-
tively, with d = dim~F
(4~I) Theorem. Let p : Z ~ X be a bundle with a free fibre preser-
ving action of G over a connected base space X , and let F be a
fibre of p . Suppose that Hdn~ ~ ~ is freely generated by cn(F)
and HIF = 0 for i > dn. Suppose that hi(X) operates trivially
on H*F . Let q : V ~ Y be an orientable vector space bundle of
orthogonal representations of G whose Euler class is non-zero and
with dimFV = k . Let f : Z ~ V be a fibre preserving map. Then the
characteristic homomorphism
^ • Hi+d(n-k)~f Pn_k(Af) : HIX
is injective for every i .
(4.2) Remark. This theorem applies, ~or instance, if the fibre F
is the unit sphere S d(n+1)-1 in F n+1 and the action of G on Z
and V is the standard scalar multiplication. It is a parametrized
(or fibrewise) extension of a theorem proved in [ 7] in the same sense
as the results of [5] and [6] are parametrized extensions of the clas-
sical theorems of Borsuk-Ulam and Yang.
(4.35 Corollary. The covering dimension of Af is at least
dim X + d(n-k) + d - I .
.dim X +d(n-k)~ ~ 0 and the orbit map This is so because n f
Af ~ Af is a bundle with fibre G = S d-~
(4.4) The kernel of a linear map. Suppose that p : W ~ X and
q : V ~ Y are vector space bundles over • with fibre dimensions
dimEW = n , dimEV = k , and suppose that f : W ~ V is a fibre pre-
148
serving linear map. If f is of a constant rank, then the kernel of
f is a subbundle of W of a fibre dimension (over £ ) at least n-k
and hence its total space is of a covering dimension at least
dim X + d(n-k) Corollary (4.3) can be used to obtain the same num-
ber as a lower bound for dim Kerf , even if the rank of f is not
constant, as follows. Suppose that the bundle is furnished with a norm.
With the standard scalar multiplication action, f is equivariant, and x
a non-zero vector w if W is in the kernel of f if and only if Ix---[
belongs to AflsW , where S W is the unit shphere bundle in W .
Then Ker f - (0-section) ~ AflsW x ~ and by (4.3) the covering dimen-
sion of Ker f is at least dim X + d(n-k) This lower bound can
also be obtained more directly, without using (4.1).
5. Proof of (4.1).
We can assume that f is equivariant; otherwise we can replace
f by Av f , as in Section I. Thus Af = f-10 , where 0 is the
zero section in V . A A
Let p = Pn_k(Af) : Hix ~ Hi+d(n-k)A . We will construct a tran-
sfer homomorphism, t : HqAf ~ Hq-d(n-k)x f, and show that tp = X ,
where is the Euler number of V ; by the assumption, X ~ 0 . The
construction of t will be similar to that used in [9].
By the continuity of the cohomology theory we are using, it suffi-
ces to show that for any invariant neighborhood N of Af in Z
there is a transfer map ~ : H*Af ~ H*X such that tNP N = X , where A PN = p(N) : H*X ~ H*N .
Consider the equivariant map of pairs f : (Z,Z-Af) ~ (V,V O)
and the excision map e : (N,N-Af) ~ (Z,Z-Af). Denote by j the
inclusion map Z ~ (Z,Z-Af) or V ~ (V,V o) Define t N to be the
composite map U f~U G (V)
tN : Hi+d(n-k)~ ~ nG'i+d(n-k)~ ~ H~+dn(N,N_Af)
• j* • .
[* H~+dn(z,Z_A ) ~ H~+dnz ~ Hi+dn~ ~# HiX .
In this sequence, fN : (N~N-Af) ~ (V,V o) is the restriction
of f to N We identify HGZ with H*[ (since Z is free) and
thus consider c(Z) as either a class in H~ or in H*Z . According-
ly, we replace p~ by p* The map p# is the Gottlieb "integration
along a fibre"; we quote some of its properties:
(5.1) Proposition [4, p. 40]. Let p : E ~ X be a bundle with a
fibre F such that Hi+mF = Hi( • ) for i ~ 0 . Then there exists
natural homomorphism p# : HP+mE ÷ HPx such that:
t49
(5.2) If X = • then p# is the given isomorphism HmF ~ H°(') ;
(5.3) If x ~ H*X and z E H*E then p~((p*x).z) = x.(p#z)
We continue with the proof of (4.1). Let x e H*X . Then
~pNx = p#j*e*-1~p~Nx).cn-k(N) • f~UG(V)) =
= p#j*((p~x)-cn-k(z).f*UG(V)) = p~((p*x).cn-k(z)-f*U~(V))
= x.p#(cn-k(z).f*U~(V))
We claim that f*U' (V) = xck(z) . It suffices to check this against
every fibre Z x over x e X . Let y = fx, let i x : zx+ Z and
i y : V y + V be the fibre inclusions and let fx : Z x V x ÷ be the
induced map. Then
ix*f*U~(V) = fx*iY*u~(v ) = fx, U, (V y) = ck(z x)
= iX*ck(z) = ix, ck(Z)
Therefore tNPN x = x'p#(cn-k(z)'xck(z)) = Xx'p#cn(z) Since
Hdn~ is freely generated by cn(F), for each fibre Z x , p#cn(zX) = I - n
by (5.2). Hence p#c (Z) = I and tN~NX = XX .
This completes the proof.
Re ferences
I. Conner, P. E. and Floyd, E. E.: Fixed point free involutions and
equivariant maps. Bull. Amer. Math. Soc. 66 (1960), 416-441.
2. Fadell, E. R. and Rabinowitz, P. H.: Generalized cohomological
index theories for Lie group actions with an application to
bifurcation questions for Hamiltonian systems. Invent. Math.
45 (1978), 139-174.
3. Fadell, E. R., Husseini, S. and Rabinowitz, P. H.: Borsuk-Ulam
theorems for arbitrary S 1 actions and applications. Trans.
Amer. Math. Soc. 275 (!982), 345-360.
4. Gottlieb, D. H.: Fibre bundles and the Euler characteristic.
J. Differential Geometry 10 (1975), 39-48.
5. Jaworowski, J.: A continuous version of the Borsuk-Ulam theorem.
Proc. Amer. Math. Soc. 82 (1981), 112-114.
6. Jaworowski, J.: Fibre preserving maps of sphere bundles into
vector space bundles. Proc. of the Fixed Point Theory Workshop,
Sherbrooke, 1980; Lecture Notes in Mathematics, vol. 886,
Springer-Verlag, 1981.
7. Jaworowski, J.: The set of balanced orbits of maps of S I and
S 3 actions. To be published in Proc. Amer. Math. Soc.
150
8. Liulevicius, A.: Borsuk-Ulam theorems for spherical space forms.
Proceedings of the Northwestern Homotopy Theory Conference (Evan-
ston, Ill., 1982). Contemp. Math. 19 (1983), 189-192.
9. Nakaoka, M.: Equivariant point theorems for fibre-preserving
maps. Preprint. A
10. Yang, C. T.: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo
and Dyson, I. Ann. of Math. 70 (1954), 262-282.
Jan Jaworowski
Department of Mathematics
Indiana University
Bloomington, IN 47405
U. S. A.
INVOLUTIONS ON 2-HANDLEBODIES
Joanna Kania-Bartoszynska Mathematical Institute Polish Academy of Sciences Sniadeckich 8, P.O. Box 137 00-950 Warsaw, Poland
In this paper we classify actions of Z 2 on orientable and nonorientable
handlebodies of genus 2. We use a method of splitting involutions on
2-handlebodies to involutions on handlebodies of lower genus. All of
the considered objects and morphisms are from the PL (piecewise linear)
category.
A 2-handlebody is a 3-manifold H which contains 2 disjoint, properly
embedded 2-cells DIeD 2 such that the result of cutting H along DIVD 2
is a 3-cell.
Involutions (i.e. homeomorphisms of period 2) are classified up to con-
jugation by a homeomorphism; i.e. two involutions h,g of a 2-handlebody
H are conjugate if there exists a homeomorphism f : H--~H such that
h = fogof -1
It turns out that the involutions on 2-handlebodies are classified by
their fixed-points sets together with their position in a handlebody.
More precisely:
Theorem
Two involutions h I and h 2 on 2-handlebodies H I and H 2 respectively are
conjugate if and only if there exists a homeomorphism of pairs
(Hi,Fix h I) ~ (H2,Fix h 2)
Possible fixed-points sets of Z2-actions on 2-handlebodies can be found
using Smith theory. It turns out that for every such set there is an
involution of 2-handlebody realizing it, Using the constructions de-
scribed in this paper we can verify that there are 17 conjugacy classes
of involutions on an orientable 2-handlebody and 28 classes on a non-
orientable handlebody.
The involutions are listed in the appendix.
The above theorem was already proved for O-dimensional fixed-points sets
by J.H. Przytycki (see [P-I]~ thm.2.1); for orientation-preserving in-
volutions with homogeneously 2-dimensional fixed-points sets it was
proved by R.B. Nelson (IN-I] and IN-2]).
For the rest of this paper let H denote a 2-handlebody (both orientable
152
or not), Hor-Orientable 2-handlebody, Hnon-nonorientable 2-handlebody,
Dn-n-disk (i.e. n-cell), T-solid torus, Ks-solid Klein bottle, ~-M~bius
strip. (M,h) denotes involution h on a manifold M. Fix h denotes a
fixed-points set of a map h.
To prove the classification theorem we shall split involutions on 2-
handlebodies to involutions on 3-disks, solid tori and solid Klein bot-
tles. To do this we look for a 2-disk D in H which is either preserved
by an involution or is disjoint with its image~ Then we analyse the
situation obtained by removing that disk and its image from H. The
existence of such disk has been proved by P.K. Kim and J.L. Tollefson
in the following lemma (see [K-T], lemma 3).
Lemma
Let h be an involution on a compact manifold M. Suppose that there
exists a 2-disk D in M such that ~ D lies in a given component ~i M of
a boundary ~M and ~D does not bound a disk in ~i M, Then there exists
a disk S properly embedded in M with the properties:
(1) ~s ¢ ~i M ,
(2) ~ S does not bound a disk in ~i M,
(3) either h(S) ~ S = ~ or h(S) = S and S is in
general position with respect to Fix h.
The proof can be found in [K-~ or in ~-~ • It is worth mentioning
that this lemma was generalized by W.H. Meeks III and S.T. Yau for
actions of any finite group of homeomorphisms; they used minimal sur-
face techniques. The purely topological proof was given by A.L. Edmonds.
Obviously, 2-handlebodies satisfy the assumptions of the Kim-Tollefson
lemma. Let h be an involution on Ho There exists a properly embedded
2-disk S with properties (1)-(3) of the lermna. Clearly, involution h
acts on H - (U Lsh(U)) , where U is a small regular neighborhood of S
in H which is either h-invariant (in the case of h(S) = S) or disjoint
with h(U) (in the other case).
It follows that any involution h : H --~H is obtained in one of the 5
constructions described below.
Assume first that h~S) ~ S = ~ We have to consider 3 cases depending
on the number of components of H - (S ~h(S)).
The result of cutting H along S v h(S) is a ball D 3.
153
Then h is obtained from an involution i of a ball D3 by an identification
of two pairs of 2-disks on the boundary of D 3 : D 1 with D 2 and i(Dl) with
i(D2), where
D IN D 2 = @ , Dj ~ Fix i = @ for j = 1,2 ,
i(D I) ~ D 2 = @ = D 1 ~ i(D 2)
To identify D 1 with D 2 and i(Dl) with i(D2) we use a homeomorphism
f : D 1 V i(Dl)---> D 2 v i(D2)
which commutes with the involution h.
Denote the result of this construction by
(D3,i)
D I ~ D 2
Notice that if f changes orientation we obtain an involution of an ori-
entable handlebody. If f preserves orientation we obtain an involution
of H non
Observe also that conjugacy class of the involution (D3,i)
D I ~ D 2
DI remains the same for the different choice of DI,D 2 and f as long as ] lies on the same side of Fix i as D. (j = 1,2), and f,f' are in the
] same orientation class.
II.
The result of cutting H along S ~h(S) has two components i.e.
H - (S v h(S)) = D 3 u M ,
where M is a solid torus T or a solid Klein bottle Ks.
In this case h is obtained from an involution i : D3--> D 3 and an invo-
lution j : M---~ M by identifying a 2-disk D 1 C ~ D 3 with a 2-disk
154
D2C ~M and by identifying their images i(Dl), j(D 2) using a homeomor-
phism f : DIKJi(DI)---> D2 ~ J(D2) such that
f'i = jof
D 1 and D 2 have to satisfy
D I ~ Fix i = 9, D 2 ~ Fix j =
Denote the involution obtained in such way by
(D3,i) ~ (M,j)
D 1 f D 2
Notice that if M = Ks we obtain an involution of a nonorientable 2-han-
dlebody as well as in the case when M = T and one of the involutions
i,j preserves and the other changes orientation. If M = T and either
both involutions i,j oreserve or both change orientation (D3'i)D~21
,j) is an involution of H or
Observe also that the conjugacy class of an involution obtained in this
construction does not depend on the choice of DI,D 2 and f.
~ D3
Fig. 2
III.
The result of cutting H along S ~ h (S) has three components:
H - (S~jh(S)) = D3~,*MI~JM 2 ,
i where both MI,M 2 are solid tori or both Mis are solid Klein bottles.
It is easy to see that in this case Fix h is equal to the fixed-points
set of an involution i on D 3. Involution h is conjugate either to a
central symmetry, to a line symmetry or to a plane symmetry in Fix h.
D 3 T
Fi~Lre 3
Fix i / S i ~ " ~
" ,#;_
/ I /
T
155
Now let h(S) = S
We can assume that h does not exchange sides of S. If it does then for
U - an h-invariant regular neighborhood of S in H we have
U ~ [-i,i] ~ S, where S = {0] S.
If we put S O = _G-I? ~ S then h(S0) = _~17_ X S so h (So) ~ S O = ~Y and h
could be obtained by one of the constructions described above. Again
we have to consider two cases depending on the number of components of
H - S ,
IV.
The result of cutting H along S is connected. Then
H - S = M ,
where M is a solid torus T or a solid Klein bottle Ks.
In this case h is obtained from an involution j : M--->M by an identi-
fication of a 2-disk DIC ~ M with a 2-disk D 2 C ~ M using a homeomorphism
f : DI---> D 2 commuting with j. Disks DIeD 2 are chosen in such way that
D I ~ D 2 = ~ , J(Di) = D i for i=1,2.
Denote this involution by
(M,j)
D I ~ D 2
X
156
Fig.4
Observe that for M = T the conjugacy class of (M,j) f depends only
D I = D 2
on the orientation class of f and fIFix j For M - Ks the conjugacy
class of the involutions obtained in this construction remains the same
for a different choice of f as long as both f,f' either preserve or T change local orientation on Dis and either both preserve or change local
orientation on Di~ Fix j
Let D{,D~ C ~ M be the different choices of 2-disks such that
D~ ~ D~ = / j(Di) = D~ for i=1,2 i ' l "
Observe that if there exists an isotopy ~ of M taking D i to D~l (i=1,2)
which is also an isotopy of Fix j (and thus takes Di/~ Fix j to DinFix j)
then (M,j) f and (M,j) {of are conjugate. D I = D 2 D i : D~
V.
S disconnects H i.e.
H - S = M I~_TM 2 ,
where M. is either a solid torus T or a solid Klein bottle Ks. l Then h is obtained from involutions J l
an identification of a 2-disk D Ic ~ M I
a homeomorphism f : DI--- ~ D 2 such that
f°Jl = J2 °f
2-disks D iC ~ M i , i=1,2 have to be chosen so that
" MI---) MI ' J2 : M2 ---) M2 by
with a 2-disk D2C~ M 2 using
Ji(Di) = Di for i=1,2 .
F i x J2 ~ ' . . ~ig.5
157
T I Y 2
Clearly the conjugacy class of the obtained involution does not change
for a different choice of 2-disks DIC _~M I 'D~C~M2L -- if there exist
isotopies ~ i of (Hi,Fix ji ) taking (D i,D i t~ Fix ji ) to (D~,D~I /~ Fix ji )
for i=l, 2 .
So we have reduced our problem to pasting together involutions of 3-
disks, solid Klein bottles and solid tori, and to checking which con-
structions give us involutions from the same conjugacy class. Fortu-
nately, involutions on handlebodies of lower genus are already classified.
Theorem
Involutions of D 3 are orthogonal up to conjugation.
Proof follows from C.R. Livesay theorem (see eLi]) and Smith Hypothesis
proved by F. Waldhausen (see [Wa]). Q
Thus the only involutions of D 3 are central symmetry, line symmetry and
plane syrmnetry. Denote them by il,i 2 and i 3 respectively.
Let solid torus T be represented as
T = S Ix D 2 = ~ x D2/,~ , where
D e= {zeC : I zl~l] ,
(t,y) ~- (t+l,y)
Solid torus T can be also described as
T A = ~{ x D2/r..A , where
(t,y) "-k (t+l,-y)
Denote T* = T/~, where x ~ y iff ( x = y or x = j(y) ).
158
Theorem
Every involution of the solid torus has one of the following forms (up
to conjugation:
I) Involutions preserving orientation.
a) Jla : T--9 T , Jla(t,y) = (t,~) ,
= S 1 D 2 Fix Jla , T* = sly
b) Jlb : T--9 T , Jlb(t,y) = (t+½,-y)
Fix Jlb =~f , T* = S I× D 2
e) Jle : T--~ T , Jle(t,y) = (l-t,y)
Fix Jlc = DILjDI , T* = D 3 ( tJ denotes disjoint sum)
2) Involutions changing orientation.
a) J2a : T--~ T , J2a(t,y) = (t+½,y)
Fix J2a =~f , T* = Ks
b) J2b : T---~ T , J2b(t,y) = (t,~)
= SIx D 1 T* = D2X S 1 Fix J2b
c) J2c : T ) T , J2c(t,y) = (l-t,y)
Fix J2c = D2LjD2 , T* = DIx D 2
d) J2d : T--~ T , J2d(t,y) = (l-t,-y)
Fix J2d = two points
e) J2e : TA--gTA ' J2e (t'y) = (t+l,-y)
Fix J2e = Mb , T* = Ks
f) J2f : TA---~TA ' J2f (t'y) = (l-t,-y)
Fix J2f = p°intUjD2
Proof : see [P-2] , theorem 6.5, I O
Let solid Klein bottle be represented as
Ks = ~D2/,~ , where (t,y) ~ (t+l,7)
Theorem
Every involution of solid Klein bottle has one of the following forms
(up to conjugation):
I) K 1 : Ks---) Ks , Kl(t,y) = (t+l,-y)
Fix K 1 = S 1 , Ks ~ = Ks
159
2) K 2 : Ks----)Ks , K2(t,y ) = (t+l,y)
Fix K I = SIx D I , Ks* = SIxD 2
3) K 3 : Ks---) Ks , K3(t,y) = (t+l,-y)
Fix K 3 = ~ , Ks* = Ks
4) K 4 : Ks--'~Ks , K4(t,y ) = (l-t,-~)
Fix K 4 = DILJpoint
5) K 5 : Ks--> Ks , KD(t,y ) = (l-t,~)
Proof : see [P-2] , theorem 6.5 , II
Possible fixed-points sets of involutions on 2-handlebodies can be found
using Smith theory (see [FI], thm. 4.3 and 4.4). If we denote a fixed-
points set of an involution on 2-handlebody H by F then the following
have to be satisfied
rk Hi(F;Z2 ) ~ ~-- rk Hi(H;Z2 ) for any integer n
(H;Z2) ~ ~ <F;Z2) rood 2
Thus only the following cases may occur:
i) rk H0(F;Z2 ) = i , rk HI(F;Z2 ) 0
2) rk H0(F;Z2 ) = i , rk HI(F;Z2 ) = 2
3) rk H0(F;Z2 ) = 2 , rk HI(F;Z2 ) i
4) rk H0(F;Z2 ) = 3 , rk HI(F;Z2 ) 0
It is easy now to list all possible fixed-points sets of involutions on
2-handlebodies.
For each of these sets we check which constructions give us an involu-
tion with such fixed-points set. In all cases it is seen immediately
that the results of different constructions of an involution for a given
pair (H,Fix h) are conjugate. To show it we use a technique of cutting
H along some suitably chosen properly embedded 2-disk and along its image.
Example
Consider the case of (H, Fix h) = (Hor,pointt~ annulus).
Involutions with such fixed-points set can be obtained only by the fol-
lowing constructions:
160
I) Construction II , for (D 3, i I) , (T,J2b)
(H,h) = (D3,il) ~ (r,J2b) D I - D 2
i . e .
Fix j
I Fix i
T
F i g . 6
2) Construction IV , for M = T , j = J2f "
(H,h) = (T,J2f)DI ~ D2 , where f : DI~-> D 2 changes orientation on
D 1 but locally preserves it on DI~ Fix J2f
3) Construction V, for M 1 = T , Jl = J2f and M 2 = T , J2 = J2b ' i..e..
(H,h) = (T,J2f) ~ (T,J2 b) D I = D 2
We will prove that the three involutions described above are conjugate.
2f)D I ~ (T j it suffices to find (D3,il) f , 2b ) To show that (T,j ~ D2 DI = D2
a properly embedded 2-disk D C T disjoint with D 1 and D 2, disjoint with
Fix J2f and disjoint with its image j2f(D)
2f)Dl along D vJ2f(D) is a The result of cutting (Hor,h) = T,j ~ D2
disjoint sum of a solid torus with the involution which has an annulus
as a fixed-points set and a 3-disk with central symmetry.
2f)Dl could have been obtained by i) (see figure 7). Thus (T,j ~ D2
161
Figure 7
~ S [ ~ D I C FIx h
The proof that 3) r~-'l) is analogous : we cut (T,J2f)D~__ D (T,J2b)
1 = 2
along D~JJ2f(D) , where D is a 2-disk properly embedded in (T,J2 f) and
such that D~D 1 =~ , D~Fix J2f =Y ' D~J2f(D) =~
F i g . 8
T 1 T 2
For all the other fixed-points sets we Droceed in the same way.
It turns out that there are 17 conjugacy classes of involutions on an
orientable 2-handlebody and 28 conjugacy classes of involutions on a
162
nonorientable 2-handlebody.
The involutions with their fixed-points sets are listed in the aopendix.
This paper is based on my Master's thesis. I would like to express my
deepest gratitude to my advisor Stefan Jackowski and to J6~ek Przytycki
for their invaluable help.
APPENDIX
Observe first that there is no fixed-points free involution on H since
the Euler characteristics of H is odd.
Denote by U(Fix h) a regular neighborhood of Fix h in H. Pi denotes
a point, t./ denotes disjoint sum.
Fix h (H,h) Description of (H,h)
point p (Hor h I)
(Hno n , h i )
central symmetry in Fix h I
central symmetry in Fix h I
Pl L/ P2 U P3 (Hor'h2) (D3,il)DI _~D2(T,J2 d)
D 1 (Hor,h 3)
(Hnon,h 2)
line symmetry in Fix h 3
line symmetry in Fix h 2
DILj S I (Hor,h 4)
(Hnon,h 3)
(D3,i2) ~ (T,Jla) D I = D 2
(D3,i 2) ~(Ks,KI) D 1 = D 2
DILl DILj D I (Hor,h 5) (D3,i2) ~ (T,Jlc)
D 1 = D 2
D 2 , H - D 2 is (Hor,h6)
not connected (Hnon, h4)
plane symmetry in Fix h 6
plane symmetry in Fix h 4
D 2 , H - D 2 is
connected
trinion (i.e. D 2
with two holes)
H - Fix h
orientable
163
(Hor,h 7)
(Hnon,~5)
(D3,i 3) ~--~(T,J2a) D I = D 2
(D3,i 3) ~ (r,Jlb) D I = D 2
trinion
(Hor,h 8)
(Hno n , h 6 )
(T,J2b)DI ~ D2 , where f
changes orientation but
locally oreserves it on
D I ~ Fix J2b
(Ks,K2) f , where f D I = D 2
changes local orientation
on D I, preserves local orien-
tation on D I ~ Fix K 2
H - Fix h
nonorientable
(Hno n , h 7 ) (Ks,K2) f , where f D I = D 2
preserves local orientation
on D I and on DIF~ Fix K 2
Klein bottle with
a hole, R - Fix h
connected
(Hor,h 9)
(Hnon,h 8)
(TA'J2e)DI =~D2(TA'J2e )
K3)DI , where f (Ks, ~ D2
locally preserves orientation
on D I and on DI f%Fix K 3
Klein bottle with
a hole, H - Fix h
is not connected
(Hno n , h 9 ) K3)DI , where f <Ks, ~ D2
locally changes orientation
on D I but preserves it on
D I ~ Fix K 3
164
Moblus strip with
a hole, H - Fix h
is connected
(Hor,hl0)
(Hnon,hl0)
(TA,J2e)D~__f D2(T'J2b )
K3)DI , locally f (Ks, ~ D2
changes orientation on D I and
on D I ~ Fix K 3
MSbius strip with
a hole, H - Fix h
is not connected
(Hnon, h i i ) (Ks,K3) f , locally f D I = D 2
preserves orientation on D I
and changes it on D I~ Fix K 3
D2LjSI~ D I (Hor,hll)
(Hnon,hl2)
(D3,i3) ~ (T,J2b) ~I = ~2
(D3,i3)D~=f D2(Ks'K2 )
D2,, Mb (Hor,hl2)
(Hnon,hl3)
(D3,i~) ~ (TA,J2e) J DI m D2
(D3,i~)~"~ (Ks,K 3) DIE D 2
D2Lj D2Lj D 2 (Hor,hl3) (D3,i3)~-----~" (T,J2c) D I ~ D 2
point LJS I
U(Fix h) is
orientable
(Hno n , h14) (D 3,i I)DI~= D2(T,Jla )
point L_JS I U(Fix h) is
nonorientable
(Hnon ' hi 5 ) (D3,il) ~ (Ks,K I) D I = D 2
D2LjS I
H - (D2~jS I) is
orientable
(Hnon,hl6) (D 3,i~) ~ (T,Jla) D I ~-D 2
D 2 U S I
H - (D2~ S I)
neno rient ab I e
is
165
(Hno n ,h17) ( D3 i3)Dl =~D2(Ks,K I)
point~ SIx D I (Hor,hl4)
(Hnon,hl8)
(D3,il)D~__f D2(T'J2b )
(D3,il)D~=f D2(Ks'K2 )
point~-J~ (Hor ,h15)
(Hno n ,h19)
(D 3 i ) ~(TA,J2e) ' i Dl f D2
( D3 il)D~=f D2(Ks,K3 )
DIL_J SIx D I
U (Fix h) is
orientable
(Hno n ,h20) (D3,i2)~_f ~ (T,J2b) Ul - u2
DIL.j SIx D I
U(Fix h) is
nonorientable
(Hnon,h21) (D 3 ,i9) ~ (Ks ,K2) - D I = D 2
DI L.JMb
U(Fix h) is
orientable
(Hno n , h22 ) (TA,J 2e) (D3,i2)DI f D2
D I ~_JMb
U(Fix h) is
nonorientab le
(Hno n ,h23) (Ks,K3) (D3'i2)DI ~ D 2
point~ DI~/ D 2 (Hno n , h24) (D3,i3)~ (Ks,K 4) D I ~ D 2
point ~ D2, ~ D 2 (Hor,hl6) (T, j 2c ) (D3,il)DI f D2
point~ point hJ D 2
t66
(Hor,hl7) (D3,in) ~ (T,J2d) D I -= D 2
Doint L.J DILj D 1 (Nnon,~25) (D3,i2)D~f=f D2(Ks,K4)
point~-~point~D I (Hnon,hL6) (DB,il)Dl =~D2(Ks,K 4)
DILj DIL-jD 2 (Hnon,h~7) (D3,i2)DI =~D2(Ks,K 5)
DIL-I D2L-JD2 (Hn°n'hL8) (DB'iB)D~ =f D2(Ks'K5)
References
[FI]
[G-L]
[K-T]
[Li]
[M]
IN-l]
[N-2]
[P-1]
[P-2]
[Wa ]
E.E. Floyd, Periodic Maps via Smith Theory, in: A. Borel,
Seminar on Transformation Groups, Annals of >lath. Studies
46, Princeton, New Jersey 1960.
C. McGordon, R.A. Litherland, Incompressible Surfaces in
Branched Coverings, preprint P.K. Kim, J.L. Tollefson, Splitting the PL-involutions of
Nonprime 3-manifolds, Michigan Math. J. 27 (1980) C.R. Livesay, Involutions with Two Fixed Points on the
Three-sphere, Annals of Math., vol 78, N ° 3 (1963)
R. Myers, Free Involutions on Lens Spaces, Topology, vol
20, 1981 R.B. Nelson, Some Fiber Preserving Involutions of Orient-
able 3-dimensional Handlebodies, preprint
R.B. Nelson, A Unique Decomposition of Involutions of
Handlebodies, preprint
J.H. Przytycki, Zn-aCtions on Some 2-and 3-manifolds,
Proc. of the Inter. Conf. on Geometric Topology, P~,
Warszawa 1980. J.H. Przytycki, Actions of Z n on Some Surface-bundles
over S I, Colloquium Mathematicum vol. XLVII, Fasc. 2,
1982
F. Waldhausen, Uber Involutionen der 3-sDhare, Topology 8, 1969.
NORMAL COMBINATORICS OF G-ACTIONS ON MANIFOLDS
Gabriel Katz
Department of Mathematics, Ben Gurion University, Beer-Sheva 84105, Israel
This paper is the first in a series of papers developing a certain approach to
the following general problem. What are the relations between the combinatorics of
smooth G-actions on (closed) manifolds,in particular between the normal representa-
tions to fixed point sets, and global invariants (one can think about multisignatures
as a model example) of different strata in the stratification of a manifold by the
sets of points of different slice-types?
We have a pretty complete understanding of this problem for the special case
G = ~n' p an odd prime. The answer is in terms of nontrivial numerical invariants,
in particular~ it depends essentially on the first factor h I of the class number
of the cyclotomic field ~Ce2~i/P). In this way one gets, for example, interesting
conditions on the normal representations which can arise from exotic actions on
~CP)-h°m°i°gy complex projective spaces.
Our point is that to answer the question stated in the beginning it is very
useful to organize all compact smooth G-manifolds into a ring, identifying G-
manifolds having "similar" (bordant) combinatorial data with the "similar" lists of
global invariants [4]. This can be viewed as an analogue of the classical relation-
ship between the Burnside ring ~(G) (which is a result of a Grothendick's construc-
tion, applied to finite G-sets) and the set of equivalence classes of G-CW-complexes
[2]. The last equivaleneedeals with the Euler characteristics of different strata
in the stratification of the CW-complex by different orbit-types. So, roughly speak-
ing, the idea is to replace in the classical context the orbit-type stratification
by slice-type one, and the Euler characteristic of strata by the corresponding Witt
or multisignature invariants of different slice-types. For these purposes, one has
to create the "discrete" objects, playing the same role wfth respect to the new
context as finite G-sets do with respect to the classical one. We call these objects
normal G-portraits (.see Definitions A and B). Similar, but different, notations
were considered by Dovermann-Petrie in the framing of their G-surgery program [3].
Our definitions are more accurately adjusted to the category of smooth G-actions.
The present paper is the foundation of the program described above.
In fact, any smooth G-action on a compact manifold M produce~a normal G-
portrait WM" Roughly speaking, WM is a collection of the following data (which
satisfy certain relations): I) the list of subgroups G of G, i.e. the x
stationary groups of the points x in M; 2) the list of Gx-representations ¢x
168
G (~x is determined by the G~action on the fiber of the normal bundle v(M x, M)
G over x); 3) the list of groups which leave components of M X Cx E M) invariant and
which are maximal w~th respect to this property; 4) the partial ordering on the set
of components of M x x E M, induced by inclusion.
It is known that there is a significant difference between the possibility of
realizing data i), 3), 4) in the category of G-manifolds and in the category of
G-CW-complexes.
For examples the partially ordered set of subgroups of ~ ~p,q,r are pqr
distinct primes), represented on Figure I, is realizable as the set of stationary
groups on some connected ~ -CW-complex. But from the representation theory and pqr
data 2) it follows that this picture is not realizable on connected ~ -manifolds pqr
[3]. In contrast to this, the partially ordered set on Figure 2 can be realized
on a G-manifold. We assume that the inclusions of various stationary groups on
these diagrams correspond with the inclusionsof the closures of the appropriate
orbit-types of the action.
1
/\ 77
77pq 7/pr Pq
\ / \ Z
pqr
1
2Z
7Z pr
J 7/pq r
Fig. 1 Fig. 2
-action on the set of The idea here is simple: the comhinatorics of the G x Gx
components of M Gy ~y E M), containing the component of x E M , is the same as
the combinatorics of the linear G -action on the underlying space of the representa- x
tion @ (defined above). Basically, this observation is formalized in the notation "x
of a normal G-portrait Csee Definition B). General normal combinatorics are the
result of gluing combinatorics of linear representations together.
It turns out that the notation of normal G-portraits is adequate to describe
the combinatorics of G-actions. Namely, any compact smooth G-manifold determines a
normal G-portrait (Lemma 2).
Our main result (Theorem) states that any normal G-portrait ~ can be realized
by a smooth G-action on a compact manifold M .
169
Moreover, one can construct this manifold M~ with homology concentrated only
in dimension 2, and the closure of each set of a given slice-type also has a similar
homological structure.
Our construction allows us to "minimize" the fundamental groups of different
components of the slice-type stratification of M . This is important if one wishes
to use M as a basis for an equivariant version of Wall's construction [9] with the
purpose of realizing geometrically equivariant surgery obstructions.
If we make no restrictions on the dimension of M, then there is no difference
between the realization of a given normal combinatorial structure on a closed G-
manifold or on a compact G-manifold (with ~M realizing the same normal G-portrait
as M does).
Under certain weak orientability assumptions (all the representations ~x taken
to be SO-representations) one can prove (see Corollary) that such normal G-portraits
are realizable on G-manifolds of the homotopy type of a bouquet of 2-dimensional
spheres. Moreover, if all ~x are complex, then any fixed point set will also be
of the homotopy type of VkCS~).
These general results should be compared with more precise results obtained by
other authors in important special cases (of G-actions on disks). We would like to
mention two results of this sort. T. Petrie proved that any list of complex
representations (up to some stabilization*), satisfying some necessary Oliver type
conditions and Smith theory restrictions, are realizable as normal representations
to the G-fixed points on some G-disk for G-abelian [9] (c.f. Pawa~owski [8] and Tsai
[I0]). The geometrical construction that we use to prove our Theorem also requires
some weak (+S-dimensional) stabilization not "in the normal direction to fixed
point sets" as in [9], but in the "tangential one". In our approach we are flexible
with dimensions of fixed point sets, but rigid with codimensions and normal
representat%ons.
The second result is due to K. Pawalowski [8]. For finite Gp the following
conditions are equivalent: (i) for any smooth G-action on a disk D, the tangential
representations at any two G-fixed points are isomorphic, (i i) for any smooth G-
action on a disk D, all the components of D G have the same dimension, Ciii) all
the elements of G have prime power order. This theorem shows that for G with
all the elements of prime power order, the normal portraits of G-actions on disks
are the result from gluing a few copies of the G-portrait of a linear G-representa-
tion together.
Thus, it is well understood that normal G-portraits which are realizable on
contractible G-manifolds Con G-disks) satisfy quite strong restrictions (see, e.g.
*Unfortunately this stabilization destroys the original combinatorics of these representations.
170
[7]). In contrast to this, as we mentioned above, any normal oriented G-portrait
can be realized on a 1-connected G-manifold with non-trivial homology only in
dimension 2. This 2-dimensional homology group, as a g[G]-module, is not projective
in general (so, our construction does not assGciate a projective obstruction with a
given normal G-portrait as one might expect).
In [4] using the results of this paper we will show that any normal G-portrait
together with an arbitrary list of multisignatures (or Witt invariants), para-
metrized by ~, is realizable on smooth G-manifolds with boundary. For G-manifolds
with boundary this completes the algebraization of the general problem stated in the
very beginning. The analysis of closed G-manifolds is more complicated and leads to
different integrality theorems.
I am grateful to J. Shaneson for stimulating discussions and to K.H. Dovermann
and J. Shaneson for their help in making this text more readable.
Let M be a compact smooth manifold with a smooth right action of a finite
group G on it. We will describe a stratification of M, defined by the G-action.
Let H be a subgroup of G. Denote by °}~ the set {x C MIG x = H}, where
G is the stationary group of the point x. Let "M H be the closure of °M H in X
M. It is a closed and open subset in M H = {x 6 MIG x ~ H} and a compact manifold.
In fact, "b~ consists of those connected components of b~, which have a dense
subset with the stationary subgroup H.
Consider the set ~., which by definition is the connected component set
~0(~ °MH). If codim[~, "M K) > 1 for any "M H c "M K, then ~M coincides with HOG
Exam~!_e. Let G = ~12 and M = CP 4. Consider the G-action, which in homo-
geneous coordinates (z0:zl:z2:Zs:Z4) is given by the formula:
(Zo:Zl:Z2:Z3:z4)g = (Xz0:~2Zl:~4z2:~3z3:X9z4). Here g is a generator of ~12
and X = exp(~i/6). The components of the set M H, where H~I2 5is a subgroup,
are in one-one correspondence with the nontrivial eigenspaces in ~ of a generator
of H. Considering ('~-stratification we are selecting only components of M H
which have H as a stationary group of a generic point. k
Figure 3 describes ~ = ~0('~ "M g ), k = 0,1,2,3,4,6. The elements of ~ are
denoted by vertices of the graph and the inclusion of components one into another by
arrows (the directions of arrows are opposite to the inclusion). The right side of
the picture describes the partially-ordered set SCgI2) of all subgroups in ~12"
The horizontal arrows, pointing from ~ to S(~12) , associate with each component
the stationary group of its generic point.
_LL HcG
171
m
) / J
__> {gO}
{ }
--> {g}
Fig. 3_
s(zl2)
Now we are going to axiomatize the properties illustrated by this example very
much in the style of [3], [7]. We do this by introducing further structure on the
set ~.
Let G be finite and let S(G) denote the set of all subgroups of G. The
group G acts on SCG ) by the conjugation: Ad :H ÷ g-iHg for any H E S[G). g
Let ~ be a finite partially-ordered right G-set with a G-map p: ~ ÷ S(G).
The map p is consistent with the partial order > in S(G) in the following
sense: for any two elements ~ > B of ~, the group pea) is a proper subgroup
of p(6). As usual, > means > and ~.
Denote by G the stationary subgroup of ~ E
action on ~). We assume that p C~) c G .
(with respect to the G-
Moreover, an isomorphism class of an orthogonal representation ~ :p(~) + O(V )
is associated with every element ~ E ~, and we assume that the following two
properties hold.
I. The representations (~ } are consistent with the G-action on
172
in the following sense: for any a 6 ~ and g C G, the representation Ad
-I ~ P(~g) ' g ...... ~ P C~) ~ O[Va) is isomorphic to the representation ~g. In
particular, @a and 9a o Ad -i are isomorphic for any g E G . g
If. The representations {t } are consistent with the partial order in ~ in
the following sense. For any two elements a ~ B, in the canonical decomposition
of the pCB)-representation ReSp($)(~a) into the direct sum of the trivial summund
and its orthogonal complement, the latter is isomorphic to t~. By the definition,
for any maximal a E ~, the space V is O-dimensional.
Remark. In fact, property I describes a nontrivial relationship between G,
p(~) and ~ . Let N [p(a)] denote the subgroup of elements in the normalizer
N(p(a)) which preserve the character of *a under the action by conjugation. By
I, G has to be a subgroup of N [p(a)].
Property II implies V p(a) = {0} for any element a in ~.
Definition A. A partially ordered right G-set ~ with a G-map p (as above)
and with a list of representations {~ ) satisfying Properties I and II, we will a
call a discrete portrait of a G-action, or more briefly a G-portrait. One can find
this notion (with minor changes) in [3], [7] under the name of POG-set.
One can replace orthogonal groups O(V ) in the previous definition by the
classical groups SO(Va) , U(V ) (or any other classical Lie groups). The corre-
sponding discrete portraits of G-action will be called (correspondingly] G-portraits
with an oriented orthogonal or complex structure.
The following definition plays the central role in our considerations.
Definition B. A (discrete) G-portrait ~ is called normal if for every
E 7, the G-map p maps the partially-ordered G -set ~ = (B E ~IB > a} ~ >~
isomorphically onto I@ a l . H e r e a f t e r , I ~ t d e n o t e s A p a r t ~ ' a l l y - o r d e r e d G - s e t o f
subgroups o f G which are s t a t i o n a r y groups of v e c t o r s v E V with r e s p e c t to
the @~(pCa)) -ac t ion ( G ac t s on I ~ t by the c o n j u g a t i o n ) .
In the following lemma we are underlying a few properties of normal G-portraits.
Lemma I. For any normal G-portrait ~ the following holds:
i) for any three elements a,B,y E ~ such that a > y, B > y, there exists
unique element 6 E ~ with the properties 5 ~ a, 6 >_ ~ and p(~) = p(a) N p(B).
2) as an immediate consequence of i), for any ~ E ~, there is a unique maximal
element in the set
3) for any a C 7, there is not more than one element ~ E ~>a with a
given value P C~) = H E S(G).
4) for any two elements B >_ a, the group G B N G is the normalizer
173
N G (pC~)) of gO6) in G . C~
To p rove p o i n t 1) o f t h e lemma c o n s i d e r t h e s e t ~ y. By t h e d e f i n i t i o n o f
normal G-portrait it is isomorphic to I~yl and the isomorphism P:~>_y ÷ [~yl is
G -isovariant and order-preserving. So it is enough to show that if p(m) 2(6) C~ ~
are stationary groups of some vectors in V with respect to the p(y)-action, Y
then p(~) N p(~) is also a stationary group of this action. Consider subspaces
V p(~) V p(B) and V p(~)ApCB) in V . It is easy to see that the stationary group y ' y Y Y
of a generic point in V p(~)Np(B) is precisely p(~) N p(B). Y
Point 3) of the lemma just reflects the fact that the restriction of the map
at ~>~ is a one-one map onto l~I c S(G)
Point 4) is also quite simple. If p (B) is a stationary group of some vector
in V~ and g E NGaCO(8)), then p(Bg) = g-lpcB)g = PEg). Because Sg also
belongs to ~>~ for g E G , and because p($g) = ~(6), by point 3) one concludes
that Bg = B,-which means that g E G B. Now if g E G B N G , then p(Bg) = p(B).
So, g-lp(B)g = p(B) and g E NG(P(B)) N G~ ~ N G (gEB)). Lemma 1 is proved.
In particular, Lemma 1 shows that G-portrait in Figure 1 in the introduction is
not normal. More precisely, it is impossible to introduce any normal structure in
the partially ordered set of subgroups of %qr' described on Figure I.
Lemma 2. Every compact smooth manifoZd M with a .smooth G-action (G is
finite) determines a normal discrete G-portrait ~M" Ff the normal bundles
v('~,M) are oriented (or have a complex structure) for all H E S(G) and if G
acts on them o~entation-preserving (or preserving the complex structure), then ~M
will have oriented orthogonal (or complex) structure.
Proof. The manifold M determines the set ~ = ~0( ~ °MH) as it was HCS CG)
described above. An element ~ C ~ is associated with any connected component
°@ in °W~
By the definition, p(e) = H. The group G acts on ~ by permuting the
components {°MH}. So, the stationary group G of an element a £ ~ is, in fact,
the maximal subgroup in G keeping °~ invariant. One can check that the map
p:~ + S(G) is a G-equivariant map. The partial order in ~ is induced by the
inclusion of components {'~{} one into another. It is clear that this order is
consistent with the G-action on ~, and by the map p it is also consistent with
the natural partial order in S(G).
For x 6 °M~=, the G-action on M defines a representation :~x:H ÷ O(Vx)
(or an H-representation into S0(Vx) , U6Vx) ) in the fiber V x of the normal
~('~au M) over x. Because °M H is connected, the isormorphism class bundle of c~
~x does not depend on x 6 o~{ (and even on x £ "~). We put ~ = ~ for ~ X
some x 6 °M H. It follows easily from the Slice Theorem that Properties I and II
174
in Definition A both hold, as well as that the portrait ~ is normal ~see
Definition B).
Let ~ be a normal G-portrait. Denote by cd the real dimension of
Let cd(~) be max cd .
Now we are able to formulate the main result.
Theorem. a). Any normal G-portrait ~ is realizable on a compact smooth G-
manifold W of a G-homotopy type of a 2-dimensional CW-co,rplex. The boundary M
of W realizes the same G-portrait.
b). If ~ is oriented (has a complex structure) one can construct W and
all "W H [H E S(G)) to be oriented manifolds Ccorrespondingly all v('W H, W)
~ave equivariant complex st2~cture).
c). Assume ~ is oriented. Let Z~a denote the centralizer of the group
~a(p(a)) in" O(V ) (correspondingly in S0(V )or in U~Va~), and Zo@ ~ denote the
connected component of the unit in Z@a. Then one can assume the fundamental groups
~iC°M~ (~)) ~I L ~ J to be isomorphic to Z~/Zo~ ~, In particular, if all ~
are complex representations, then one can realize ~ on a manifold of a G-homotopy
type of a 2-dimensional CW-complex with one-connected components °W 0(~) °M ~(~)
for each a E ~ .
d). The dimension of W, satisfying a), b), c), can be any natural
n > cd~) + 5. If the condition ed = cdC~) implies G = p(a) and the condition
cd < cd(~) implies cda --< cd¢) - 5, then one can construct w of any dimension
n >_ cd[~), fn this case the G-portrait of M will ~e ~@, where
@ = {a E ~Icd = cd~)}, and ~I[=M ~)) ~ Z~/ZQt~ for any a E ~0. The set
"W ~(~) is a point for any ~ E @.
Corollary. a). Let ~ will be a normal oriented G-portrait. Let cd > 2 a for any ~ E ~ which is not a maximal element. Then ~ is realizable on a
compact oriented G-manifold of a homotopy type of a bouquet of 2-dimensional
spheres.
b). If, in addition, for any ~ E ~, ~0(Z@a) = i and for any two elements
> ~, cda-cd B > 2, then one can realize ~ on a manifold W of a G-homotopy
type of a 2-dimensional CW-con~plex and each component "W p (~) will be of homotopy
type of a bouquet of 2-dimensional spheres.
Before we will prove the theorem we need to describe a classifying space for
certain type of G-vector bundles. More precisely, let H be normal in G and
$:E(~) ÷ X be a G-vector bundle, satisfying the properties:
i) "E<~) H = X ,
2) G/H acts freely on X,
175
3) The H-representations in the fibers ~x are isomorphic to a given
representation ~:H ÷ 0CV),
Depending on context, 3) can be replaced by:
3a) $ is an oriented vector bundle, G-action on E C$ ) preserves the
orientation of fibers Sx' and ~ is an H-representation in S0(V).
3b) ~ is a complex vector bundle, G-action on EC~) preserves the complex
structure of fibers, and # is a unitary H-representation in U(V).
The natural problem to classify G-bundles satisfying 1)-3) was first studied
by Conner and Floyd [I]. An explicit classification of general equivariant bundles
of this type is given in [6]. See also [5] for abelian G. Actually, in the case
of G-vector bundles one may use an idea due to tom Dieck: the associated principal
A-bundle P over X A = 0[V), S0(V), U(V), etc ..... with the induced G-action on
it may be viewed as a A×G - space with a single orbit type H~ = {(a,g) 6 AxG I
g 6 H and a = ~(g)}. Thus using the slice theorem, P is the associated bundle H
over X/G of the principal NAX G (H~)/Hg-bundle P ~ with fiber AxG/H~. In fact
one has the following exact sequence 0 ÷ Z* ÷ NAxG(H~)/H ~ + G/H + 0, where Z~
is the centralizer of ~ in A. Note that NAx G (H~)/H~ is the group of kxG-
equivalences of AXG/H~. Hence NA× G (H~)/H~ can be viewed as the centralizer of
a representation ~: G + {group of A-equivalences of AxG/H~} ~ Ax H G. The last
group is in the same time the group of A-equivalences of V×HG and, in fact, is
isomorphic to the Wreath product __AIS n of A with the symmetric group S n ,
n = IG/HI. Let us denote NAx G (ll~)/H~ by Z~. Since X = P/A~ using the exact
sequence above, one has the following lemma.
L emma 3[6]. Isomorphism classes of G-vector bundles over a G-space X with
a single orbit type (H), satisfying the properties 1)-3) are in one-to-one
correspondence with the homotopy classes of lifts of a classifying map f:
X/G ÷ B(G/H) to BZ~ = B[NAxG(H~)/H~].
Now we will prove the following lemma.
Lemma 4. ~ere exists a connected 2-dimensional CG/H)-CW-complex X 2 with a
G-vector bundle ~ over it, satisfying the properties 1)-3) ~or 3a),3b)). The
fundamental group ~I(X2] is isomorphic to Z@/Z0~, where Z0@ denotes the
connected component of 1 in the centralizer Z~.
Remark 5. In the case, when the short exact sequence
0 ÷ Z~/Z09 ÷ Z~/Zo~ ÷ G/H ~ 0
splits, one can construct X 2 to be one-connected. In particular, if
H2(G/H; Zg/Z0~) = O, this can be done. Recall that in the case A = 0(V),
Z~/Zo~ = @ ~2' so that if, for example, ]G/H I is odd, X 2 can be taken simply-
connected.
176
Proof. The right G-action on VXHG will produce some representation
• :G + ISOA(V×HG).
Consider the fibration 8:BZ~ ÷ KCG/H,I) with fiber BZ9 induced by the
extension 0 + Z9 ÷ Z~ ÷ G/H ÷ 0.
Let T denote the fundamental group ~I(BZ~). Choose some finite presenta-
tion of T. Let y2 be a 2-dimensional connected CW-complex, realizing this
presentation. Let us take a map s:Y 2 ÷ BZ~ inducing an isomorphism of the
fundamental groups. By Lemma 3, s induces a G-vector bundle ~ over some space
X 2 over y2. This covering X 2 ÷ y2 (with fiber G/H), induced by the map
y2 @os ~ K(G/H.I), corresponds to the subgroup ~' in ~I~Y 2) which is the kernel
of the map (Oos),:~l~Y2 ] ÷ ~I(KCG/H,I)). So, ~I(X 2) is isomorphic to the
fundamental group of the fiber BZ~. The last group is isomorphic to the group
Z~/Zo~. Lemma 4 is proved.
Now we are able to prove the main theorem. The proof goes by induction.
Let ~ be a given normal G-portrait and @ a closed G-invariant subset in it
(by "closed" we mean tkat if ~ < ~ then B C @ for any a 6 @).
Suppose there exist a compact smooth G-manifold @W and its boundary @M
both realizing the same normal G-portrait @~, and the following list of properties
is satisfied.
in
i. There is a G-map ~:9~ -~ ~, such that:
a) ^ is onto and p(~) = pC&) for any cz 6 8~;
b) the partial order in @~ is the "pull-back image" of the partial order
7: for any ~,B 6 @z, c~ > B if ~ > ~;*
c) the map is a G-isomorphism of G-sets ~ -i(@) and @.
2. For any ~ 6 ~ the representations ~ and ~ are isomorphic
3. @W has a G-homotopy type of a 2-dimensional CW-complex and if ~ is
oriented, ~IL @- ..... "~P(~)) ~ ~I(@~MPC~)) ~ Z~/Zo~ ~ for every ~ E ^-i(@).
4. The dimension of @W can be any natural n ~ cd[~)+5. In the case when
the condition cd~ = cd[~) implies G~ = p (~) (~ £ z) and the condition
cd~ < cd(~) implies cd~ <_cdC~)-5, one can construct @W of any dimension not less
than cd(~).
5. If all 9^ are oriented orthogonal (or unitary) representations [in other
words, ~ is oriented (has complex structure)], then the G-action on @W is
orientation-preserving, moreover, all normal bundles v(~'W~ c~) ~ , @W) are oriented
(have a complex structure), and the G-action preserves this preferred orientation
(complex structure).
*but is not a one-to-one map, ~ = ~ does not imply ~ = B.
177
Let B 6 ~'-@ be an element, such that any element ~ < ~ belongs to @.
The inductive step will be to construct a new G-manifold @,W with the G-
portrait @,~ also satisfying all the properties 1.-5. for @ replaced by
@' = @ U ~G c ~ (~G denotes G-orbit of ~ 6 ~).
Using Lemma 4, we can construct a connected 2-dimensional G~-CW-complex X~ B
with G^-vector bundle ~^ over it, satisfying the following properties: B
2 3) the p(~)-representation in i) "E(~ )P(~) = X~; 2) G~/p(B) acts freely on X~;
the fibers of ~ is isomorphic to ¢~. Moreover, ~l~X~ ) ~ Z¢~/Zo¢ ~.
Let us take an imbedding of X~/G^ into the euclidean space of the dimension ~B
n-dim ~^ = n-cd^. According to our assumption about n, n-cd^ > 5. Denote by Z~ B 2 n-c~ ~ --
a regular neighborhood of X~/G~ in ~ ".
One can extend the classifying map X~/G~ + BZ~^B to a map Z^ ÷ BZP^, and in
this way to extend the bundle ~~ from X~ to the corresponding ~/ p(~-covering space U^ over Z~. Denote by g^ this extension. It is obvious that ~
s a t i s f i e ~ t he same p r o p e r t i e s 1 ) -3 as g^ does , and t he base o f ~^ i s an a B
o r i e n t a b t e G~-mani fo ld U^. Note t h a t i f ~ i s o r i e n t e d @as a complex s t r u c t u r e ) , ^ ^ ~ p ^
t hen g ~ w i l l be o r i e n t e d ( w i l l be complex) t oo . Moreover, f o r y > g , "E(g~) (Y)
wi i1 a l s o be o r i e n t e d (complex) a c c o r d i n g t o t h e d e f i n i t i o n of an o r i e n t e d (complex)
G-portrait.
Let B = @~ denotes the preimage of ~ by the map ^. By the property l,a)
and 2) of the induction assumption, p(B) = p(~), ,~ ~ ,B for any B £ B.
set "~(~) = oB6 B (8'~(B)) in @M. It is G^-invariant. Now consider the
Let D~ ) stand for the corresponding disk bundle. It is possible to form an
equivariant connected sum of D~) ×G G and @W by attaching equivariantly l-
handles one boundary component to ~U~× G G c ~D~) × G and the other to
(@'MP) ×G~ G c@M (see Figure 4).
To make this construction let us consider the decomposition of the set B into
different G^-orbits. For each G^-orbit we are picking up a representative B and a
point x$ in SM p(~) . Let xBG ~ denote the G~-orbit of x$ in SM p(~) (this
(B)~G~) ' be some point in 3U^ c ~D~).~ Then h
orbit is G^-isomorphic to p . Let xB
x~G$~ is also isomorphic to p(B)~G~.~ Moreover, if D will be some p(B)-invariant
neighborhood of x~ in ~D~), then the two G-sets Dx~G ~ ~D~)XG^G and
DxsG c @M are equivariantly diffeomorphic. We are using, of course, that fact that
178
Y @W --
91)( ~)×G G
D(~-~-~ )× ~u~ G~ G
~U~XG~G
I)(~)XG~G iMp(B)
Fig. 4
by the inductive assumption ~B ~ ~x~ and ~ ~ ~x~ are isomorphic 0(~)-representa-
tions.
of
In the case, when all ~^ are oriented orthogonal (or unitary) representations
p (~), this diffeomorphism is orientation-reversing.
So, we can realize a 1-dimensional G-surgery on the 0-dimensional sphere !
__II xB. Let us repeat this procedure for each G^-orbit in B. Denote by @,W' X~ B
the result of these surgeries.
We claim that @,W' satisfies all the properties of the induction assumption,
except for the property 3). In fact, the G-portraits @~ and @,~ of @W and
@,W' differ only by the "collapse of the set B to the element ~" and by gluing
together (@~)~Bg' (@~)>~'g for any two elements B,~' E'B and for any g E G.
So, this 1-dimensional G-surgery induces a map AB:@~ ÷ @,~, identifying the
elements of the G~g-Set{y E @~IY ~ Bg} (g E G) with the corresponding elements of
the Gag-Set {yIE @,~Iy~> AB(Bg)}. The last set is isomorphic to
One can show that A B is order-preserving and P EY) = P(AB~Y)) for y E @~.
Therefore, the original map ^:@7 ÷ ~ factors through AB, and one can define a
canonical map ^':@,~ ÷ ~ such that = ^'oA B.
The new map ^' satisfies the same properties i, 2, 4, 5 as ^ does, but
for the new closed subset @' = @ U ~G. It is still onto and, obviously,
p(~) = p(~') for any ~ E @,~. Because A B identifies only incomparable elements
179
in @~, one can see that ~ > ~ if an_d only if ~' > ~ for any ~B E @,~. zt is
clear that ^' is an isomorphism of the G-sets C^')-Ic@ ') and @'.
It follows from the geometry of the previous construction that ~ ~ ~, for
W I IDC~ G any ~ E @,~ (recall that we are connecting the components in @ )XG~
with isomorphic representations of the corresponding stationary groups).
The dimensional assumptions (property 4 of the induction assumptions) cannot be
destroyed by surgery on t~e Boundary.
An important remark Nas to be made. Namely, we claim t~t the G-portraits of
o,W' and its boundary @,M' are the same. Zn fact, by connecting @W and
^-l(o) c O~. Recall that, By the construction,
U~ with tI~e boundary ~U^. If dim U^ > 2,
Therefore "[~DC~ consists only
DCE~)XG^G we did not c~ange the set B
DR) is a bundle over the manifold
then ~U~ is nonempty and connected.
of one component as does "D(~) p[~).
The group p(y] is a stationary group of G^-action on ~DC~ ~) if and only if
it is a stationary group of p(~)-action on the space of the representation ~. On
the other hand, "~[D(~)] p(~) = "[D(~I~U~) U~D(~)]P(Y) is connected. So, G~-
portraits of and aro isomorphic to i* i
By 1-dimensional G-surgeries we have connected all the components "" M) p(Bg) L@ pg , ^
($ E B, g E G), with the component '~[D(~)x G G] PcBg) . Therefore every component
• , (~) B [@,W ]$ being the space of a vector bundle over U^, has nonempty and connected
intersection with the boundary "[@,M']~ 6~). Hence, the G-portraits of o,M' and
o,W' are isomorphic.
Now we would like to have some control on the fundamental groups of the sets
(@,W) ' (@'"' ;B , where B C @,~ has its image ~' = ~ C ~.
The manifold @,W' has the G-homotopy type of a 2-dimensional G-(W-complex Y.
Therefore there exists an equivariant retraction rt:o,W' ÷ otW', 0 < t < I, such
that r 0 = id, rl:o,W' ÷ Y. Moreover, r t is an isovariant G-map, inducing the
identity map of O,~ into itself for all t, except t = i. So, there is an iso-
variant, combinatoric preserving map rtl:@,W'~Y ÷ @,W'~-Y, O < t < i, such that -1
r t o r t = id.
If dim(o,W')P(B) ~ 5, any loop and homotopy of it in °(@,W')~(B)/G B can be
o , ~CB) removed away from [Y N Co,w ) ]/G$. By the map rtl/G , t is close to I, this
loop or any homotopy of it are mapped into a regular neighborhood of of M,~p(B)/n
180
o , p (F) So, ~ [o( ~4'~P(B)/G l is isomorphic to ~i[ (@,W)B /G6]" I ~ "e' ~B S ~
The normal G -bundle of of w,~p(B) in @,W' determines a homotopy class of
the map o(@,W,)~(6)/GB÷ BZ'~' B. Consider the kernel K of the induced map
° ' ~ ( ~ ) / G ~ ] ~i[ (o,W) + ~I[BZ~$] of the corresponding fundamental groups.
o , ~(B)/G ~ if (e,M) is orientable (see property 5 of the induction's
assumptions), the normal bundle of any loop i:S 1 ÷ [°(e,N')~CB)]/G is trivial,
and one can do surgery on the immersion class of i(sl). If i(sl) B belongs to the
kernel K, one can extend the map [° ,-p(B) (@,M)5 ]/GB + BZTB to the 2-handle
D2×D d(B)-2 attached by the map i (d(g) is the dimension of "(@,W'] p(8) and we
use here the fact that d(B) > 4). This extension produces an extension of the o T ~ I normal Gs-bundle of (@,M) (6) in @,M to a G6-bundle v 6 over
o , p (~) [ (D2xDd(6)-2)×p(B)Gs) ] U ~x id [ ( e , M ) S ] .
,~ o , ) p ( B ) The map i is a lifting on the (@,M of the imbedding i. This lifting is o , o(B) + possible because i(S I) 6 K and the covering (@,M)8
o(@,M,)80(~)/GB is induced by the map into K(GB/p(B ),I), which factors through the
map (@,M')~(S)/G~ ÷ BZ.~s.
(SI×Dd(6)-2) ÷ o . , .p (6 ) The attaching imbedding ~xid: xP[6)G~ CO 'M )6 can be extended
G-equivariantly to a G-imbedding of (SlxDd(B)-2)Xp(~)G into
U ° M'~P(6g)l c ~6XG6G gEG [ (0,,., ~g j o,M'. In this way one can extend the bundle-system
over 2-handles (D2×Dd(~)-2)Xp~8)G and form a new G-manifold
@,W" = @,W' U# [DCv6)× G]. llere ¢ denotes a G-imbedding of G B
( D v6 t ( s lxDd(S)_2)×p(B)G @,M'.
Let us r e p e a t t h i s p r o c e d u r e , k i l l i n g s t e p by s t e p a l l e l emen t s of the k e r n e l
K. be t @,W d e n o t e the r e s u l t i n g G - m a n i f o l d s .
I t i s obv ious t h a t a 2 - s u r g e r y on t he boundary does no t a f f e c t t he c o m b i n a t o r i c s
o f a G-mani fo ld ( i f t h e d imens ion o f t he s u r g e r e d component i s > 2) . T h e r e f o r e the
G-portraits of @,W and e,M = 3(@,W) are still e,~. Moreover, @,W is G-
orientable (the normal bundles system has a complex G-structure) if e,W' is (we
used oriented orthogonal (or unitary) bundles in the process of G-surgery).
o P[8) ,M)p (S) But now ~i[ (@,W)~ ] 7 ~i[°6@ ] are subgroups of ~I(BZ~), where
BZ'f~ ÷ BZ~B is the GB/p (~)-covering induced from the universal G6/p(B)-covering
over K(G$/p(S),I) by the canonical map BZ~' S ÷ KQ~(~),I). By Lemma 3, BZT 6
is homotopy equivalent to BZ:~$, and therefore ~I(BZ#8} = ~o(Z~] ~ Z~s/Zo~ ~. In
181
fact, by the construction of US c D(~), the fundamental groups of °(@,W)~ ~)
and °(@,M)~ ~) are isomorphic to Z~/Zo¢ ~.
Since we did equivariant 2-surgeries on the boundary, the resulting manifold
@,W still will be of the G-homotopy type of a 2-dimensional G-CW-complex.
The induction step @ ÷ 8' = ~G U @ of th~ theorem is proved.
Now we have to prove the basic statement of the induction•
Let @ be the set of all minimal elements in ~. Let @~ be the G-set
• ^ _ + ~. Define P(~) = 0(~) for ~L~>a There is an obvious onto-map : ~ 7>a
any a 6 @~.
The partial order in @7 is induced by the partial order in ~: a >
only if ~ > ~. The G-action on ~ also induces a G-action on @~. By the
if and
definition, ~g is [~-Ic~g)] N ~g for B C 7~ and g C G. This makes sense
^ + ~ is a one-one map for any ~ 6 @. because :~>~
Let ~ be ~ for any g E @~. It is clear that under these definitions,
@~ also becomes a normal G-portrait.
The map ~:@7 + ~ induces an equivariant isomorphism of the sets of minimal
elements in @7 and ~.
Consider the compact G-manifold @W ~ I_! D(L) ×G G where ~ is a chosen
representative in each G-orbit in @. By the construction, the portrait of the
G-action on @W is @~. As we mentioned before, the property 4 of the unduction
assumption implies that ~(OW) has the same G-portrait as @W does• The only
exception could be if we want to realize an element ~ 6 @ with the maximal
dim ~ by O-dimensional (but not by ~ 5-dimensional) components in @W. In this
case the G-portrait of the boundary ~(@W) will differ from the portrait of @W
by the elements {~ 6 @} with the maximal dim ~ . The Theorem is proved.
The proof of the Corollary now follows easily. If ~ is realizable on a G-
orientable manifold W of a G-hometopy type of a 2-dimensional CW-complex, then
one can equivariantly attach 2-handles to the "free part" of the top strate of W
(or even of ~W) to kill the fundamental group of the set °W of generic points
in W.
If cd > 2 for every nonmaximal a 6 ~, then codim(W'-°W) in W is greater
than 2, and W will be 1-connected. So, one can construct W of the homotopy type
of a bouquet of 2-spheres.
If 70~Z~ ) = 1 for any ~ E ~, then each component °~W~(~) is one-
connected by the Theorem, and if, in addition, dim ~ - dim ~ > 2 for any ~ > ~,
th~n "W O(e) is of the b_omotopy type of a bouquet of 2~dimensional spheres. This
ends the Corollaryts proof.
182
References
[11 Conner P.E., Floyd E.E., Maps of Odd Period, Ann. of Math. 84, 132-156 (1966).
[2] tom Dieck T., Transformation Groups and Representation Theory, Lecture Notes, in Math., 766 Springer-Verlag (1979).
[3] Dovermann K.H., Petrie T., G -Surgery II. Memoirs of A.M.S., Vol. 37, N. 260 (1982).
[4] Katz G., Witt Analogs of the Burnside Ring and Integrality Theorems I & II, to appear in Amer. J. of Math.
[5] Kosniowski C., Actions of Finite Abelian Groups. Research Notes in Math. Pitman, 1978.
[6] Lashof R., Equivariant Bundles over a Single Orbit Type, IIl. J. Math. 28, 34-42 (1984).
[7] Oliver R., Petrie T., G-CW-Surgery and K0(ZG ). Mathematiseh~ Zei~0 179, 11-42 (1982).
[8] Pawalowski K., Group Actions with Inequivalent Representations of Fixed Points, Math. Z., 187, 29-47 (1984).
[9] Petrie T. Isotropy Representations of Actions on Disks. Preprint, (1982).
~0] Tsai Y.D., Isotropy Representations of Nonabelian Finite Group Actions, Proc. of the Conference on Group Actions on Manifolds (Boulder, Colorado, 1983), Contemp. Math. 36, 269-298 (1985).
Topological invariance of equivariant
rational Pontrjagin classes
Dedicated to the memory of Andrzej Jankowski and Wojtek Pulikowski
K. Kawakubo Department of Mathematics
Osaka University Toyonaka Osaka 560/Japan
i. Introduction.
In [7], Milnor showed that the integral Pontrjagin classes of an open
manifold are not topological invariants. Afterward Novikov showed
topological invariance of the rational Pontrjagin classes [9].
In [3], we defined equivariant Pontrjagin classes and equivariant
Gysin homomorphisms. Concerning these concepts, we studied equivariant
Riemann-Roch type theorems and localization theorems in general.
The purpose of the present paper is to show topological invariance
of the equivariant rational Pontrjagin classes and to give some applica-
tions connected with the equivariant Gysin homomorphisms.
Let G be a compact Lie group. Given a right G-space A and a left
G-space B, G acts on A × B by
g o (a , b) = (ag -I , gb) g E G , a @ A , b @ B .
The quotient space of the action on A × B is denoted by
A × B . G
Denote by
G ) EG ) BG
the universal principal G-bundle. For a G-vector bundle ~ ) X
over a G-space X , we associate a vector bundle:
EG × ~ ~ EG x X . G G
Then we define our equivariant rational total Pontrjagin class
PG({) by
PG(~) : P(EG x ~) C H*(EG × X ; ~) G G
Research supported in part by Grant-in-Aid for Scientific Research.
184
where ~ is the field of rational ntunbers and P(EG × ~) is the G
classical rational total Pontrjagin class of the bundle EG x ~ G
EG × X . G
Similarly we define our equivariant total Stiefel-Whitney class
W G ( ~ ) by
WG(~) = W(EG x ~) C H*(EG x X ; ~2 ) G G
where Z 2 is the field Z/2Z of order 2 and W(EG x ~) is the G
classical total Stiefel-Whitney class of the bundle EG x ~ ) EG x X . G G
For G-spaces X , Y and for a G-map f : X > Y , we denote by
fG the map
fG = id x f : EG x X > EG x y Q
G G G For a G-manifold M , we denote by T(M) the tangent G-vector bundle
of M .
Then our main theorem of the present paper is the following.
Theorem 1. Let M 1 , M 2 be compact smooth G-manifolds and f : M 1
-- ~ M 2 a G-homeomorphism. Then we have
PG(T(MI)) = fGPG (T(M 2)
* denotes the induced homomorphism where fG
* H* fG : (EG × M 2 , ~) ~ H*(EG x M 1 ; ~) G G
The author wishes to thank Professor Z. Yoslmura for enlightening
him on cohomology of infinite CW-complexes.
2. Approximation by manifolds
Let G be an arbitrary compact Lie group. By the classical result
[2], G is isomorphic to a closed subgroup of an orthogonal group O(k)
for k sufficiently large. We can suppose that G C O(k) For any
non negative integer n , we regard O(k) (resp. O(n)') as the closed
subgroup
(rasp. [ < Ik 0 B E I o }l of O(k + n) , where I denotes the unit matrix of degree s .
s the sugroups O(k) and O(n)' of O(k + n)
identify their direct product O(k) × O(n)'
Then
commute; and one may
with the subgroup
185
of 0 (k + n)
Let
o) } 0 B A C O(k) , B C O(n)
Since G C O(k) , the same is true of G × O(n)'
EG n = O(k + n)/O(n) '
BG n : O(k + n)/G x O(n)'
be left coset spaces. As is well-known, EG n and BG n inherit unique
smooth structures such that the projections O(k + n) ~ EG n ,
O(k + n) ~ BG n are smooth maps and that they have smooth local
sections. Moreover by the inclusions
G C O(k) c O(k + n) ,
G acts on EG n freely and smoothly so that the ordinary smooth
structure on the orbit space EGn/G coincides with that of BG n and
that the projection p : EG n > BG n gives a principal G-bundle.
According to [i0], we have
~ (EG n) = 0 for 0 < i < n - 1 .
Namely the bundle above is n-universal in the sense of [i0].
The correspondence
A l ) ( A 0 1 0 1
gives rise to an inclusion map
O(k + n) ....... > O(k + n + I)
Clearly this inclusion map induces the following inclusion maps
EGn+I ' Jn : BGn '> BGn+I ~n : EGn
and the following diagram
EG n ~n> EGn+I
Jn BGn+l BG n >
.-r-- is commutative. Then 3 n is a bundle map of the principal bundles.
Let EG (resp. BG) denote the direct limit (or union) of the sequence
EG 1 C EG 2 c EG 3 C -.- ,
(resp. BG 1 C BG 2 C BG 3 C ... )
Then the induced projection map p : EG ..... ) BG gives a universal
principal G-bundle.
186
Let M be a smooth G-manifold. Since G acts freely and smoothly
on EG n , the quotient space
EG n × M G
inherits the smooth structure. Then observe that the following is a
smooth fiber bundle
M ~ EG n × M ~ > BG n G
where ~ is induced from the projection map EG n × M ) EG n
Since G acts on the tangent bundle T(M) as a group of bundle
automorphisms, we get the bundle along the fibers [i]
EG n × T(M) ) EG n × M
G G
of the above fibration.
Then the following lemma is well-known [i].
Lemma 2.
T(EG n × M) ~ EG n x T(M) ~ ~!T(BG n)
G G ! n
H e r e ~ s t a n d s f o r a b u n d l e i s o m o r p h i s m a n d T (BG )
i n d u c e d b u n d l e o f T ( B G n) v i a t h e m a p ~r .
denotes the
3. Topological invariance of equivariant rational Pontrjagin classes.
Let M 1 , M 2 be G-manifolds and f : M 1 ) M 2 a G-homeomorphism.
In §2, we showed that EG n × M 1 and EG n × M 2 are smooth G-manifolds G G
for any non negative integer n . It is clear that f induces a
homeomorphism
fG n = id × f : EG n × M 1 > EG n × M 2 G G G
Then we first show the following lenur~a on which Theorem 1 is based.
Lemma 3. n,
P(EG n x T(MI) ) = fG P(EGn × T(M2)) G G
Proof. Notice first that the rational total Pontrjagin class
satisfies the product formula:
P (~ ~ n) = P(~) "P (n)
for vector bundles ~ • n over X in general.
Consider the following commutative diagram:
187
n
EG n × M1 fG ) EG n × M2 G G
BGn ,,, i d > BG n
Then we have
n. ! _1 n n!~T(BG )) fG m(~2 T(BGn)) = P(fG
It follows from Lemma 2 that
! = p (~iT(BGn))
n. fG P(T(EG n × M 2))
G n*
= fG P (EGn x T(M2) @ ~T(BGn)) G
i n = fGn*{P(EGn GX T(M2)).P(~2T(BG ) }
= f *P(EG n x T(M2)).f G p(~T(BGn)) G
= f~*p (EG n × T (M 2) ) .P (~T (BG n) ) G
On the other hand, we have
P(T(EG n x M1) ) G
= P ( E G n x T(M1) ( ~ ~ T ( B G n ) ) G
! n = P ( E G n x T ( M 1 ) ) o P ( ~ T ( B G ) )
G
According to [9], there holds
n. p(T(EG n x MI) ) = fG P(T(EGn × M2))
G G
Combining the above results, we have
n n, P(EGn × T(MI)) "P(~IT(BG )) = fG (P(EGn × TMz))'P(~!IT(BGn))
G G i
Since P(niT(BG n)) is invertible, we have
n, P ( E G n x TM1) : fG P ( E G n x TM2)
G G
This makes the proof of Lemma 3 complete.
Remark.
map
n Milnor's example means that fG does not induce a bundle
T(EG n x MI) ~ T(EG n x M2) G G
in general.
188
Lemma 4. For a compact G-manifold M , the natural map
: lim (EG n × M) ) (lim EG n) × M = EG × M > G > G G
is a homeomorphism.
Proof. Consider the following commutative diagram:
lim (EG n x M) ~ • EG × M
lim (EG n x M) ~ > EG x M > G G
where $ is also the natural map, li~ Pn is induced from the projection
maps Pn : EGn x M > EG n × M and p is also the projection map. G
Clearly both ~ and $ are bijective maps.
In the following, we employ the terminology of Steenrod [ii]. Since
EG n is a closed subset of EG n+l for each n , the sequence
EG 1 C EG 2 C EG 3 C .-. ,
is an expanding sequence of spaces {EG n} The union EG = lim EG n
is given the weak topology. Namely a subset A of EG is closed if
A A EG n is closed in EG n for every n .
AS is well-known EG has a CW-complex structure such that each
EG n is a finite CW-subcomplex. It turns out that EG is a compactly
generated space. Hence EG is a filtered space as well.
Since M is a finite CW-complex, M is also a filtered space by
setting M. = M (n = 1,2,3,-.. ) l
We now get the product EG × M filtered by n
(EG × M) n U EG l x Mn_ i = EG n x M . i=0
It follows from Theorem 10.3 of [ii] that the product space EG × M
of filtered spaces has the topology of the union
lim (EG × M) = lim (EG n x M)
Remark that the topology on EG × M is given by the associated compactly
generated space k(EG x M) where x denotes the product with the C C
usual cartesian topology. However the topology EG x M coincides with c
k(EG x M) , since EG x M is a CW-complex. C O
It follows that EG x M coincides with the usual cartesian topology.
Thus we have shown that
: lim (EG n x M) > EG × M }
189
is a homeomorphism.
In order to prove Lemma 4, it suffices to show that the topology
lim (EG n × M) coincides with the quotient topology via the surjective G
map
lim Pn : lim (EG n x M) ~ lim (EG n × M) > "> G
Let C be a subset of lim (EG n × M) By definition, (lim ~ pn)-l(c) ) G
is closed if and only if
(lira pn)-l(c) Q (EG n × M)
is closed in EG n × M for every n . Clearly there holds
(lira pn)-l(c) N (EG n x M) = pnl(C N (EG n x M)) G
have that (lim pn)-l(c) is closed if and only if p~l(c N Hence we
(EG n × M)) is closed in EG n × M for every n . Since EG n × M has G G
the quotient topology via the projection map Pn : EGn × M • EG n × M , G
-i - Pn (C N (EG n × M)) is closed in EG n × M if and only if C N (EG n × M) G G
is closed in EG n × M . Furthermore C N (EG n × M) is closed in G G
EG n × M for every n if and only if C is closed in lim (EG n × M) G • " G
by definition.
Putting all this together, we have that (lim pn)-l(c) is closed if
and only if C is closed in lim (EG n × M) Namely lim (EG n × M) G .... > G
has the quotient topology via the map lim Pn "
This makes the proof of Lemma 4 complete.
We are now in a position to prove Theorem i. Consider the following
commutative diagram:
7~ 1 1
EG n × T(M I) G
I ,n 11
EG n × M .... G I
\~ EG n ~ T (M 2)
EG n x M 2 G
> EG × T(M I) G
EG x M I G
.n 12
~. ) EG × T(M 2)__
n 3.
2 > EG x M 2
G
where the horizontal arrows are induced from the inclusion map EG n
EG and give bundle maps. Note that there are no bundle maps
190
EG n × T(M I) G
EG × T(M 1) G G
i n g e n e r a l .
I t f o l l o w s f rom t h e a b o v e d i a g r a m t h a t
. n * . f , (T (M2)) 11 GPG
= fG* n* "i 2 PG (T(M2))
n, = fG P(EGn x T(M2))
G
= P(EG n x T(MI) ) G
.n* = l I PG(T(M1 ))
) EG n × T(M 2) , G
> EG × T(M 2)
(Lemma 3)
According to Proposition 4 of [13], the following homomorphism
: H*(lim (EG n × M I) ; ~) > lim H*(EG n x MI ; ~) > G < G
is an isomorphism.
By virtue of Lemma 4, we have an isomorphism
~* : H*(EG × M I ; ~) ) H*(lim (EG n × M I) ; ~) G ~ G
It turns out that the composition
~.~* : H*(EG × M I ; Q) > lim H*(EG n × M I ; ~) G ~ G
is an isomorphism.
Since there holds
i~*(f~PG(T(M2))- PG(T(MI)) = 0
for any n , we may assert that
,.~*(f~PG(T(M2))- PG(T(MI) ) = 0 .
Consequently we have
f~ PG(T(M2))-PG(T(MI)) = 0 .
This makes the proof of Theorem 1 complete.
4. G-homotopy type invariance off equivariant stiefel-Whitney classes.
In [3] and [5], we showed G-homotopy type invariance of equivariant
Stiefel-Whitney classes in different ways. In this section, we shall
give the third proof of it. Namely we show the following theorem.
Theorem 5. Let M 1 , M 2 be closed G-manifolds and f : M 1 > M 2
a G-homotopy equivalence. Then we have
191
w G (T (M l) )
where f ~ : H*(EG x M 2 ; ~2) G
h o m o m o r p h i s m i n d u c e d f r o m fG :
Proof. It is clear that f
f~ = id × f G
= f~WG(T(M 2)
H*(EG × M 1 ; Z 2) denotes the G
EG × M 1 > EG × M 2 . G G
induces a homotopy equivalence
EG n x M 1 ~ EG n x M 2 G G
for any n . Then the same technique as the proof of Lemma 3 applies
to prove the following lentma.
Lemma 6. W(EG n × T(M1)) = f~*W(EG n × T(M2)) G G
By making use of Lemmas 2 and 6, we can show the following equality
.n, , = iX,WG(T )) 11 fGWG (T (M 2) ) (M 1
as in the proof of Theorem 1 where iX* denotes the induced homomorphism
iX* H* : (EG × M 1 ; ~2 ) ~ H*(EG n × M 1 ; ~2 ) G G
AS is well-known, the following homomorphism
H* : (lim(EG n × MI); ~2 ) ~ lim H*(EG n × M 1 ; ~2 ) G ~---- G
is an isomorphism as well (see for example [12]).
Furthermore by virtue of Lemma 4, we have an isomorphism
}* : H*(EG × M 1 ; ~2 ) ) H*(li~ (EG n × M I) ; ~2 ) G G
Hence the rest of the proof is the same as that of Theorem I.
5. Topological invariance of equivariant genera.
Let G be a compact Lie group and hG( ) an equivariant multipli-
cative cohomology theory. Let M and N be closed hG-oriented G-
manifolds. Then for a G-map f : M ~ N we defined an equivariant
Gysin homomorphism
f! : hG(M) > hG(N)
in general [3]. Concerning the equivariant Gysin homomorphism f! ,
we got a localization theorem and an equivariant Riemann-Roch theorem
and so on.
We now make use of the equivariant cohomology theory H*(EG x M ; Q) G
as hG(M) When N is a point with trivial G-action, our equivariant
Gysin homomorphism
192
f! : H*(EG × M ; ~) > H*(BG ; ~) G
is called an index homomorphism and is denoted by Ind. Using the index
homomorphism, we define equivariant Pontrjagin numbers as follows. Let
m be a positive integer and I : i I ---i k a partition of m . Then
for a vector bundle ~ > X , we set
PI (~) : Pi I (~) "'" Pi k (~)
where Pi (~) are the ordinary rational Pontrjagin classes. Let M ]
be a closed oriented G-manifold such that the G-action is orientation
preserving. Then M is H*(EG × - ; ~) oriented and we have G
Ind : H*(EG × M ; Q) > H* (BG ; ~) G
We now define our equivariant Pontrjagin number PGI(M) by
PGI(M) = Ind PI(EG × T(M)) E H*(BG ; ~) G
Note that even if m is larger than dim M/4 , PGI(M) makes sense and
gives us important informations in general.
In this section, we will show that equivariant Pontrjagin numbers
are topological invariants under some conditions. Accordingly equivariant
genera defined by equivariant Pontrjagin numbers are also topological
invariants.
We now prepare some lemmas whose proofs are easy excercises.
where f~ :
Namely f!
Lemma 7. Let M 1 and M 2 be closed oriented manifolds and f : M 1
M 2 a degree 1 map. Then we have
f! • f* : id
H*(M I) > H*(M 2) denotes the ordinary Gysin homomorphism.
is defined by the following commutative diagram
f~
H*(M I) > H*(M 2)
f . He(M) --~ H.(M 2)
where D denote the Poincar6 duality isomorphisms and
induced homomorphism of homology groups.
f, denotes the
Lemma 8. Suppose that EG n is an oriented manifold and that G
acts on EG n preserving the orientation for every n . Let M 1 and
M 2 be closed oriented G-manifolds such that the G-actions on M 1 and
M 2 are orientation preserving. Let f : M 1 > M 2 be an orientation
193
preserving G-homeomorphism~ Then EG n × M 1 and EG n ~ M 2 inherit G
the orientations so that
fG n = id G × f : EGn G × M1 > EGn G × M2
is an orientation preserving homeomorphism.
By combining Lemmas 3, 7 and 8, we shall show the following lemma.
Lemma 9. Under the conditions of Lemma 8, we have
n fG!PI(EGn × T(M1)) = PI(EGn × T(M2))
G G n n
where fG! d e n o t e s t h e o r d i n a r y G y s i n homomorph i sm o f fG :
) EG n × M 2 G
Proof. It follows from Lemmas 7 and 8 that
fG!n .fGn*PI(EGn G × T(M2)) = PI(EGn G × T(M2))
On the other hand, by virtue of Lem~a 3, we have
fGn*PI(EGn G × T(M2)) : PI(EGn G × T(MI))
Hence we obtain the reguired equality.
EG n × M 1 G
Theorem 10. Under the conditions of Lemma 8, we have
f,PI(EG × T(MI)) : PI(EG × T(M2)) " G G
Proof. As in the proof of Lem~a 4.1 in [4], one verifies the
commutativity of the following diagram:
fl H* H* (EG × M I ; ~) " > (EG × M 2 ; ~)
G G ~.n, I.n*
ii fn 12
H* (EG n x M1 ; ~) __ G!> H* (EG n × M 2 ; ~) G G
where i n* are induced from the inclusion maps (j = 1,2) ]
From this, we have
.n* 12 "f!Pi (EG × T(MI))
G n .n,
= fG!11 Pl (EG × T(MI)) G
n (EG n x T(MI) ) = f G ! P I G
Hence by v i r t u e o f Lemma 9, we have
194
i2*(f,Pi(EG × T(MI) ) - PI(EG x T(M2))) • G G
n (EG n x T ( M 1 ) ) - P I ( E G n x T(M2) ) = fG!PI G G
: 0 .
Since H*(EG × M 2 ; Q) ~ lira H* (EG n × M 2 ; ~) , we may assert that G '~ G
f!PI(EG × T(MI)) = PI(EG × T(M2)) G G
Theorem ll. Under the conditions of Lemma 8, we have
PGI(MI) : PGI(M2) ,
for any partition I
Proof. Since our equivariant Gysin homomorphism has the functional
property ((iii) of Lemma 2.2 in [3]), we have the following commutative
diagram:
H*(EG × M 1 ; ~) G
f~ " Z n d ~
H* (BG ; ~)
J H*(EG x M 2 ; @) /
G
Hence by Theorem i0, we have
PGI(MI) = Ind PI(EG x T(MI) ) G
: Ind f,PI(EG x T(MI) ) : Ind PI(EG × T(M2)) - G G
= PGI (M2)
This completes the proof of Theorem ii.
It follows from Theorem ii that any equivariant genera defined by
equivariant Pontrjagin classes are topological invariants. In the
following, we pick up one of them.
Let B be a multiplicative sequence in the sense of [8]. Then as
an application of Theorem ii, we have the following corollary.
C orollar~ 12. Under the conditions of Lemma 8, we have
BG(M I) = BG(M 2)
where 8G(Mi) are defined by Ind ~(EG × T(Mi)) (i = 1,2) G
Concerning the localization theorem and the equivariant Riemann-
Roch type theorem in [3], we have similar formulae.
195
We conclude the present paper giving the following conjecture which
seems to be an application of Theorem ii.
Conjecture. 1 S -homeomorphic sl-manifolds are sl-bordant.
References.
i. A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. J. Math., 80, 458-538 (1958).
2. C. Chevalley, Theory of Lie groups, Princeton Univ. Press, 1946. 3. K. Kawakubo, Eguivariant Riemann-Roch theorems, localization and
formal group law, Osaka J. Math., 17, 531-571 (1980). 4. K. Kawakubo, Global and local equivariant characteristic numbers
of G-manifolds, J. Math. Soc. Japan, 32, 301-323 (1980). 5. K. Kawakubo, Compact Lie group actions and fiber homotopy type,
J. Math. Soc. Japan, 33, 295-321 (1981). 6. J. Milnor, On axiomatic homology theory, Pacific J. Math., 12,
337-341 (1962). 7. J. Milnor, Microbundles: I, Topology 3 (Suppl. I), 53-80 (1964). 8. J. Milnor and J. Stasheff, Characteristic classes, Ann. of Math.
Stud. Princeton Univ. Press, 1974. 9. S. P. Novikov, Topological invariance of rational Pontrjagin
classes, Doklady Tom 163, 921-923 (1965). i0. N. Steenrod, Topology of fiber bundles, Princeton Univ. Press,
1951. ii. N. Steenrod, A convenient category of topological spaces, Michigan
Math. J., 14, 133-152 (1967). 12. Z. Yosimura, On cohomology theories of infinite CW-complexes, I,
Publ. RIMS, Kyoto Univ., 8, 295-310 (1972/73). 13. Z. Yosimura, On cohomology theories of infinite CW-complexes, III,
Publ. RIMS, Kyoto Univ., 9, 683-706 (1974).
ON THE EXISTENCE OF ACYCLIC r COMPLEXES OF THE LOWEST POSSIBLE DIMENSION
by
Tadeusz Kozniewski
Department of Mathematics, University of Warsaw, PK1N IXp, 00-901 Warszawa, Poland
Introduction.
Let r be a discrete group which contains a torsion-free subgroup of f inite index. By
a r complex we wil l understand a proper F CW complex (i.e. a F CW complex which has all
isotropy groups finite). In the present paper we investigate connections between the
existence of Ep acyclic (or contractible), f lnite dimensional r complexes and the
following algebraic properties of the group r. We say that r has cohomological dimesion
n ( cd r = n ) i f pdEyE : n, where 77 has tr iv ial EF module structure and for any ring A
and any A module M PdAM denotes the projective dimesion of M,i.e. the length of the
shortest A projective resolution of M. The group r has virtual comologlcal dimesion n
( vcd r = n) if there exists a subgroup A of f inite index in F such that cd A = n. vcd F is
well defined ( i t does not depend on A, see [ l t ] ) . For every prime p one defines
CdpY :pd E r77p and VCdpr = CdpA for any torsion-free subgroup of f inite index in Y.
Observe that if X is a Ep acyctic, f inite dimensional r complex then i t follows from
Smith theory that for every f inite p subgroup P in r and every torsion-free subgroup A in
N(P)/P (where N(P) denotes the normalizer of P in r ) the cellular chains C,(xP)®Ep form a
EpZ~-free resolution of 7/p. Therefore VCdp N(P)/P ~; dim X p, in particular VCdp F ~ dim X.
The f i rs t results in the opposite direction, i,e. results showing that i f cd r = n (resp.
vcdp r = n) then there exists a contractible (resp. Ep acyc]ic) r complex of dimension n
were proved (for n ) 3) by Eilenberg and Ganea (see [6]) and by Quillen (see [g]). Our goal
is to generalize these results to the case n = VCdp r or n = vcd £.
197
For a given prime p we wil l say that a F complex is of type p if all i ts isotropy
groups are p groups. Also we wil l say that a F complex is of prime power type if the
order of i ts every isotropy group is a power of a prime (prime may vary from one
i sotropy group to another). To phrase our theorems we wil l use the posets:
~'H(F) = {KIK is a f inite subgroup of F and H g K},
~H,p(r) = {K I K is a f inite p subgroup of !r and H ~ K}.
By homology of a poset we mean the homology of i ts geometric realization.
We also use the notion of reduced equivariant cohomology ~iF(x;B) of a !r complex.
'~i For any F complex X and any 7/F module B H r(X;B) is defined as H t+ I(HomEF(C,(Px),B))
where PX denotes the canonical projection map EFxX---~ Erxpt , C.(Px) is the algebraic
mapping cone of (px). : C.(E!rxX)-I)C.(Elrxpt) and E!r is the universal cover of a CW
complex of type K(F, 1 ). Then we have
COROLLARY 3.1 Let VCdp F = k ;~ 2. Then the conditions (1) and (2) below are equivalent:
(1) There exists a k dimensional 77p acyclic F complex of type p
(2) For every f inite p subgruop H in I" we have:
(a) Hk(gH,p(F);77) = O,
"~k (b) H A(~TH,p(!r);B) = 0 for some subgroup A of f inite index in N(H)/H
and every 2EpA module B.
We also get
COROLLARY 3.2 If there exists a contractible k dimensional I ~ complex of prime power
type then the conditions (2) of 3.1 are satisfied.
A partial converse to Corollary 3.2 is given by
PROPOSITION 3.3 Assume that vcd F = k ~ 2 and that for every prime p conditions (2) of
3.1 are satisfied. Then there exists a contractible k+l dimensional [' complex of prime
power type.
The paper is organized as fottows, In § 1 we give conditions for the projectivity of
modules over group rings, In § 2 we construct F complexes with the property that their
fixed poit sets are 77p acyclic and have dimensions prescribed by a given function k from
a set of f inite subgroups in I" to integers ;~ 2, In § 3 we apply these constructions to the
question of the existence of 77p acyclic (resp. contractible) r complexes of dimension
198
equal to VCdp r (resp. vcd F).
The paper is a revised version of a part of the author's doctoral dissertation which
was wri t ten under direction of Professor Frank Connolly and submitted to the University
of Notre Dame in 1985. The author would like to express deep gratitude to Professor
Connolly for his help and encouragement.
§ I. Projective modules over group rings.
We start with algebraic lemmas which give conditions for project iv i ty of modules
over group rings.
1.1 LEMMA. If r is any group and A is a subgroup of f ini te index in F then A contains a
subgroup A' which is normal in F and has f inite index in F.
Proof: Let A be a subgroup of f inite index in F. Define A' = flge(F/A) gag-1
normal in F and has f ini te index in F.
• Then A' is
[]
1.2 LEMMA. Generalized projective criterion.
Let F be a group, let Z~ be a subgroup of f inite index in r and let R be a commutative ring
with unit element t#0. Let M be an RF module. Then the conditions (1) and (2) below are
equivalent:
(1) M is RC projective,
(2) M is RA projective and PdRrM < ~.
Proof: ( 1 ) * (2) is clear.
(2) * ( 1 ). We start with the following two claims:
Claim 1. For any RF module A
ExtiRF(R,A) ~ Hi(F;A).
Proof of Claim 1 Let F. be any 7/r projective resolution of 77. Then R ®77 F. is an RF
projective resolution of R and ExtiRF(R,A) ~ Hi(HomRF(R ®77 F . , A)) ~ Hi(Hom7/F(F.,A)) =
Hi(F;A) which proves Claim 1.
199
Claim 2. For any two RF modules N and L such that N is R projective we have:
ExtiRF(N,L) ~ Hi(F; HomR(N,L)).
Proof of Claim 2 : Let F. be any RF projective resolution of R. Then for each F i we have
H°mRF(Fi ®R N, L) ~ HomRF(F i , HomR(N,L)). N is R projective so the functor HomR(N, ) is
exact and consequently the functor HOmRF(Fi, HomR(N, )) is exact. Therefore the
functor HOmRF(F i ®R N, ) is exact, so F i ®R N is projective. This shows that F. ®R N is
an RF projective resolution of N and we have:
ExtiRF(N,L) = Hi(HomRF(F. ® N, L)) ~ Hi(HomRF(F., HomR(N,L)) = ExtiRF(R, HomR(N,L))
Hi(F; HomR(N,L)).
The last isomorphism follows from Claim 1 and ends the proof of Claim 2.
Now observe that by Lemma 1.1 we may assume here that A is normal in r (if not
replace A by a smaller subgroup which is normal and has finite index in F). Denote the
quotient group G = F/A and let tT:F i_~ G be the canonical epimorphism. For every
subgroup H of G denote F(H) = 11"- 1 (H).
Claim 3. If M is an RF module which is RA projective then for every RF module N
ExtiRF(M,N) ~ Hi(G; HomRA(M,N))
and more generally
ExtiRF(H)(M,N) ~ Hi(H; HomRF(H)(M,N))
Proof of Claim 3: Consider the Lyndon - Hochschlld - 5erre spectral sequence for z& < r
and the RF module HomR(M,N).
EPq 2 = HP(G; Hq(A; HomR(M,N))) -= HP(G; ExtqRA(M,N)) =
I HP(G; HomRA(M,N)) if q = 0
0 i fq>O.
The f i rs t isomorphism follows from Claim 2. The fact that all lines except q = 0 are 0
follows from RA projectivity of M, Therefore we get:
HP(F; HomR(M,N)) ~ HP(G; HomRA(M,N)). This combined with Claim 2 prove the f i rst
isomorphism of Claim 3. The proof of the second isomorphism is analogous.
Now observe that PdRFM<~ implies PdRF(H)M<~. Therefore by the second
isomorphism of Claim 3 we get that for each subgroup H in F Hi(H; HomRz~(M,N)) = 0 for
200
big i. It follows ([10], Theorem 4.12) that the EG module HOmRA(M,N) is cohomologically
trivial, in particular ExtlR(H,N)= HI(G; HomRA(M,N))= 0 which proves RF projectivity
of M because N is arbitrary. []
1.3 LEMMA. Assume that VCdp F < ~. Let M be a ~pF module. If for every f inite subgroup
H in F M is 7/pH projective, then HI(F;M) = 0 for big i.
Proof: Let K be a f inite dimensional, 7/p acyclic F CW complex (see [9]). Then there is a
Leray type spectral sequence
EPq 2 = HP(K/F ; {Hq(F~ ; M)}) =~ H*F(K;M) ~ H*(F;M)
where F~ denotes the isotropy group of a cell ~ in K (see e.g. [3]). Because all isotropy
groups are f inite we get that EPq 2 = 0 if p > dim K or q > O. Therefore Hi(F;M) = 0 for
i > dim K.
El
t.4 LEMMA Assume that VCdp F < ~. Let A be a subgroup of f inite index in F. Let M be a
EpF module which is 7/pA projective and 7/pp projective for every f inite p group P in F.
Then M is ~pr projective.
Proof: By Lemma I. 1 we may as well assume that A is normal in F. Denote G = F/A and
for every f inite subgroup H in G denote F(H) = Tr- 1 (H) where Tr : F ~ G is the natural
projection. Let N be a 7/pl- module. M is 7/pA projective, so by Lemma 1.2, Claim 3:
Exti~l~r(M,N) ~ Hi(G; Hom~I~A(M,N)).
It is therefore enough to show that Hom~gA(M,N) is G cohomologically trivial. By [10],
Theorem 4.12, i t is then enough to show that Hom~I~(M,N) is H cohomologically tr iv ial
for every q group H in G, where H ranges over all primes.
If q ~ p this is clear since Hom~EI~A(M,N) is torsion prime to q. If q = p consider the
subgroup F(H). F(H) does not contain torsion other than p-torsion. But i f P is a f inite p
group in F(H) then for i > 0 Hi(p; Hom~(M,N))~ Exti~p(M,N) = 0 because M is 7/pp
projective. 5o we can apply Lemma 1.3 to F(H) and Hom~p(M,N) and we get that
201
Hi(F(H); HomT/p(M,N))= 0 for big i. Lemma 1.2, Claims 2 and 3 says:
Hi(F(H); Hom7/p(M,N)) ~ Hi(H; HomEpA(M,N)). So for every p group H in G
Hi(H; HomEpA(M,N)) = 0 for big i and therefore Hom~pA(M,N) is G cohomologically trivial.
[]
To construct contractible I" complexes we wil l need the following fact:
1.5 PROPOSITION. Let X be n dimensional, n-1 connected I" complex, where
n ~ vcd E" - 1 . Assume that for each prime p and each finite, nontrivial p group P in r X P
is 7/p acyclic. Then Hn(X) is a projective 77F module.
Proof: A EF module is projective if it is projective over some subgroup of finite index
and over all finite p subgroups, for all primes p, ([5], Corollary 4. l,b).
Let A be a torsion-free subgroup of finite index in F. Then A acts freely on X and
C,(X) - the cellular chain complex of X is a complex of free 7/A modules. X is n
dimensional, n-I connected, so
0 --~ Hn(X) --~ Cn(X) --~ C n_ I(X) --~... --~ Co(X) --* 7/--~ 0
is a resolution of 7/ in which all Ci(X) i = 0 .... ,n are 7/A free. vcd F ~ n+i implies
cd A ~ n+l and therefore Hn(X) is 7/A projectve by the generalized Schanuel's lemma
(e.g. [4], Chapter VIII, Lemma 4.4).
Now let p be a prime and let P be a finite p group in F. Let S be the singular set of
the P complex X./3 is 7/p acyclic (by Mayer - Vietoris sequence and induction). Therefore
for every i ~i(X;7/p) -~ Hi(X~6;77 p) and we get that Hn(X)®7/p ~ Hn(X;7/p) ~ Hn(X,~;7/p) is the
only nonzero homology group of a free, n dimensional 7/pp chain complex C,(X,~)®7/p. tt
follows ([13], Lemma 2.3) that Hn(X;7/p) is 7/pp projective. But Hn(X) is also 7/ free, so
Hn(X) is 77p projective.
0
§ 2. r complexes with fixed point sets having prescribed dimensions.
2.1 LEMMA. Let X be a ]~ complex which has dimension < n and is n -2 connected, n ;~ 2.
202
Then the conditions ( 1 ) and (2) below are equivalent:
(1) There exists a 7/p acyclic, n dimensional iT complex Z containing X as a subcomplex
and such that Z - X is free
(2) Hn(X;7/p) = 0 and Hn_I(X;77 p) is 7]pF projective.
N X Proof: (1 ) * (2). For every i Hi(Z,X;7]p)~Hi_I(;7]p). It fol lows that Hn(X;77p)=O
(because Z is n dimensional) and it fol lows that Hn(Z,X;7/p) is the only nonzero homology
group of a free, n dimensional 7/pF chain complex C,(Z,X)®77p. Therefore Hn(Z,X;7/p) is
7/pF projective ([t 3] Lemma 2.3).
(2 )* (1) . Hn_I(X;7/p) is ~7pF projective. Therefore by "Eilenberg tr ick" (see e.g. [4]
Chapter VIII, Lemma 2.7) there exists a free 7/pF module F such that H n_ I(X;7]p)@F is 7/pF
free. Attach t r iv ia l ly free F cells of dimension n -1 to X, one for each basis element of F.
We obtain a new n dimensional F complex, X', which is n - 2 connected, has
Hn- 1 (X';77p) ~ Hn_ 1 (X;77p)@F and Hn(X';E p) = O. Use the epimorphism
1Tn- 1 (X') -" H n_ 1 (X') ~ H n_ 1 (X')®7/p ~ H n_ 1 (X';7/p)
to represent basis elements of the free 7/pF module Hn_l(X';7/p) by continuous maps
S n-1 --~ X' and use these maps to attach free r cells of dimension n to X'. The new F
complex, Z, obtained this way s t i l l is n - 2 connected. Moreover
6 : Hn(Z,X';7] p) --~ Hn_ 1 (X';7/p) is an isomorphism which implies that:
Hn_ t(Z;7/p) = 0 = Hn(Z;7/p), so Z is 7/p acyclic.
[]
This lemma has an obvious analogue when 7]p is replaced by 7/(see [7], Lemma 1.3).
Now let X be a F complex and let (~(X) be the singular set of X. It was proved in [5]
that there exists a [" map f : ~(X) --~ I~{1}(F)I such that for every f in i te subgroup H in i-
f restr ic ts to N(H)/H map fH : ~H (X) ~ I~H(F)I, where O'H(X) = {xeX I Fx~H}. It is
specially easy to construct the map f in the case when X is a F simplicial complex.
Namely: let X' denotes the barycentric subdivision of X. If 5 is a vertex in X' (i.e. a
simplex in X) define f((~) = ]-~ = the isotropy group of (~. If (~1 < (~2 < . " < (~k is a
203
simplex in X' then F(~c D F(~zZ> .. . D F(~ k . Therefore f is a simplicial map. Also for every
g • F f(gc) = Fg<~ = (F(~)g = (f(<~))g so f is a F map. For the general construction see [5],
Lemma 2.4. Another way of identifying (~(X) and I~{ l}(r) l is given in [4], Chapter IX,
Lemma t 1.2.
Observe that if there is a prime p such that all isotropy groups in X are p groups
then f . OH(X) . i~ I~H,p(F)I. Also, note that i f for every K • ~FH(F) X K is acyclic (resp. 7/p
acyctic) then by Mayer-Vietoris sequence and induction we get that f is a homology
equivalence (resp. a 7/p homology equivalence).
Let's f ix a prime p. To formulate our next resoult we need the following notation. Let
k be a function from the set of all f inite subgroups of F to integers ) 2. Assume that k
satisfies:
(A) For each H k(H) ) VCdp N(H)/H,
(B) tf H < K then k(H) ) k(K),
(C) For each H and each g e F k(H) = k(Hg).
The following theorem gives the necessary and sufficient conditions for the
existence of a F complex which has all fixed point sets 7/p acyclic and of dimensions
prescribed by the function k.
2.2 THEOREM. Let VCdp F < ~. Then the conditions (1) and (2) below are equivalent:
(1) There exists a F complex X such that for every f inite subgroup H in F X H is 7/p
acyclic and dim X H = k(H),
(2) For every f inite subgruop H in F
(a) Hk(H)(~'H(F);7/p) = O,
(b) There exists a subgroup A of finite index in N(H)/H such that
~k(H)A(ZFH(F);B) = 0 for every 7ZpA module B.
Proof: ( I ) ~ (2). Let H be a finlte subgroup of F and let k = k(H). In the exact sequence:
Hk+ I(xH,(~H(X); T/p)----) Hk((~H(X); T/p)---> Hk(XH; T/p)
the first group is 0 because dim X H = k and the last group is 0 because X H is Zp acyclic
204
So Hk(~FH(£); Ep) :'- Hk((~H(X); 7/p) = 0 which proves 2 (a). Now let W = N(H)/H and let B
be any EpW module, Then in the exact sequence
Hkw(XH; B)--~ Hkw(CH(X), B)--~ H k+ 1w(XH,(~H(X); B)
the f i rs t group is Hk(w;B), the second is Hkw(~H(F); B) and because (xH,CH(X)) is a
free W complex the third group is isomorphic to Hk+I(xH/w,(~H(X)/W; B) which is 0
because dim X H = k. So we get that Hk(w; B ) ~ Hk(~H(tr); B) is an epimorphism This
proves 2 (b).
(2) ~, (I). Let n be an integer bigger or equal to the order of every finite subgroup of
r. we wil l construct a sequence x n c Xn_ 1 c .. . c x 1 of F complexes which satisfies:
(i) xiH is Ep acyclic for all finite subgroups H of r such that IHI ~ i,
(ii) If H is a subgroup of r such that IHI = i then all open £ cells of type H
lie in X i - Xi+ i and have dimensions < k(H).
In particular X = X 1 wil l satisfy condition ( 1 ) of 2.2.
To prove the existence of the sequence we wil l procede by induction. Let IHI = i,
k = k(H). First observe that Xi+IH = (~H(Xi+I) = UK~ H X K and therefore dim Xi+l H ~ k
by the inductive assumption. Also by the inductive assumption Xi+ 1K is 7/p acyclic for
each K ~ H and therefore H.(O'H(Xi+I); Ep) = H.(~H(F); Ep). Now attach ceils of type H
and of dimension<k-i to Xi+ 1 to get that Xi+ 1 is k-2 connected. We sti l l have
Hk(XI+ 1H; ~p) ~ HK(~TH(r); 7/p) and the last group is 0 by 2 (a). So to end the construction
of X i such that H.(xiH; Ep) = 0 it is enough to prove that H k_ l(Xi+ 1H; Ep) is Ep(N(H)/H)
projective &emma 2.1 ).
By Lemma 1.2 to prove that H k_ l(Xi+ 1H; Ep) is Ep(N(H)/H) projective it is enough
to show that Hk_I(Xi+IH; Ep) is:
t ° EpA projective (where A is some subgroup of f inite index in N(H)/H),
20 PdEp(N(H)/H) Hk- i (Xi+ t H; Ep) ~ ,~.
Proof of l°: Assume that A is torsion free. Let f : Xi+t H ~ EA be a classifying map.
205
Then Hk_l(Xi+lH; 7@) -= Hk(f; Ep) which is projective provided Hk+l(f; B) = 0 for all
ZpA modules B ([t3], Lemrna 2.3). Consider the exact sequence:
Hk(A;B) ~ Hk(xi+ 1H; B) ~ H k+ l(f; B) --~ H k+ I(A;B).
The last group is 0 (because Cdp A ~ k) and Hk(xi+ IH; B) = HkA(xi÷ tH; B)
HkA(~FH(F); B) so the condition 2 (b) gives H k+ l(f; B) = O.
Proof of 20: Let N ~> max(k, dim Z) for some F complex Z which has the property that all
its fixed point sets are 77p acyclic and highly connected (for the existence of Z see e.g.
[9]). Let X (k- 1 ) = Xi ÷ 1 and for j ~ k let x(J ) be a j - 1 connected N(H)/H complex obtalned
from X (j- 1 ) by attaching free N(H)/H cells of dimension j.
Then HN(X(N),x(N-1);7@)E2-~... ~ Hk(X(k),x(k-1 );77p)--~ H k_ l(X(k-1 ) ;7@)~ 0 is an N-k
stage free 7]p(N(H)/H) resolution of Hk_I(Xi+IH; 77p) with ker6 N ~ HN(X(N); 77p). To end
the proof we will show that HN(X(N); 77p) is a projective 77p(N(H)/H) module. Let
f : X (N)--~ Z H be a classifying map. o'(f), : H,((~(x(N)); 77p)~ H,(o'(ZH); 77p) is an
isomorphism, so HN(X(N); Z/p) ~ HN÷ l(f; 7@) -= HN÷ l(f,o(f); 77p). This is the f irst nonzero
homology group of a free 77p(N(H)/H) complex C.(f,~(f))®7/p and for any 7/p(N(H)/H)
module B HN+2(f,(~(f); B) = 0 because dim (f,~(f)) = N÷I. Therefore HN(x(N); 7@) is
7/p(N(H)/H) projective by[ l 3], Lemma 2.3.
If we perform the above construction on Xi+ 1 for all subgroups H such that IHI = i we
wilt get X i in our sequence. This ends the proof of the existence of the sequence and the
proof of the theorem,
[3
2.3 DEFINITION
(a) Let p be a prime. A r complex X is of type p if all isotropy groups of X are p groups.
(b) A F complex is of prime power type if every isotropy group of X has prime power
order.
Fix a prime p, Our goal is to glve the necessary and sufficient conditions for the
existence of a 77p acyclic F complex of type p such that dim X = VCdp F. To do this we
206
wil l f i rs t consider the analog of Theorem 2.2 which wi l l take into account only f inite p
subgroups of £. Let k be a function from the set of all f inite p subgroups of £ (trivial
subgroup included) to the set of integers ;~ 2 and assume that k satisfies conditions (A),
(B), (C) above. Then we have:
2.4 THEOREM. Let VCdp Ir < ~. The conditions ( 1 ) and (2) below are equivalent:
(1) There exists a 7/p acyclic !r complex X of type p such that for every finite p group H
in [" dim X H = k(H),
(2) For evey finite p group H in £
(a) Hk(H)(~rH,p(£); 7/p) = O,
(b) There exists a subgroup A of finite index in N(H)/H such that
"t~k(H)z~(~H,p(F); B) = 0 for every 7/p module B.
Proof: The methods of the proof are similar to Theorem 2.2, so we wil l only point out the
differences.
(1)* (2) is like in the proof of Theorem 2.2.
(2) • (1). As before we wil l construct a sequence of F complexes
X ncXn_ 1 c . . . c X I but now we require:
(i) Xi H is 7/p acyclic for all f inite p subgroups H of r such that IHI ~ i,
(l i) X i - Xi, 1 consists of open lr cells of type H ond of dimensions ~ K(H),
where H runs over all p subgroups H in F which have [HI = i.
In particular note that the only subgroups of i r which have nonempty fixed point sets are
finite p subgroups.
The proof of the inductive step in the construction of the sequence is based, as
before, on 7/p(N(H)/H) projectivity of Hk_I(Xi+IH; T/p). To prove that this module is
77p(N(H)/H) projective we use again Lemma 1.2. The proof that Hk_I(Xi+IH; T/p) is 7/pA
projective remains the same. The proof that pd~p(N(H)/H)H k_ l(Xl+ 1H; Z/p) < ~ requires
a new argument. As before we want to show that HN(X(N);7/p) is 7/p(N(H)/H) projective.
First note that the generalized Schanuel's lemma implies that HN(x(N);Tz p) is 7/pA
207
projective because X (N) is N dimensional, N-1 connected and Cdp A < N. Moreover for
every finite p group P in N(H)/H HN(X(N)) is 7/p projective by Proposition 1.5. So
HN(X(N);7/p) ~ HN(X(N))®2E p is 7/pp projective. Now we can use Lemma 1.4 to conclude
that HN(X(N);77 p) is 7/p(N(H)/H) projective and therefore
PdEp(N(H)/H) Hk- 1 (Xi + 1 H; 7/p) < ~.
So we get that Hk_I(Xi+IH; 7/p) is 77p(N(H)/H) projective and the rest of the proof
proceeds as in 2.2. []
§ 3. Ep acyclic F complexes and contractible F complexes,
Let k = VCdp F (resp. k = vcd lr). We wil l apply the resoults of the previous section
to examine the question of the existence of a 77p acyclic (resp. contractible) F complex of
dimension k. First note that the following is an immediate consequence of Theorem 2.4.
3.1 COROLLARY. Let VCdp F = k ;~ 2. The conditions ( 1 ) and (2) below are equivalent:
(1) There exists a k dimensional, 7/p acyclic F complex of type p,
(2) For every finite p group H in F we have:
(a) Hk(~H,p(F); 77p) = 0
(b) ~kACFH,p(F); B) = 0 for some subgroup A of finite index in N(H)/H and every
7/pA module B.
Note that it follows from Theorem 2.4 that if VCdp F < ~ then there exists a finite
dimensional, 7/p acyclic F complex of type p.
We also get
3.2 COROLLARY. If there exists a contractible k dimensional F complex of prime power
type then for every prime p the conditions (2) of 3.1 are satisfied.
208
Proof: If X is a contractible r complex of prime power type then for every prime p such
that F has nontrivial p torsion Xp = {x~Xl r x is a nontrivial p group} is a F subcomplex of
type p. The corollary now follows from Theorem 2.4. []
We can use Proposition 1.5 to give a partial converse to 3.2
3.3 PROPOSITION. Assume that vcd F = k t> 2 and that for every prime p conditions (2) of
3.1 are satisfied. Then there exists a contractible, k+l dimensional F complex of prime
power type.
Proof: For every prime p such that F has nontrivial p torsion use Corollary 3.1 to
construct a k dimensional 7@ acycllc F complex Xp of type p (N.B. there are only f ini tely
many such primes p). Attach free r cells to I JXp to get a k dimensional, k-1 connected
r complex X'. i t follows from Proposition 1.5 that Hk(X') is 77F projective. The existence
of a contractible [" complex of dimension k+l follows now from the arguments which are
analogous to the proof of Lemma 2.1.
0
References.
1. R. Bieri, Homological dimension of discrete groups, Queen Mary College Mathematical
Notes, London, 1976.
2. G. Bredon, Introduction to compact transformation groups, Academic Press, New York,
1972.
3. K. S. Brown, Groups of virtually f inite dimension, Homological group theory (C. T. C.
Wall, ed.), London Math. Soc. Lecture Notes 36, Cambridge University Press,Cambridge,
1979, 27-70.
4. K.S. Brown, Cohomotogy of groups, Springer-Vertag, New York, 1982.
5. F. Connolly and T. Koznlewskl, Finiteness properties of classifying spaces of proper F
actions, to appear in: J. Pure AppI. Algebra.
6. 5. Eilenberg and T. Ganea, On the Lusternlk-Schnirelmann category of abstract groups,
Ann. of Math. 65, 1957, 517-518.
209
J •
7. T. Kozmewskl, Proper group actions on acyclic complexes, Ph. D. dissertation,
University of Notre Dame (1985)
8. R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment.
Math. Helv. 50, i 875, i 55- i 77.
9. D. Quillen, The spectrum of an equivariant cohomology ring, I, 11, Ann. of Math. 94,
1971,549-572 and 573-602.
IO.D Rim. Modules over finite groups, Ann. of Math. 69, 1959, 700-712.
l i.J-P. Serre, Cohomologie des groupes discretes, Ann. of Math. Studies 70, 1971,
77-169.
12.C.T.C. Wall, Finitness conditions for CW complexes II, Proc. Royal Soc. A275, 1966,
129-t39.
13.C.T.C. Wall, Surgery on compact manifolds, Academic Press, New York, 1970.
Unstable homotopy theory of
homotopy representations
by Erkki Laitinen
Introduction
Let G be a finite group, A homotopy representation X of G is a G - CW
-complex such that for each subgroup H of G the fixed point set X H is a finite-
dimensional CW-complex homotopy equivalent to a sphere of the same dimension. The
stable theory of homotopy representations has been well explored by tom Dieck and
Petrie. We shall initiate here an unstable theory.
We first describe the main problems, which all compare two homotopy represen-
tations X and Y of G. A starting point for the whole paper was the cancellation
problem
A. If X and Y are stably G-homotopy equivalent, are they G-homotopy equi-
valent?
More generally, we should give invariants which decide the classification problem
Bo When are X and Y (stably) G-homotopy equivalent?
An obvious invariant of the G-homotopy type of a homotopy representation X is the
dimension function Dim X, which assigns to each subgroup H the dimension of X H.
Let X and Y be homotopy representations with the same dimension function. Then
a G-map f: X + Y induces maps fH: X H + yH between spheres of the same dimensions.
After a choice of orientations we may attach to f the degree function d(f) whose
value at the subgroup H is the degree of fH. It turns out that the degree func-
tion determines the stable homotopy class of a G-map. Hence the analogue of the
cancellation problem for maps is
C. Are G-maps f and g: X ~ Y with the same degree function G-homotopic?
Finally a G-map f: X + Y is a G-hometopy equivalence if and only if def fH = ±I
for each subgroup H of G, so the classification problem B is a special case
of problem
D. What are the possible degree functions of a G-map f: X ~ Y?
We shall answer the problems A - D in reverse order.
A fundamental example of a homotopy representation is a linear G-s~here, the
unit sphere of an orthogonal representation of G on a vector space. It is elemen-
tary to see that a linear G-sphere admits a triangulation as a finite simplicial
G-complex, and can therefore be considered as a finite homotopy representation (i.e.
211
it is finite as a CW-complex).
If X is a linear G-sphere, the celebrated theorem of Segal tells that G-maps
f: X + X are stably classified by the degree function d(f) which may take
arbitrary values in the Burnside ring A(G). There have been two approaches to
Segal's theorem: transversality and equivariant K-theory. They rely on smooth (and
even analytic) techniques. We propose in section 1 an alternative based on a new
type of equivariant Lefschetz class [A(f)] defined for equivariant self-maps
f: X ~ X of finite G-CW-complexes. The class [A(f)] lies in the Burnside ring
A(G) and its characters are the Lefschetz numbers A(fH). The existence of this
class immediately shows that the degrees deg fH satisfy the usual Burnside ring
congruences, when X is a finite homotopy representation. In fact it suffices that
each fixed point set X H has the R-homology of some sphere with any ring R of
coefficients. This kind of a finite G - CW-complex is called a finite R-homology
representation.
Theorem I. Let G be a finite group and let R be any commutative ring. If X is
a finite R-homology representation of G then
deg fH ~ _ ~ ¢(IK/HI)deg fK mod IWHIR, H < G
for all G-maps f: X + X.
(The summation is over those subgroups K of G which correspond to non-triv-
ial cyclic subgroups K/H of the Weyl group WH = NH/H of H, and ¢ denotes the
Euler function.)
When X and Y are different homotopy representations with the same dimension
function, some care is needed in choosing the orientations. Section 2 is devoted
to this question. In general it seems to be impossible to orient all fixed point
sets coherently. Instead we orient the spheres X H and yH only for a sufficiently
small collection of subgroups H. A subgroup H is called an essential isotropy
~ of X if X H + ~ and dim X K < dim X H for each subgroup K strictly larger
than H. If X and Y have the same dimension function we orient them by first
choosing a set of representatives of the conjugacy classes of essential isotropy
groups and by then fixing orientations of X H and yH for this set of subgroups H.
Then a G-map f: X + Y has well-defined degrees deg fH for the chosen subgroups
H, and we show that they can be uniquely extended to all subgroups H by requiring
that deg fH = deg fK either when H and K are conjugate or when H ! K and
dim X H = dim X K.
In section 3 we show following tom Dieck-Petrie [9] that there always exist
G-maps g: Y + X with invertible degrees, i.e. deg gH is prime to IGI for each
subgroup H of G. Composing an arbitrary G-map f: X + Y with such a g we are
back in the situation of Theorem 1 and get congruences for the degrees deg fH
212
Conversely constructing G-maps with preassigned degrees by equivariant obstruction
theory as in [8] we prove
Theorem 2. Let X and Y be finite homotopy representations of a finite group G
with the same dimension function n. There exist integers nH, K such that the
congruences
deg fH ~ _ ~ nH,Kdeg fK mod IWHI, H !G
hold for all G-maps f: X ÷ Y. Conversely. given a collection of integers d = (d H)
satisfying these congruences there exists a G-map f: X + Y with deg fH = d H for
each H ~ G if and only if d fulfils the unstability conditions
i) d H = 1 when n(H) = 1
ii) d H = i, 0 or -I when n(H) = 0
iii) d H = d K when n(H) = n(K) and H < K.
(Here d H is assumed to be constant on conjugacy classes, and n(H) = -i means
X H = yH = ~.) Tom Dieck and Petrie [8] prove a stable version of Theorem 2 for
complex linear G-spheres, and Tornehave [20] proves it for real linear G-spheres.
Both papers determine the numbers nH, K explicitly. We can only say that nH, K =
~(iK/Hi)deg gK/deg gH mod IWHI where g: Y ÷ X is any fixed G-map with invertible
degrees.
Two G-maps f,g: X + Y with the same degree function need not be G-homotopic
even when X and Y equal the same linear G-sphere. However, for nilpotent groups
the situation is satisfying:
Theorem 3. Let G be a finite nilpotent group and let X and
homotopy representations of G with the same dimension function
f,g: X + Y are G-homotopic if and only if
i) deg fH = deg gH for each H < G
ii) fH = gH when n(H) = 0
(note that X H = yH = S ° when n(H) = 0.)
Y be finite
n. Two G-maps
Tornehave [20] proves Theorem 3 for linear G-spheres. Our proof proceeds by
comparison with the linear case. A crucial fact is tom Dieck's theorem that the
dimension function of any homotopy representation of a 2-group is linear [6].
If G is not nilpotent we must impose stability conditions on the dimension
function to guarantee a conclusion of the type of Theorem 3. In particular we get
the following generalization of Segal's theorem: the stable mapping group WG(X,X)
is isomorphic to the Burnside ring A(G) for any finite homotopy representation X
of a finite group G°
In section 4 we study the problem of G-homotopy equivalence of two homotopy
representations X and Y with the same dimension function. Choose a G-map
213
g: Y+X
function
classes of subgroups of G. Since g has invertible degrees, d(g) defines an
element in C(G) x, the group of units of the quotient ring ~(G) = C(G)/IGIC(G).
The invariant which distinguishes the G-homotopy type will be the value of d(g) in
some quotient group of C(G) x, depending on the situation.
Consider first the oriented case. We assume that X and Y have fixed
orientations and require that G-homotopy equivalences are oriented, i.e. have degree
i on each fixed point set. For two choices of g: Y + X the degree functions
differ by the reduction of some invertible element in the Burnside ring. Hence the
value of d(g) in the oriented Pieard group
with invertible degrees. By our orientation conventions the degree
d(g) lies in C(G), the group of integral-valued functions on the conjugacy
Inv (G) = C(G)X/A(G) x
of tom Dieck and Petrie is well-defined and it may be denoted by D°r(x,Y).
Theorem 4. Let X and Y be finite homotopy representations of a finite group G
with the same dimension function. Choose orientations for X and Y. The following
conditions are equivalent:
i) X and Y are oriented G-homotopy equivalent
ii) X and Y are stably oriented G-homotopy equivalent
iii) D°r(x,Y) = i in Inv (G).
Theorem 4 was known earlier for complex linear G-spheres: tom Dieck [3] proves
the equivalence of i) and ii) and tom Dieck and Petrie [81 the equivalence of ii)
and iii).
The reason why the oriented cancellation law holds is easily explained. Indeed,
the congruences of Theorem 2 are stable and they can be determined from a stable G-
map g: Y + X with invertible degrees. If X and Y are stably oriented G-
homotopy equivalent, we can find a stable map g with all degrees equal to i, and
then the congruences become the Burnside ring congruences. The constant function I
satisfies them and is clearly unstable, so there exists an oriented G-homotopy
equivalence f: X + Y.
The oriented case is more complicated. A change of orientations of X and Y
multiplies the value of d(g) by a unit ~ = (e H) where E H = ±i for each subgroup
H of G. If the group of such units is denoted by C x, the value of d(g) in the
Picard group
Pic (G) = C(G)X/A(G)xC ×
is well-defined and denoted by D(X,Y). This invariant unfortunately detects only
stable G-homotopy type, The unstable Picard group Picn(G) is obtained by replacing
214
all three groups ~(G)X,A(G) x and C x in the definition of Pic (G) by the sub-
groups determined by the unstability conditions of Theorem 2. Let Dn(X,Y) be the
value of d(g) in Pic (G). n
Theorem 5. Let X and Y be finite homotopy representations of a finite group G
with the same dimension funcion n. Then
i) X and Y are stably G-homotopy equivalent if and only if D(X,Y) = 1
in Pic (G)
ii) X and Y are G-homotopy equivalent if and only if Dn(X,Y) = 1 in
Pic (G). n
The stable part i) is a result of tom Dieck and Petrie [9]. There is a canoni-
cal map Pic n (G) + Pic (G) which is neither injective nor surjective in general.
Doing a little computation in the Burnside ring, we show that nilpotent groups are
again singled out:
Theorem 6. Let G be a finite nilpotent group with an abelian Sylow 2-subgroup and
let X and Y be finite homotopy representations of G. If X and Y are stably
G-homotopy equivalent they are G-bomotopy equivalent,
Rothenberg [16] proves Theorem 6 for linear G-spheres and abelian groups G.
At present one knows that it holds in the linear case for a wide variety of groups,
e.g. all groups of odd order [21]. In fact, no counterexamples is known to the
cancellation law of linear G-spheres. We give an example which shows that the
cancellation law fails as soon as we step outside the linear category. If p and
q are distinct odd primes and G is a metacyclic group of order pq, we show that
there exist two free smooth actions of G on a sphere which are stably but not un-
stably G-homotopy equivalent (Example 4.11). This contradicts some results in
Rothenberg [17].
We have tried to keep the paper rather self-contained at the expence of length.
Except for some examples, only basic knowledge of G - CW-complexes and obstruction
theory is needed. A special feature is the absence of linear representation theory
which is replaced by permutation representation theory, that is, the Burnside ring.
The assumption of the finiteness of the homotopy representations is due to the
elementary treatment of the equivariant Lefschetz class. It can be removed from all
theorems,
215
i. The equivariant Lefschetz class
Let G be a finite group. We define in this section an equivariant Lefschetz
class [A(f)] for equivariant self-maps f: X + X of finite G - CW-complexes X.
It lies in the Burnside ring A(G) and its characters coincide with the ordinary
Lefschetz numbers A(f K) of the mappings fK: X K ÷ X K. Hence the well-known
congruences between the characters of an element of the Burnside ring also hold
for the Lefschetz numbers A(fK). In the case of a homology representation X
this gives congruences for the mapping degrees deg fK.
Let us first recall the definition and some basic properties of the Burnside
ring (see [5, Ch. i]). The isomorphism classes of finite G-sets form a semiring
A+(G) under disjoint union and Cartesian product, and the Burnside ring A(G) is
the universal ring associated to A+(G). Let H be a subgroup of G. The function
which assigns to each G-set S the number of H-fixed points XH(S) = IsHI induces
a ring homomorphism XH: A(G) ÷ Z, called a character. Clearly XH = XH,, when
H and H' are conjugate. Let ~(G) denote the set of conjugacy classes of sub-
groups of G. The maps XH combine to give an injective ring homomorphism
X = (XH): A(G) + ~ Z ~(G)
whose image is characterized by the congruences
XH(X) ~ - I ¢(IK/HI)XK(X) mod IWHI, H ! G (i.i)
HAK<G K/H cyclic ~ 1
where ~ denotes the Euler function and WH = NG(H)/H.
As the congruences (I.I) are central for this paper, we shall indicate a proof.
It is enough to consider the case where H = i, WH = G and x is a G-set S.
Assume first that S is a transitive G-set S = G/L. Counting the number of
elements of the set
X = {(g,s)[gs = s} c G × S
in two ways, first according to g and then according to s, gives
I SZI = ~ IGsl = IG/LIILI = IG]-
z~G s~S
More generally, decomposing an arbitrary G-set S into G-orbits we get the formula
Isgl = IS/GIIGI
zEG
(i.2)
216
known already to Burnside [2, Th. VII p. 191]. This implies (I.I) and it shows also
that the nature of the congruences is purely combinatorial, although they are usually
derived from the theory of linear representations.
Let X be a finite G - CW-complex, briefly a G-complex. Then the set S n of
(ordinary) n-cells of X is a finite G-set. Following tom Dieck, we define
Definition 1.3. The equivariant Euler characteristic of a finite G-complex X is
the element
IX] = I -l)mSm
m
of the Burns ide r i n g A(G).
It follows at once from the definitions that
×H[X] = x(X H) for HiG. (1.4)
Indeed, the subcomplex X H of the CW-complex X has a cell decomposition (S~), so
the ordinary Euler characteristic of X H is
m H ×(x H) = z (-I) ISml = ×~[x]. m
Two elements x and y in A(G) are equal precisely when XH(X) = XH(Y) for all
H J G. By (1.4) it follows for any pair of finite G-complexes X and Y that
[X] = [Y] if and only if ×(X H) = ×(yH) for each H J G. (1.5)
This shows that [X] is an invariant of the G-homotopy type of X. In particular
it does not depend on the G - CW-structure (S). One could also define A(G) as m
the set of equivalence classes of finite G-complexes under the relation (1.5), and
this definition of the Burnside ring works also for compact Lie groups, see [5, Ch.
5.5].
Corollary 1.6. Let G be a finite group. If X is a finite G-complex then
x(X g) = x(X/G) IG I. L
g~G
Proof. Combine (1.4) to (1.2).
Let X be a finite G-complex with cell structure (Sm). If
G-map, we may assume up to G-homotopy that it is cellular. Then
fm: Cm(X) + Cm(X) between the integral cellular chain groups
f: X~X is a
f induces maps
217
Cm(X) = Hm(Xm,xm-I;z ) ~ Z[Sm].
If c E S m is an m-cell of X let nf(x) E Z be the coefficient of c in fm(C)
Cm(X) with respect to the basis S m. By equivariance nf(gc) = nf(c) for each
cell gc in the orbit of c and we may denote unambiguously nf(Gc) = nf(c).
Decompose that G-set S into orbits S = U G and define the equivariant m m S /G c
trace of fm: Cm(X) ÷ Cm(X) by m
TrG(f m) = E nf(Gc)Gc ~ A(G). S /G m
Definition 1.7. Let X be a finite G-complex. Th___~e equivariant Lefschetz class of
a G-map f: X + X is the element
[A(f)] = ~ (-l)mTrG(fm)
of the Burnside ring A(G).
Proposition 1.8. The class [A(f)] depends only on the G-homotopy class of
It satisfies XH[A(f)] = A(f H) for all subgroups H J G.
f.
Proof. The character Xe[A(f) ] is
Xe[A(f)] = Z (-I) TM E nf(Gc)IGc I = 2 (-l)mTr(fm: Cm(X) ÷ Cm(X)) = A(f), m S /G m
m
the ordinary Lefschetz number of f: X + X computed from cellular chains. More
generally XH[A(f)] = A(f H) for all H J G since C,(X H) = Z[S~]. If f is G-
homotopic to g, then fH and gH are homotopic and A(f H) = A(g H) for all
H ~ G. This implies that [A(f)] = [A(g)].
Corollary 1.9. Let G be a finite group and let X be a finite G-complex. Then
A(fH) ~ - I ~(IK/HI)A(fK) mod IWHI, H ! G,
H~K<G K/H-cyclic ~i
for any G-map f: X ~ X. D
It is easy to compute the Lefschetz number of the action g: X + X of group
element g ~ G. The result is the Lefschetz fixed point formula
A(g) = X(xg), g ~ G. (I.I0)
218
Indeed, g induces a permutation representation on
Cm(X) + Cm(X)) = IS~I and summing up gives
A(g) = Z (-l)mlSmgl = ×(Xg),
Cm(X) = Z[Sm]. Hence Tr(gm:
Remarks. i. W, Marzantowicz informed the author in the Poznafi conference of his
unpublished thesis [14] where he had developed an equivariant Lefschetz class for
finite G-complexes along somewhat different lines. See also [15, Th. 1.2].
2. The equivariant Euler and Lefschetz classes can be defined for finite groups
G and finite-dimensional G-complexes X provided each fixed point set X H has
finitely generated integral homology. The formulae (1.4), (1.6), (1.8), (1.9) and
(i.i0) remain valid. The idea is to approximate the cellular chain complex by
finitely generated projective complexes over the orbit category O G. For this and
a generalization to arbitrary G-spaces, see [13].
3. S. Illman pointed out to the author that the construction of the equivariant
Lefschetz class can be modified to cover finite G-complexes where G is a compact
Lie group. Let X be a finite G-complex with equivariant m-cells S m = (dj) d.
corresponding to G/H. × D m, and let X m denote the equivariant m-skeleton. Then ]
Hm(Xm,xm-I;z) ~ ~ H (D m x G/Hj,S m-I × G/Hj;Z) ~ ~{) Ho(G/Hj;Z). d. ES m d.~S J m j m
Hence the group Cm(X) = Hm(Xm,xm-I;z) is free abelian with a basis consisting of
the path components of G/Hj, dj ~ S m, (The homology of the chain complex C,(X)
is H,(X/Go;Z) where G o < G is the identity component.)
Define now the equivariant trace of a cellular G-map f: X ÷ X by
TrGf m = Z nf(dj)[G/Hj] ~ A(G) d . ES
J m
where nf(dj) is the coefficient of any component c. of G/H, in f,(cj). Then J 3
the equivariant Lefschetz class [A(f)] = ~ (-l)mTrGfm ~ A(G) satisfies XH[A(f)] =
A(f H) for every closed subgroup H i G such that WH is finite. A crucial point
is that X H is always a finite WH-complex.
Let R be a commutative ring (with I). An R-homology representation of G
is a finite-dimensional G-complex X such that for each subgroup H of G the
fixed point set X H has the R-homology of a sphere S n(H), i.e. H~(xH;R)
H,(sn(H);R). We set n(H) = -I when X H = @. It is not reGuired that X H is n(H)
dimensional as a CW-complex. An R-homology representation X is finite if X is
a finite G-complex.
Every self-map f: X ÷ X of an R-homology representation has degrees deg fH
in R defined for all subgroups H i G by
219
Hn(H)(xH;R) f~(x) = deg fH'x, x ~ ~ R
for n(H) ~ 0 and by the convention deg fH = I when X H = ~.
Theorem I. Let G be a finite group and let R be a commutative ring. If X is
a finite R-homology representation of G then
deg fH ~ _ ~ ~(iK/Hl)deg fK
HqK<G K/H cyclic #I
for all G-maps f: X + X.
Proof. Assume first that R = Z. Since
Euler class of X satisfies by (1.4)
mod IWHIR
H,(X H) ~ H,(S n(H)) the equivariant
XH[X ] = 1 + (-I) n(H), H _< G.
Similarly from (1.8) we get the formula
XH[A(f) ] = I + (-l)n(H)deg fH, H ! G
for the equivariant Lefschetz class. Hence the product
{f} = ([X] - l)([A(f)] - i) c A(G)
has characters XH{f} = deg fH, and the claim follows from the Burnside ring
congruences (i.i).
If R is arbitrary, IX], [A(f)] and {f} are still defined as elements of
A(G) and we must show that the R-degree deg fH is the image of the integer
×H{f} under the canonical homomorphism Z ~ R. Denote by C n = Cn(X,R) the
cellular chain of group of X with coefficients in R in dimension n. As S
finite CW-complex each C is a finitely generated free R-module. Let Z and n n
denote the cycle and boundary subgroups of C n. The fact that Hn = Hn(X,R) is
free in all dimensions n implies by induction starting from dimension 0 that
the sequences
is a
B n
0 ~ Zn+ 1 + Cn+ 1 ~ Bn + 0, 0 + Bn + Zn ÷ Hn ÷ 0
split. Hence B and Z are finitely generated projective R-modules for all n. n n
The Bourbaki trace is therefore defined for the R-modules Cn,Zn,B n and H n.
The usual proof of the Hopf trace formula then works and we can compute the Euler
and Lefschetz classes either from homology or from chains. On the chain level
they correspond to the elements IX] and [A(f)~ and on homology we have the
220
same situation as in the case R = Z.
In order to get congruences for G-maps f: X + Y between two different repre-
sentations we have to construct suitable maps g: Y + X. This is a problem in
obstruction theory, so we must restrict the dimensions and connectivity of the fixed
point sets. We shall work with the homotopy representations of tom Dieck and Petrie
[9].
Definition i.ii, Let G be a finite group. A hom0topy representation X of G
is a finite-dimensional G-complex such that for each subgroup H of G the fixed
point set X H is an n(H)-dimensional CW-complex homotopy equivalent to S n(H).
We set n(H) = -I when X H is empty. The homotopy representation X is finite
if X is a finite CW-complex,
A homotopy representation is a Z-homology representation. Hence Theorem 1
implies
Corollary 1.12. If X is a finite homotopy representation then any G-map f: X + X
satisfies the congruences of Theorem 1 with R = Z. D
As remarked above, the finiteness assumption on X can be removed from Theorem
1 and Corollary 1.12.
As an illustration of Theorem I we give simple proofs of two classical results,
one of which is used in the sequel. We need the following fact from obstruction
theory: If (X,A) is a relative n-dimensional G-complex such that G acts freely
on X \ A and if Y is an (n-l)-connected G-space, then any G-map A + G can be
extended equivariantly over X. This is easily proved by induction over cells (no
obstruction groups are necessary).
Proposition 1.13. (Smith) Let p be a prime and let X and Y be finite homotopy Z Z
representations of Z such that dim X = dim Y and dim X p = dim Y P. If P
f: X + Y is equivariant, then
Z deg f ~ 0 (mod p) if and only if deg f p ~ 0 (mod p),
Z Proof. Since both X p and
n(Z ) S P (or empty) we can choose
extension g: Y ~ X exists since
~i(X) = 0 for i < dim Y. Z
deg h £ deg h p (mod p) or
Z Y P are homotopy equivalent to the same sphere
Z Z gl: Y p + X p with deg gl = I. An equivariant
Z (Y,Y P) is a relatively free Z -complex and
P By Theorem I the composite h = g o f: X + X satisfies
Z Z deg g deg f ~ deg gl deg f p = deg f p (mod p).
221
Z If deg f p } 0 (mod p) then clearly de g f } 0 (mod p). Applying this to the map
g we see that deg g ~ 0 (mod p), whence the other implication.
Proposition 1.14 (Borsuk-Ulam) Let G be a non-trivial finite group and let X
and Y be free finite homotopy representations of G. If f: X + Y is equivariant
then dim X < dim Y.
Proof. It is enough to asume that G has prime order p. If dim X > dim Y,
we can find a G-map g: Y + X as above. Theorem 1 implies
Z deg h ~ deg h p (mod p)
for the composite h = g o f. But
deg h = 0 since up to homotopy h
contradiction shows that dim X < dim Y. a
The idea of this proof of the Borsuk-Ulam theorem is due to Dold [i0].
we don't need any manifold structure in computing the degrees.
Z deg h p = i since the actions are free and
S n S m S n factors as ~ ~ with m < n. The
However
222
2. The isotropy ~roup structure an ~ orientation
This preliminary section is concerned with the isotropy group structure of
homotopy representations. We introduce the concept of an essential isotropy group.
It is an isotropy group H such that the fixed point sets X K have strictly
smaller dimension than X H when K > H. The essential isotropy groups are needed
in order to define orientations for arbitrary homotopy representations. Finally
we discuss certain manifold-like conditions on the isotropy groups used by tom Dieck
in his work on dimension functions, and we show that for nilpotent groups they are
always satisfied.
Let G be a finite group and let X be a homotopy representation of G.
Hence each fixed point set X H is an n(H)-dimensional CW-complex homotopy equivalent
to S n(H). If f: X + X is equivariant then the degrees deg fH satisfy the
congruences of Theorem i. The next lemma implies further relations.
Lemma 2.1. Let X be a homotopy representation of a finite group G. If n(H) =
n(K) for subgroups H > K of G then the inclusion X K c X H is a homotopy
equivalence. Each subgroup H of G is contained in a unique maximal subgroup
with n(H) = n(H).
Proof. Let H ! K be subgroups with n(H) = n(K) = n. If n = -I, then
X H = X K is empty and if n = 0 then X H = X K consists of two points. Assume
n > 0. As X K and X H are CW-complexes of the homotopy type of S n, it suffices
to show that the inclusion i: X K c X H induces an isomorphism on integral homology.
The exact sequence of the pair (xH,x K) contains the portion
0 ~ Hn(XK) ----+ Hn(XH) ~ Hn(XH,x K) + 0
where the first two groups are Z. The third group is torsion free since (xH,x K)
is an n-dimensional relative CW-complex. It follows that i, is an isomorphism.
If K 1 and K 2 contain H and n(H) = n(K I) = n(K 2) = n, let K = <KI,K2>
be the subgroup generated by K 1 and K 2, Then X K = X K1N X K2 and the second
claim follows if we can show that n(K) = n for all possible choices of K 1 and
K1 xKI xK2 K1 xK2 X H K 2. Assume n > 0 and let i: X c U and j: X U c denote the
inclusions. The composite
Hn(XKI) i, Hn(XKI U X K2) ~ Hn(XH)
is an isomorphism by the first part of the proof. Hence j, is surjective. But
j, is injective as well since the CW-pair (xH,x K1U X K2) is n-dimensional. There-
fore j, and i, are isomorphisms. The Mayer-Vietoris sequence
223
K1 K2) K 1 xK2) 0 + Hn(X K) + Hn(X ) ~ Hn(X + Hn(X U +...
now shows that H (X K) = Z so that n(K) = n.
If ~ is not empty then H is an isotropy group such that the singular set
x >~= {x ~ XIG x > ~}
of X ~ is a subcomplex of smaller dimension than X ~. These isotropy groups resemble
the principal isotropy groups in smooth G-manifolds.
Definition 2.2. Let X be a homotopy representation of a finite group. A subgroup
H of G is called an essential isotropy group of X if H = H and X H + ~. The
set of essential isotropy groups is denoted by EssIso (X).
For linear representation spheres or more generally for locally smooth G-mani-
folds all isotropy groups are essential, since all fixed point sets X H are connected
manifolds so that X K c X H and dim X K = dim X H imply X K = X H. The set EssIso (X)
depends only on the dimension function Dim X: @(G) ÷ Z,
Dim X(H) = dim X H, H < G,
and it will also be denoted Iso (Dim X) as in [6]. It follows that Esslso (X) =
EssIso (y) when X and Y are G-homotopy equivalent.
It is now clear that a G-map f: X + X satisfies
The unstability conditions 2.3
i) deg fH = i when n(H) = -I
ii) deg fH = i, 0 or -i, when n(H) = 0
iii) deg fH = deg fH.
The congruences of Theorem 1 and the unstability conditions 2.3 turn out to be
necessary and sufficient conditions for the existence of a G-map f: X + X with
given degrees deg fH We shall prove a more general version for maps f: X ÷ Y
between two homotopy representations X and Y such that Dim X = Dim Y, i.e.
dim X H = dim yH for each H < G. (2.4)
But here it is no longer clear how to orient the fixed point spaces X H and yH
coherently.
If X H # 0 then the Weyl grop WH = NH/H acts on X H and therefore on
~n(H)cxH;z) = Z. Let e~(g) = 1 (resp. -I) if g ~ NH preserves (resp. changes)
a generator of ~n(H)(xH;z). This defines the orientation homomorphism
224
X eH: WH + {_+I}
X H @ and we agree that e~ : I when X H = @, cf. [9, 1.7]. If Dim X = when
Dim Y we have the following important observation of tom Dieck and Petrie, left
unproved in [9, p. 135].
Lemma 2.5. If X and Y are finite homotopy representations of X Y
Dim Y then e H = e H for each subgroup H of G.
G with Dim X =
Proof. We may assume that H = 1 and WH = G. Denote n(g) = dim X g for
g ~ G and let n = dim X. The Lefschetz number of g is A(g) = 1 + (-l)ne~(g).
By the Lefschetz fixed point formula (l.i0) it equals ×(X g) = I + (-i) n(g). Hence
e~(g) = (-i) n-n(g), g ~ G,
is determined by he (co)dimension function of X. D
Let X and Y be homotopy representations of G with Dim X = Dim Y and let
f: X ~ Y be a G-map. To define deg fH we must orient X H and yH. If there
xgHg -1 g ~ NH with eH(g )x = -i then X H = but left translation by g exists
changes the orientation. Hence it is difficult to orient the subspaces X H X
coherently unless all orientation homomorphisms e H are trivial. Instead of
requiring this we choose one subgroup H from each conjugacy class of EssIso (X) =
Iso (n), where n = Dim X is the dimension function. Let ~n(G) be the set of the
chosen representatives.
Definition 2.6. An orientation of a homotopy representation X of G is a choice
of generator of ~n(H)(xH;z) for each H in ~n(G), n = Dim X.
If X and Y are oriented by using the same set #n(G) then a G-map
f: X + Y has degrees deg fK for subgroups K ~ #n(G). We define deg fH for all
subgroups H as follows. The group H is conjugate to a unique group K in
~n(G), say_ H = gKg -I. The left t_ranslation by g induces a homeomorphism
1 : X K ~ X H and the inclusion X H c X H is a homotopy equivalence by lemma 2.1. g
We transport the orientation of X K to X H along the composite homotopy equivalence
1 x K : x H.
If yH is oriented by using the same translation and inclusion, we get deg fH =
deg fK. Another choice of g may result to different orientations of X H and yH, X Y
but since e K = e K (2.5) they are either both preserved or both reversed, and
deg fH remains unchanged.
With these conventions we conclude
225
Proposition 2.7. Let X and Y be finite homotopy representations of G with
Dim X = Dim Y = n. Orient X and Y using the same set of representatives of
conjugacy classes in Iso (n). Then any G-map f: X + Y has well-defined degrees
deg fH for each subgroup H of G. The degrees deg fH depend only on the
conjugacy class of H and they satisfy the unstability conditions 2.3. D
One of the principal motives for studying homotopy representations is the
construction of complexes which can be used as first approximation to smooth or PL
actions on spheres. Therefore it is desirable that the isotropy group structure of
a homotopy representation X would resemble that of an action on a genuine sphere.
Consider the following two conditions:
(A) EssIso (X) = Iso (X), i.e. for any isotropy group H of X
L > H implies n(L) < n(H).
(B) Iso (X) is closed under intersections.
They are both satisfied if X is a locally smooth G-manifold, since in that case the
fixed point sets are connected submanifolds, and therefore X H = X H for any H ! G.
Remark. Our terminology follows tom Dieck and Petrie [9]. Tom Dieck uses later
a more restrictive notion of homotopy representation where conditions (A) and (B) are
required as a part of definition [6, (1.4),(1.5) p. 231].
We first note that condition (A) always implies condition (B).
Proposition 2.8. If X is a homotopy representation and
Iso (X). is closed under intersections.
EssIso (X) = Iso (X) then
Proof. Assume that EssIso (X) = Iso #X~. In order to prove that Iso (X) is
closed under intersections it suffices to show that each subgroup H i G is contain-
ed in a unique minimal isotropy group, viz. H. If H ~ Iso (X) the claim is
obvious. If H ~ Iso (X) then X H = U X L where the union is over all isotropy
groups L > H. Let n = n(H). Then dim X H = n and dim X L < n for other
isotropy groups L > H. We claim that they all contain H.
Otherwise, let K ~ Iso (_X) be minimal with respect to H < K, H ! K. Then
m = n(K) < n and dim X K N X H < m by condition (A). If L ~ Iso (X), L > K and
L + K, the~ dim X K n X L < m. Indeed, if H i L then X L c X ~ and dim X K N X L J
dim X K N X H < m. On the other hand if H ! L then KL > K by the minimality of K
and dim X K N X L = dim X KL < m by condition (A). Hence Jf we denote Y = U X L,
union over isotropy groups L > H different from K, we have X H = X K U Y and
dim (X K N Y) < m. The Mayer-Vietoris sequence
0 = Hm(XK N Y) + Hm(X K) • Hm(Y) ~ Hm(XH) = 0
now leads to contradiction since H (X K) m Z. m
226
If Esslso (X) = Iso (X), we can thus characterize H either as the minimal
isotropy group containing H or as the maximal subgroup of G such that X H = X H.
It is easy to give an example which demonstrates that condition (B) does not
imply condition (A). Let X be the union of a circle S 1 with trivial G-action
and IGI copies of the unit interval freely permuted by G, glued together at one
end to a common base point x o ~ S I. Then Iso (X) = {I,G}
is closed under intersection but the isotropy group 1 is ~ f ~
not essential since X 1 = X and X G = S 1 have dimension
i. Moreover we see that adding suitable whiskers any sub-
group H j G may appear as an isotropy group. However, X
is G-homotopy equivalent to the trivial G-space S 1 which
satisfies both (A) and (B).
We shall show that if condition (B) is modified to the homotopy invariant form
"EssIso (X) is closed under intersection" then condition (B) implies condition (A)
up to homotopy. This is done by collapsing away the inessential orbits of X.
Lemma 2.9. The following conditions on a homotopy representation
i) EssIso (X) is closed under intersection
ii) If H < K then H < K.
X are equivalent:
Proof. Let EssIso (X) be closed under intersection and let H < K. If
X K = ¢ then K = G and clearly H < K. Otherwise both H and K are essential
isotropy groups and therefore H g K is an essential isotropy group, too. Since
H < H N K < H must have H N K = H i.e. H < K.
Conversely, assume that ii) holds. Let H and K be essential isotropy
groups. Then H n K i H implies H N K i H = H and similarly H N K J K = K.
It follows that H N K < H N K, As the other inclusion holds trivially, we get
H N K = H N K. Moreover X HNK ~ X H + ~. Hence H n K is an essential isotropy
group. D
Proposition 2.10. Let X be a homotopy representation of a finite group G. If
EssIso (X) is closed under intersection then X is G-homotopy equivalent to a
homotopy representation Y which satisfies Iso (Y) = EssIso (X) and
(A) If H ~ Iso tY) and L > H then n(L) < n(H)
(B) Iso (Y) is closed under intersection.
If X is finite then Y can be chosen finite.
Proof. A family F of subgroups of G is called closed if F is closed
under conjugation and each subgroup containing a member of F belongs to F. If
X H F is a closed family, let X(F) = UHE F be the set of points of X whose
isotropy groups belong to F. For each closed family F we shall construct a G-
complex Y(F) and a G-map fF: X(F) ÷ Y(F) such that
227
a) fF is a cellular G-homotopy equivalence
b) Iso Y(F) = EssIso (X) n F.
If F = {G}, we put X(F) = Y(F) = X G and let fF be the identity. Assume
by induction that fF: X(F) + Y(F) satisfying a and b is already constructed.
Choose a maximal subgroup H not in F and let F' = F U (H). Then X(F') =
X(F) U X (H) contains X(F) as a G-subcomplex° If H ~ EssIso (X) we define
Y(F') as the adjunction space X(F') DfF Y(F). It is a G-complex since fF is a
cellular G-map. As fF is a homotopy equivalence and (X(F'),X(F)) is a CW-pair
it is a standard fact that the canonical map fF': X(F') + Y(F') is a homotopy K
equivalence [23, I 5.12]. This applies also to fF' for each K ! G. Hence fF'
is a G-homotopy equivalence which satisfies a and b.
In the case H ~ EssIso (X) we let Y(F') = Y(F). In order to construct an
extension of fF: X(F) + Y(F) to X(F') it is enough to find a cellular G-de-
formation retraction X(F') + X(F), or equivalently a cellular ~-deformation re-
traction X H ~ X >H. Hence it suffices to show that the inclusion i: X >H c X H is
a WH-homotopy equivalence.
If K/H ! WH is non-trivial then K > H and (x>H) K = (xH) K = X K so that
is the identity. Thus we are left with proving that i: X >H c X H is an ordinary
h~motopy equivalence. We have assumed that H > H. Consider the inclusions
X H c X >H c X H. The middle term is_ a finite union X >H = UK> H X K of subcomplexes
closed under intersection and X H c X H is a homotopy equivalence. If we prove
that X H n X K c X K is a homotopy equivalence for each K > H, an Easy induction
shows that X >H c X H is a homotopy equivalence. But X H fl X K = X HK and H < K
implies H < K since EssIso (X) is closed under intersections (2.9). Hence - ~K K
K < HK < KK = K which implies n(HK) = n(K). By lemma 2.1 the inclusion X c X
is a homotopy equivalence.
Finally we see that the G-cells of Y consist precisely of those G-cells of
which have type G/H with H ~ EssIso (X). Thus Y has fewer cells than X and
obviously Y is finite whenever X is finite. D
Remark. Although the map X + Y is a G-homotopy equivalence, it may be geometri-
cally complicated. Here is an example where it is not a simple G-homotopy equiva-
lence. One can realize the generator of Wh (Z 5) = Z as the equivariant Whitehead
torsion of a pair (W,x) where W is a 3-dimensional finite Z5-complex and Z 5
acts freely outside the fixed point x [12, Ex. 1.13]. Let Z 5 act trivially on
S 3 and form the wedge X = S3VW along x. Since W is contractible, X = S 3 and
Z5 S 3 X = so that X is a 3-dimensional homotopy representation of Z 5, The
inclusion y = S 3 c X is a Z5-homotopy equivalence which is not simple.
If some restrictions must be put on the isotropy group structure of a homotopy
representation, we propose the condition "EssIso (X) is closed under intersection
.K I
228
since it is G-homotopy invariant and it implies conditions (A) and (B) up to G-
homotopy. However, in this paper we need no additional assumptions. One reason
is that we obtain the sharpest results in the case of nilpotent groups and for them
the condition is automatically fulfilled as we shall see below in Proposition 2.12.
Let X be a homotopy representation of G. Then H = T is the union of all
subgroups K i G such that n(K) = n(1). Since this set is closed under conjuga-
tion, H is a normal subgroup of G. We call H the homotopy kernel of X. If
G is solvable then X is G-homotopy equivalent to the representation X H where
the action of G has kernel H in the usual sense:
Proposition 2.11. Let G be a finite solvable group. If X is a homotopy rep-
resentation of G with homotopy kernel H, then the inclusion X H c X is a G-
homotopy equivalence.
Proof. It suffices to show that the inclusion X HK c X K is an ordinary
homotopy equivalence for each subgroup K J G. Since K is solvable we can find
a tower
1 = K ° < K 1 <...< K n = K
such that K i < Ki+ 1 and Ki+I/K i has prime order for i = 0 .... ,n-l. We shall
HK. K. show by induction that X : c X i is a homotopy equivalence for all i. When
i = 0 the inclusion X H c X 1 is a homotopy equivalence by lemma 2.1 since n(H) =
n(1). Assume the claim holds for the value i. Then K i <~ Ki+ 1 and H ~ G imply
K. HK.
that Ki+ I ~ N(HKi). Hence Ki+I/K i acts on the pair (X :,X l). Now Ki+I/K i K. HI<.
Zp for some prime p and the induction assumption impl~es H,(X I,X 1;Zp) = 0.
By Smith theory the fixed point pair (X Ki+l ,X HKi+I) has also trivial Zp-homology
HKi+I c X Ki+l is a homotopy equiv- [i, III 4.1]. Hence n(Ki+ I) = n(HKi+ I) and X
alence by lemma 2.1, a
Proposition 2.12. Let X be a homotopy representation of a finite nilpotent group
G. Then EssIso (X) is closed under intersection.
Proof. We must show that H i K implies H J K (2.9). It is enough to
consider the case K = K. We fix K = K and prove the claim by downwards induc-
tion on H. If H = K or H = H the claim holds trivially. Let then H < K be
such that H < H and assume we have already proved that H < L J K implies [ i K.
Since K is nilpotent and H < K we have K 1 = NK(H) > H [ii, Th. 3.4 p. 22].
Hence the inductive assumption applies to K 1 and KI i K.
Consider X H as a homotopy representation of NH = NG(H). It has kernel
HI X H c is an NH- H I = NH N H = N~(H), and H I > H as above. The inclusion X
229
HIK 1 homotopy equivalence by Proposition 2.11. Since K I is a subgroup of NH, X c K 1
X is an ordinary homotopy equivalence so that HIK 1 ! K1 or HI J KI" Then
K1 j K gives H I J K. Hence the inductive assumption applies to Hl,too, and
HI ! K. But HI = ~ since H ! H 1 ! ~" Hence H i K.
The main result is now an immediate corollary of Propositions 2.10 and 2.12:
Proposition 2.13. Let G be a finite nilpotent group. Every homotopy representa-
tion of G is G-homotopy equivalent to a homotopy representation X such that
(A) If H ~ Iso (X) and L > H then n(L) < n(H)
(B) Iso (X) is closed under intersection.
X can be chosen finite if the original homotopy representation is finite, m
Remarks. I. If G is abelian, the proof of Proposition 2.13 simplifies consider-
ably. The only geometric imput is the fact that the fixed point set of Z acting P
on a finite-dimensional contractible complex is mod p acyclic.
2. In the ease of a p-group G Proposition 2.12 can be deduced from the work
of tom Dieck. He shows that each homotopy representation of a p-group has the same
dimension function as some linear representation sphere [6, Satz 2.6]. Since the
dimension dunction determines the essential isotropy groups and EssIso (X) = Iso (X)
is closed under intersection when X is linear, Proposition 2.12 follows for p-
groups. This argument does not apply to general nilpotent groups or even to abelian
groups since their dimension functions are only stably linear.
We close this section by an example of a homotopy representation X of G with
homotopy kernel H such that X H c X is not a G-homotopy equivalence and
EssIso (X) is not closed under intersection. It shows that some restrictions on
the group G are necessary in Propositions 2.11, 2.12 and 2.13.
Example 2.14. The binary icosahedral group I* acts on the unit quaternions S 3
by left and right multiplication. The space of the right cosets Z = S3/I * is the
Poincarg homology 3-sphere, an it inherits a smooth left action of the icosahedral
A 5 group A 5 = I*/Z 2 with precisely one fixed point Z = {eI*} (for more details,
see [I, 1.8 (A)]). Choose a small open slice U around the fixed point. It is A 5-
homeomorphic to a 3-dimensional linear representation space V of A 5. Clearly
V cannot be the trivial representation. As the degrees of the non-trivial irredu-
cible real representations of A 5 are 3, 4 and 5 [18, IB.6], V must be irre-
ducible, hence conjugate to the icosahedral representation. It follows that
dim V H = 1 for cyclic subgroups H + 1 of A 5 and dim V H = 0 for other subgroups
H ~ I. By Smith theory Z H is a mod p homology sphere when H is one of the
cyclic subgroups Zp, p = 2, 3 or 5 [I, III 5.1]. The only possibility is then
that
230
Z
g p m S I, p = 2, 3 or 5.
The normalizer of Zp in A 5 is the dihedral group D2p and
ED2p (~ZP)Z2 ~ (SI) Z2 = S ° = , p = 2, 3 or 5.
Finally D4 = Z 2 ~ Z 2 has normalizer A 4 and
A4 (ED4)Z3 (sO) Z3 S °. = ~ =
This describes the fixed point sets of all non-trivial subgroups H J A 5.
The complement Y = E \ U is an acyclic 3-manifold with boundary 8Y m S(V)
and it can be given the structure of a finite A5-complex. Then Z = Y Usy Y is A~
a homology 3-sphere and Z H m Z H for 1 < H < A 5. However Z ~ = ~ because the
A 5 fixed point E lies in U = g \ Y. The join Z*Z is simply-connected and there-
fore a 7-dimensional homotopy representation of A 5 (it is in fact homeomorphic to
S 7 by the double suspension theorem, but this in inessential). From the adjunction
space
X = (Z'Z) U SY Y
where Y lies inside one copy of Z in Z*Z and in the middle of the suspension
SY = S°*Y. X is an A5-complex in the obvious fashion. It is simply-connected
since Z*Z and SY are simply connected, and H,(X) = H,(S 7) since Y and SY
are acyclic. Hence X = S 7. We claim that X is also a 7-dimensional homotopy
Z representation of A 5. Indeed, X p is S 3 with D 2 attached along a diameter
Z xA4 so X p = S 3. Similarly X D2p or is S 1 with D 1 attached along the middle
xA4 A 5 S ° point, so X D2p = = S I. Finally X = consists of the two cone points of
SY.
Consider X as a homotopy representation of G = A 5 × Z 2 where A 5 acts as
earlier but Z 2 switches the two cones in SY and leaves Z*Z invariant. The
fixed point sets of the G-action on X are those of the A5-actions on Z*Z and X. Z
Hence the homotopy kernel of the G-action is Zo with X 2 = Z*Z. X cannot be
Z 2 G-homotopy equivalent to X since
A 5 A 5 X = X °, (xZ2) A5 = (Z'Z) = ~.
We also see that A 5 and Z 2 are essential isotropy groups but their intersection
1 is a non-essential one.
231
3. Classification of G-maps
In this section we characterize the set of fixed point degrees of a G-map
f: X ÷ Y between two finite homotopy representations with the same dimension
function. This computes the stable mapping groups WG(X,Y) since two G-maps
with the same fixed point degrees are stably G-homotopic. In particular we prove
that WG(X,X) is canonically isomorphic to the Burnside ring A(G) for any finite
homotopy representation X. Finally we show that for nilpotent groups G the
degrees deg fH already determine the G-homotopy class of f.
We shall need an unstable version of the equivariant Hopf theorem [5, Th.
8.4.1]. We start by recalling some equivariant obstruction theory. Let G be a
finite group. Assume that (X,A) is a relatively free G-complex i.e. (X,A) is
a relative G - CW-complex such that G acts freely on X \ A. If dim (X \ A) =
n ~ 1 and Y is an (n - l)-connected and n-simple G-space then every G-map
f: A + Y extends to a G-map F: A + Y and the G-homotopy classes of extensions
relative to A are classified by the equivariant cohomology group H~(X,A;~nY).
X k The group H~(X,A;~nY) is defined as follows. Let be the k-skeleton of
X relative to A and denote by
C k = Ck(X,A) = Hk(Xk,xk-l;z )
the cellular chain groups. Then C.7~ is a chain complex of free ZG-modules. The
equivariant cohomology groups H~(X,A;~) with coefficients in a ZG-module ~ are
the homology groups of the complex HOmzG(C,,v) of equivariant cochains.
For any ZG-module M let
M G = {m ~ Mlgm = m,g c G}, M G = M/<m - gmlg ~ G>
denote the modules of invariants and coinvariants. The norm N(m) = E g~G gm
induces a canonical map N: M G + M G whose kernel and cokernel are by definition
the Tate groups
AO Ho(G,M) = Ker (N: M G + MG), H (G,M) = Coker (N: M G + MG).
The unequivariant chains HomZ(Ck,~) can be considered as a G-module by defining
the translate of f: C k ÷ ~ by g ~ G to be the function gf: x ~ gf(g-lx). Then
the equivariant chains are the invariants HOmzG(Ck,~) = HomZ(Ck,~) G. But it is
easy to see that the norm homeomorphism
N: HomZ(Ck,~) G ~ HomZ(Ck,~) G
is an isomorphism because C k is ZG-free (in fact the ZG-module Homz(Ck,~) is
232
coinduced, hence cohomologically trivial). It follows that H~(X,A;~) is also the
homology of the complex of coinvariants
H~(X,A;~) ~ H,(Homz(C,,~)G).
As dim (X \ A) = n we have an exact sequence
Homz(Cn_l,~) + Homz(Cn,~) + Hn(X,A;~) ~ 0.
Applying the right exact functor M + M G gives the exact sequence
HOmz(Cn_t,~) G ~ Homz(Cn,~) G ~ Hn(X,A;~)G ~ 0.
We just saw that the cokernel of the first map is H~(X,A;~). Hence we get the
amusing formula
H~(X,A;~) = Hn(X,A;~)G (3.!I
which holds for any n-dimensional relatively free G-complex (X,A) and any ZG-module
~. Note that the coinvariants Hn(X,A;~)G cannot be replaced by the invariants since
the functor M + M G is left but not right exact. Note also that G acts on
Hn(X,A;~) by acting both on the chains and on the module ~. If H,(X,A;Z) or
has finite type over Z so that we can use the universal coefficient formula
Hn(X,A;~) ~ Hn(X,A);Z) ~ ~, Z
then G acts diagonally on the tensor product.
For any G-module M there are natural homomorphisms
t: M + MG, p: M G + M
where t is the quotient map and p is induced by the norm. In the situation of
(3.1) they induce homomorphisms
t: Hn(X,A;~) ~ H~(X,A;~), p: H~(X,A;~) ~ Hn(X,A;~).
If A = ~ then H~(X;~) = Hn(X/G;~) is the cohomology of X/G with twisted
coefficients, t is the cohomology transfer and p is induced by the covering
projection X + X/G.
Lemma 3.2. Let (X,A) be a relatively free G - CW-pair of dimension n. Assume
that Hn(x;z) ~ Z and that ~ is isomorphic to Hn(x;z) as a ZG-module. If
dim A ! n - 1 then the composite homomorphism
233
P HG(X,A;~) --'+ Hn(X,A;~) + Hn(x;~) = Z
has image IGIZ. If moreover dim A J n - 2 then H~(X,A;~) ~ Z.
Proof. If dim A ! n - I the homomorphism Hn(X,A;~) ~ Hn(X,~) is a surjec-
tion and it induces an epimorphism Hn(X,A;~)G + Hn(x;~)G . The tensor product
Hn(x;~) ~ Hn(x;z) ~Z ~ is a trivial ZG-module since G acts diagonally through the
same homomorphism ¢: G + {±i} = Aut Z on both factors. Hence Hn(x;~) G =
Hn(x;z) = Z and the norm p: Hn(x:~)G + Hn(x;~) is multiplication by IGI. The
first claim follows from the diagram
Hn(X,A;w) G ~ Hn(x;~)G ~ 0
1 P
Hn(X,A;~) + Hn(x;~) ~ 0
with exact rows.
If dim A < n - 2 then Hn(X,A;z) m Hn(x;~) ~ Z is a trivial ZG-module so that
H~(X,A;~)w ~ ZG = Z. D
We apply now these remarks to the case of homotopy representations.
Proposition 3.3. Let X and Y be finite homotopy representations of a finite
group G with the same dimension function. Then
i) there exist G-maps f: X + Y.
ii) If f: X + Y is a G-map, H ~ EssIso (X) and dim X H ~ I then for H
each integer k there is a G-map g: X + Y such that deg g =
X >H deg fH + klWH I and g coincides with f on .
iii) If dim X H > dim X >H + 2 for each H ~ Iso (H) then G-maps
f,g: X ÷ Y with deg fH = deg gH for all H i G are G-homotopic.
Proof. We construct G-maps by induction over the orbit types. In the induc-
tive step we must extend a G-map GX >H + Y to GX H or equivalently a WH-map
x>H + yH to X H. It can always be done in some way since (xH,x >H) is a relatively
free WH-complex and ~.yH = 0 for i < dim X H. This proves claim i). 1
Let a G-map f: X + Y be given. To prove claim ii) we must change
fH X H + yH outside X >H. If we can find a WH-extension gH X H + yH of
f>H x>H + yH with degree deg fH + klWHI, it can be further extended to a G-map n- H x>H ~ yH-
g: X + Y as above. The extensions rel X >H are classified by H iX , ; n ) H H
where n = n(H), and the obstruction to finding a homotopy between g and f is
precisely the difference deg gH _ deg fH. The assumptions of Le~ma 3.2 are satis-
fied. Indeed, the ZWH-modules ~ yH ~ H yH (n > i) and Hn(x H) are isomorphic n n --
234
by Lemma 2.5 and dim X >H J n - I since H ~ EssIso (X). Hence we are free to
change the degree of fH by any multiple of IWHI, and ii) follows.
If dim X H ~ dim X >H + 2 for each H ~ Iso (X) then in particular dim X H ~ 1
by the convention dim @ = -I. Since dim X >H J n - 2, Lemma 3.2 shows that the
only obstructions to constructing a G-homotopy between f and g are the differ-
ences deg fH _ deg gH and iii) follows.
The following result is crucial in deriving the mapping degree congruences.
Its proof is a direct modification of [9, Th. 3.8].
Prqposition.3.4. Let X and Y be finite homotopy representations of a finite
group G with the same dimension function. Then there exists a G-map f: X + Y
such that deg fH is prime to IGI for all H i G.
(We say that f as invertible degrees.)
Proof. By Prop. 3.3 i) there exists at least one G-map f: X + Y. We try to
correct its degrees. If X H = @ then deg fH = 1 is already prime to !G I.
Since the 0-dimensional fixed point set X H consists of two points, two such sets
are either disjoint or coincide, and we may choose f in such a way that deg fH = 1
also when X H m S °.
Assume then that dim X H > 1 and that deg fK is prime to IGI for all
- f[ K > H. If H is not an essential isotropy group then H > H and deg fH = deg
is already prime to IGI. Otherwise dim X H > dim X >H and we claim that at least
deg fH is prime to [WH I. Indeed, if p is a prime divisor of IWHI then there
exists K j G such that H 4 K and K/H ~ Zp. The K/H-map fH: X H + yH has
fixed point degree
deg (fH)K/H = deg fK ~ 0 mod p.
Hence deg fH ~ 0 mod p, too, by Proposition 1.13. For some k ~ Z the integer
deg fH + kIWH 1 is then prime to IGI. By Proposition 3.3 ii) we may modify f
outside X >(H) so that deg fK is prime to IGI for all K ~ H. u
We are now ready for the classification of the degrees deg fH for G-maps
f: X ~ Y between two homotopy representations with the same dimension function.
Let C = C(G) be the product of integers over the set of conjugacy classes of
subgroups of G. If X and Y are oriented as in Proposition 2.7, every G-map
f: X ~ Y has a well-defined degree function d(f) ~ C,
d(f)(H) = deg fH, H i G.
Theorem 2. Let X and Y be finite homotopy representations of a finite group
with the same dimension function n. There exists integers nH, K such that the
congruences
235
deg fH ~ _ ~ nH, K deg fK mod IWHI, H _< G
H4K<G K/H cyclic ~i
hold for all G-maps f: X ~ Y. Conversely, given a collection of integers d =
(d H) ~ C satisfying these congruences, there is a G-map f: X + Y with deg h H =
d H for each H J G if and only if d fulfils the unstability conditions
i) d H = i when n(H) = -I
ii) d H = i, 0 or -i when n(H) = 0
iii) d H = d K when n(H) = n(K) and H < K.
Proof. According to Proposition 3.4 there exist G-maps g: Y + X with in-
vertible degrees. We choose one of them and fix it. If f: X + Y is any G-map
then the composite g o f: X + X satisfies the congruences
deg gH deg fH ~ _ Z ~( K/H )deg gK deg fK mod [WH I
for all H J G by theorem i. The element deg gH is invertible in the ring ZIG I, 1 4
hence also in the quotient ring ZIWHI,,, and we can find integers nH, K with redidue
class
nH,K = ~(iK/Hl)deg gK/deg gH ~ ZIWHI.
Using the integers nH, K the congruences follow.
Assume then that d ~ C satisfies the congruences. If there is a G-map
f: X ÷ Y with d(f) = d then d fulfils the un~tability conditions by Proposi-
tion 2.7 (note that iii) is equivalent to d H = dH). Conversely, if the unstability
conditions hold for d, we shall construct a G-map f: X + Y with d(f) = d by
induction over orbit types. We start with conjugacy classes (H) such that
X H ~ S ° . Since two 0-dimensional fixed point sets either coincide or are disjoint
we are free to choose the degrees i, 0 and -i on them arbitrarily. Extending
over X we get a G-map f: X ~ Y with deg fH = d H when n(H) < 0.
Suppose we have obtained a G-map f: X ÷ Y such that deg fK = d K for (K) >
(H). In the induction step we modify it to a G-map f: X + Y such that deg ~K =
d K for (K) ~ (H). If H ~ EssIso (X), then H > H and
deg fH = deg fH = d ~ = d H
by Proposition 2.7 and condition iii). Hence f qualifies as f in this case.
On the other hand if H E EssIso (X) then the congruences
deg fH ~ _ E nH,Kdeg fK = _ E nH,KdK ~ d H mod IWHI
236
hold for deg fH by the first part of the proof and for d H by assumption. Using
Proposition 3.3 ii) we can modify f as desired. []
Remark. A stable version of Theorem 2 was proved for unit spheres of complex linear
representations by Petrie and tom Dieck [8, Th. 3]. It was generalized to the un-
stable situation and real representations by Tornehave [20, Th. A]. These proofs
are based on the Thom isomorphism in equivariant K-theory and they yield precise
information on the numbers nH, K. There is an alternative method using transvers-
ability which works more generally in the smooth case. Our elementary approach
to the congruences seems appropriate if one only needs the existence, not the actual
values of nH, K. In the construction of G-maps with given degrees we have followed
tom Dieck and Petrie.
Regard the Burnside ring A(G) as a subring of C as in section i. The sub-
group of C satisfying the congruences of Theorem 2 can be compactly described as
C(X,Y) = {d ~ Cld(g)d ~ A(G)} (3.5)
where g: Y + X is a fixed G-map with invertible degrees. Let IX,Y] G denote the
set of G-homotopy classes of equivariant maps f: X + Y (no base-points are consid-
ered). Theorem 2 describes the image of the degree function
d: [X,Y]G ~ C(X,Y).
As a direct corollary we get
Corollary 3.6. Let X and Y be homotopy representations of a finite group
with the same dimension function n. Then d: [X,Y]G + C(X,Y) is
i) surjective if and only if dim X G > 0 and
dim X H > dim X >H + 1 for each H < G
ii) injective if EssIso CX) is closed under intersection and
dim X H > dim X >H + 2 for each H ~ EssIso (X).
Proof. It is clear that the unstabilitv conditions vanish precisely when
condition i) holds. Assume that the conditions ii) hold. By Proposition 2.10 we
may replace X with a G-homotopy equivalent homotoDy representation Y with
Iso (Y) = EssIso (X). Then the injectiveness of d follows from Proposition 3.3
iii). u
Remark. The formulation chosen in 3.6 ii) may seem complicated. Clearly d is
injective under the single condition
237
dim X H > dim X >H + 2 for each H ~ Iso (X). (*)
If (*) holds then EssIso (X) = Iso (X) and it follows from Proposition 2.8 that
EssIso (X) is closed under intersection. Hence the conditions in 3.6 ii) are
weaker than (*), although by no means necessary.
The join X*Z of two homotopy representations is again a homotopy representa-
tion. If f: X + Y is equivariant then f*idz: X*Z + Y*Z has the same degree
function as f when product orientations with a fixed orientation of Z are used
on X*Z and Y*Z. The stable G-homotopy sets ~G(X,Y) are defined as
WG(X,Y) = li_~m [X*S(V),Y*S(V)] G V
where the limit is taken over all linear representations V. The degree function
defines a map d: WG(X,Y) ÷ C(X,Y). The set WG(X,Y) admits a group structure by
using a trivial representation as the suspension coordinate, and d is a group
homomorphism. Let
unit sphere. Then
is an isomorphism.
~G(X,Y) for every
Segal's theorem:
CG be the complex regular representation and S = S(CG) its
X*S satisfies all conditions in 3.6 and d: [X*S,Y*S] ÷ C(X,Y)
Hence [X*S(V),Y*S(V)]G is isomorphic to the stable group
V containing CG. We have arrived to the following form of
Corollary 3.7. The degree function d: ~G(X,Y) ~ C(X,Y) is an isomorphism for all
finite homotopy representations X and Y such that dim X H = dim yH for each
H<G. o
The stable group WG(X,Y) is an invariant of X and Y but the isomorphism
d and the subgroup C(X,Y) of C depend on the choice of orientation for X and
Y. If X = Y this does not matter when we use the same orientations for the source
and the target. Hence ~G(X,X) is canonically isomorphic to C(X,X) = A(G). The
Burnside ring of G, for any finite homotopy representation X of G. We denote
the group WG(X,X) by w = w G and identify it with A(G).
If X and Y are unit spheres of complex linear representations then
d: [X,Y]G + C(X,Y) is always injective by Proposition 3.6. Thornehave shows in
[20, Prop. 3.1] that this holds for real representations, too, when the group G is
The problem is to show that ~rJu(xH,x>H;~) ~ Z for each nilpotent. isotropy group
H with n = dim X H > 0. In the linear case X H = X H \ X >H is an open n-manifold
where WH acts freely, and Hn(xH,x>H;~) can be identified with Ho(XH;Z) by
duality. Hence one is reduced to study the permutation action of WH on the
components of X H, when X is linear or more generally a locally smooth G-manifold.
On arbitrary homotopy representations no kind of duality can be expected. For
example, consider the A5-space X of example 2.14. The fixed point set X H of
H = A 4 is a wedge of a circle S 1 with an interval I 1 with the middle point as
238
the wedge point, The singular set x>H = #5 consists of the two free end-points
so that HI(xH,x>H;z) = HI(sIvsI;z) = Z ~ Z but X >H does not disconnect X H.
However, homotopy representations of nilpotent groups are sufficiently close
to linear representations to admit a generalization of Tornehave's result:
Theorem 3. Let G be a finite nilpotent group and let X and
homotopy representations of G with the same dimension function
f,g: X ~ Y are G-homotopic if and only if
i) deg fH = deg gH for each H ! G
ii) fH = gH when n(H) = 0.
Y be finite
n. Two G-maps
Remark. The 0-dimensional condition has sometimes been overlooked.
The following example should make it obvious.
Let X = Y be the unit circle S 1 where G = Z 2 acts by complex conjugation.
Then the constant maps f = 1 and g = -I are not G-homotopic although all degrees
are 0, since fG and gG: S ° + S ° cannot be homotopic. In fact, it is esy to
see that [SI,SI]z2 = {±fnlfn(Z) = z n, n ~ Z}.
H Proof. If f,g: X + Y are G-homotopic then fH and g are homotopic and
have the same degree for each H < G. If dim X H = 0 then X H and yH consist
of two points, and the homotopic maps fH gH S ° ~ sO must coincide.
Conversely, let fH = gH for each H J G with dim X H = 0. Then f and g
agree on the union of 0-dimensional fixed point sets and they can be connected by
the constant homotopy. The further obstructions to constructing a G-homotopy between n ( H >H .H.
f and g are the groups H~H X ,X ;~n x ) where H is an isotropy group with
n = n(H) > 0. Since G is nilpotent we may assume that EssIso (X) = Iso (X) by
Proposition. 2.13. Hence dim X >H < dim X H - i.
As a first reduction we note that K > H implies that K 1 = K n NH > H since
G is nilpotent. Hence
X >H = U X K = U X KI
K>H NH>KI>H
is the singular set of X H considered as a WH-space. Therefore it suffices to con-
sider the case where H = 1 is the homotopy kernel of X. If dim X = n and we
denote A = X >I then dim A J n - 1 and H~(X,A;~) has rank at least 1 by
lemma 3.2. For each subgroup K J G there are epimorphisms
Hn(X,A;~) ~ ~(X,A;~) ÷ H~(X,A;~)
n by (3.1). Hence it is enough to show that HK(X,A;~) ~ Z for some subgroup K ! G.
K.
Let KI,...,K J G be the isotropy groups with dim X i = n - 1 and let
H. K. H I .... ,H 1 J G be the isotropy groups such that X j ~ X i for any i = l,...,m.
239
K°
,m X i has dimension n - 1 (or is empty) and Then A = A 1U A 2, where A 1 = Ui= 1
H. 1 j
A 2 = U~= 1X has dimension at most n - 2. Since all isotropy groups are essential, J
A ° = A I N A 2 has dimension at most n - 3. The cohomology group Hn(X,A;~) is an
extension
0 ~ Hn-I(A;~) + Hn(X,A;~) + Hn(x;~) + 0.
The Mayer-Vietoris sequence of A = A 1U A 2 shows that
consequentiy Hn(X,A;~) ~ Hn(X,AI;~).
Let K. < G be an i s o t r o p y g r o u p of X s u c h t h a t 1 - -
dim X L = n - i for any L < K. with L ~ i, since 1 -- i
X ~ S n. Let x = [X] - X(sn-l)l ~ A(Ki). Then
Hn-I(A)' ~--~ Hn-I(AI ) and
K,
dim X i = n - i. Then
is the homotopy kernel of
Xe(X) = x(S n) - x(S n-l) = ±2, XL(X) = x(S n-l) - x(S n-l) = 0 for 1 < L J K i.
The Burnside ring relations in A(K i) imply that ±2 ~ 0 mod IKil so that K i ~ Z 2.
Let K be the subgroup of G generated by Ki, i = l,...,m. The Sylow subgroup G 2
of G is normal since G is nilpotent. Hence K J G 2 is a 2-group. We shall
show that H~(X,A;~) ~ Z for a homotopy representation X = S n of a 2-group K K. K.
such that A X >I m i = = Oi= I X is a union of subcomplexes X I = sn-l.
By tom Dieck [6, Satz 2.6], X has the same dimension function as some linear
representation of K. In particular, if L is contained in the subgroups L 1 and
L. xL L 1 xL2 L 2 and dim X i = dim - 1 for i = 1,2, then X N has dimension
dim X L - 2.
Now a double induction on n = dim X and on the number m of the components K.
in A m i = Ui= 1X shows that
Hk(x,A; Z) = I free, k = n
0, k + n.
The induction starts in dimension n = i where X/A is a connected CW-complex of
dimension I, hence homotopy equivalent to a wedge of circles. The induction on m
is based on a Mayer-Vietoris argument: if B = X Km+l is not contained in A then
L. L. BnA m i = Ui= I X where L i = Km+iKi, i = l,...,m and each X i has codimension i
in B. Hence the induction hypothesis applies to the pair (B,Um=I X Li) and one
may apply the relative Mayer-Vietoris sequence of (X,A) and (X,B).
In particular, Hn(X,A;Z) is torsion free. Since the composite
H~(X,A;~) ~ Hn(X,A;~) t_~ H~(X,A;~)
240
is multiplication by IKI, a power of 2, all torsion in H~(X,A;v) is 2-torsion.
On the other hand, if S(V) is a linear representation sphere of K with Dim S(V)
= Dim X, there exists by Proposition 3.4 a K-map f: X + S(V) such that all degrees
deg fH are odd, H J K. Comparing the Mayer-Vietoris sequences used to compute
Hn(X,A;Z) and Hn(s(v),s(v)>I;z) we get an exact sequence
0 ~ Hn(s(v),s(v)>I;z) ~ Hn(X,A;Z) + C * 0 0")
where C is a torsion group of odd order. Recall that Ho(K;M) = M K for any ZK-
module M, The exact sequence of homology of the extension (*) of K-modules now gives
HI(K,C ) ,S(V)>I;~) n CK H~(StV) f* H K ( X , A ; ~ ) ~ ~ O.
The group HI(K,C) = 0 since K
the linear case it is known that
The resulting extension
is a 2-group and C is an odd torsion module. In
H$(S(V),S(V)>I,v) ~ Z [20, Proof of Prop. 3.1].
f* K 0 + Z -----+ H (X,A;~) + C K + 0
where C K is an odd torsion group shows that ~(X,A;~) m Z since it only has 2-
torsion, o
Remarks. i. The 2-group K ! G which appears in the proof is a finite group of
reflexions and we may be much more specific. From the classification of Coxeter
groups it follows that K is a direct product of an elementary abelian group (Z2)k
and dihedral groups DI,...,D I. The components of S(V) \ S(V) >I are Weyl chambers,
open n-simplices which are permuted freely and transitively by K. This implies n
that S(V)/S(V) >I ~ Vk~ K S k so that Hn(s(v),S(V)>I;z) is isomorphic to ZK as a
n ~ Z. (see Bourbaki, Groupes et K-module. Hence the group of coinvariants in H K
algebres de Lie, Ch. 4-5).
case presents some short-cuts, again. Then K ~ (z2)k is 2. The abelian
elementary abelian and Borel's dimension formula implies
dim X H = dim X - r, H m (z2)r.
Hence the representation V with Dim S(¥) = Dim X is found directly without appeal
to [6] and S(V) \ S(V) >I is easy to analyze. Of course, Borel's dimension formula
is an essential ingredient of tom Dieck's theorem.
241
4. Homotopy equivalence of homotopy representations
Let G be a finite group and let X and Y be finite homotopy representations
of G. A G-homotopy equivalence f: X + Y is oriented, if deg fH = 1 for each sub-
group H J G. We shall show that X and Y are stably oriented G-homotopy equiva-
lent if and only if they are oriented G-equivalent. A similar destabilization result
holds for ordinary G-homotopy equivalence if G is nilpotent and has abelian Sylow
2-subgroup but not in general. We give an example of smooth free actions of a meta-
cyclic group on a sphere which are not G-homotopy equivalent but become such after
adding a linear representation sphere of the same dimension.
Let X and Y be finite homotopy representations with Dim X = Dim Y. This
is clearly a necessary condition for X and Y to be stably G-homotopy equivalent.
Choose a set of representatives for the essential isotropy groups of X and orient
X and Y using this set (2.7). By theorem 2 the degree functions of G-maps
f: X + Y belong to a subgroup C(X,Y) of C and clearly IGIC is contained in
C(X,Y). Especially C(X,X) = A(G) contains IGIC and we may define
A(G) = A(G)/cC, ~ = C/eC,
where c is any multiple of IGI. Then A(G)
~x be the groups of units of the rings A(G)
~(G) fl C× = A(G) ×
xs a subring of C.
and ~. Note that
Let A(G) x and
since ~x = ~ Z x is a finite group. c
d(g) as an element of ~x.
Lemma 4.2. If G-maps g: Y + X and
d(g ' ) / d (g ) E A(G) ×.
If g: Y + X has invertible degrees, we regard
g': Y ~ X have invertible degrees then
Proof. Since d(g) is an element of the finite group ~x we can find a positive
integer k such that d(g) k = 1 in ~x Then the function d = d(g) k-I in C
belongs to C(X,Y) since
d(g)d E i + IGIC c A(G)
(see 3.5), and it also fulfils the unstability conditions since d(g) does. By
theorem 2 there exists a G-map f: X + Y with d(f) = d. In the group ~x we
have
d(g')/d(g) = d(g')a = d(g')d(f) = d(g~f) e ~(X,X) = A(G)
and 4.1 implies the claim. D
242
Following tom Dieck and Petrie [8] we define the oriented Picard group of G as
Inv (G) = [× / A(G) ×. (4.3)
It is a finite group which depends only on G, not on the multiple c of IGI used.
Let X and Y be finite homotopy representations with Dim X = Dim Y. Then we can
attach to the pair (X,Y) the invariant
D°r(x,Y) = d(g) ~ Inv (G) (4.4)
where g: Y ~ X is any map with invertible degrees. By lemma 4°2 this does not
depend of the choice of g. In fact, D°r(x,Y) is the class of the invertible
module C(X,Y) ~ mG(X,Y) over the Burnside ring A(G), but this will not be needed
in the sequel. However, D°r(x,Y) depends on the choice of orientations for X and
Y. By a stable oriented G-homotopy equivalence between X and Y we mean an
oriented G-homotopy equivalence f: X*Z + Y*Z where Z is any finite homotopy
representation.
Theorem 4. Let X and Y be finite homotopy representations of a finite group G
with the same dimension function. Choose orientations for X and Y. The follow-
ing conditions are equivalent:
i) X and Y are oriented G-homotopy equivalent
ii) X and Y are stably oriented G-homotopy equivalent
iii) D°r(x,Y) = 1 in Inv (G).
Proof. Clearly i) implies ii). Let Z be a finite homotopy representation of
G and let f: Y*Z + X*Z be an oriented G-homotopy equivalence. Choose a G-map
g: Y ~ X with invertible degrees. Since both g*id Z and f have invertible
degrees, d(g*idz)/d(f) = d(g) belongs to A(G) × by lemma 4.2. Hence D°r(x,Y) = I.
If D°r(x,Y) = I, we have d(g) ~ A(G) for any g: Y ÷ X with invertible
degrees. Then the constant degree function 1 belongs to
C(X,Y) = {dld(g)d ~ A(G)}
(3.5). Since i obviously satisfies the unstability conditions, theorem 2 shows
that there exists a G-map f: X ÷ Y with deg fH = I for each H J G. The map f
is the required oriented G-homotopy equivalence between X and Y.
Remark. Tom Dieck and Petrie proved theorem 4 for unit spheres of complex linear
representations, see [3, Th. 5] and [8, Th. 2].
Theorem 4 is useful when X and Y can be oriented in a canonical way, e.g.
when they are unit spheres of complex representations. Usually this is impossible.
However, the product orientation on X*X and Y*Y is canonical and we can state
as a corollary
243
Corollary 4.5. If X and Y are stably G-homotopy equivalent finite homotopy
representations, then X*X and Y*Y are oriented G-homotopy equivalent.
Proof. If f: X*Z ÷ Y*Z is a G-homotopy equivalence, then f*f is an
oriented stable G-homotopy equivalence between X*X and Y*Y. By theorem 4
and Y*Y are oriented G-homotopy equivalent. D
X*X
In general we must study the effect of a change in orientations of X and Y
Let g: Y + X be a G-map with invertible degrees. If new orientations are used H H
for X and Y, the degrees deg g are changed by signs ~ = il, and so the
degree function d(g) is multiplied by a unit e in C x = H{±I}. Hence the class
of d(g) in the Picard group
Pic (G) = Inv (G)/C x = ~x / X(G)×C× (4.6)
only depends on the pair (X,Y), not on the orientations. The resulting invariant
D(X,Y) = d(g) ~ Pic (G) (4.7)
detects unfortunately only stable G-homotopy equivalence. For G-homotopy equivalence
we must take into account the unstability conditions.
Let X and Y be finite homotopy representations with the same dimension
function n, i.eo n(H) = dim X H = dim ~ for H < G. Note that the essential
isotropy subgroups of X and Y can be recovered from n. If the G-map
g: Y + X has invertible degrees, then d(g) = d satisfies by (2.7)
i) d H = 1 when n(H) = -I
ii) d H = ±I when n(H) = 0
iii) d H = d H.
Hence d(g) belongs to the subgroup ~x of ~x defined by the conditions (4.8). n
Let A(G) x be the corresponding subgroup A(G) x. We see that D°r(x,Y) lies n
actually in the subgroup Inv (G) n = ~× / A(G) ×. A change of orientations of X n n
by a unit g which satisfies (4.8). Denote by C x the n
C. We thus get an unstable invariant
and Y multiples d(g)
group of such units in
(4.8)
Dn(X,Y) = d(g) ~ Pic n (G) = ~x / ~(G)×C× (4.9) n nn
and the proof of theorem 4 gives immediately
Theorem 5. Let X and Y be finite homotopy representations of a finite group G
with the same dimension function n. Then
i) X and Y are stably G-homotopy equivalent if and only if D(X,Y) = 1
in Pic (G)
ii) X and Y are G-homotopy equivalent if and only if Dn(X,Y) = i in
Pic (G). m n
244
The difference of theorems 4 and 5 is that the map Inv (G) + Inv (G) is n
always injective, whereas Pic (G) ~ Pic (G) usually has nontrivial kernel. This n
may be explained as follows: If g: Y + X is a G-map with invertible degrees and
D(X,Y) = i in Pic (G), there exists a unit s ~ C x such that x = ed(g) belongs
to A(G). Although the product d(g) = ex satisfies the unstability conditions
(4.8), the factors s and x need not satisfy them. However, if E and x can
be replaced by unstable s' and x', then Dn(X,Y) = i and X and Y are G-
homotopy equivalent. In this case ee' is a unit of A(G). Hence the difference
between stable and unstable G-homology equivalence is connected with the units of
the Burnside ring.
We shall now prove that stable G-homotopy equivalence implies ordinary G-
homotopy equivalence for homotopy representations of certain nilpotent groups.
Theorem 6. Let G be a finite nilpotent group with an abel±an Sylow 2-subgroup and
let X and Y be finite homotopy representations of G. If X and Y are stably
equivalent, they are G-homotopy equivalent.
Proof. Since X and Y are stably G-homotopy equivalent, Dim X = Dim Y.
Choose a map g: Y + X with invertible degrees. Then there exists a unit e in
C x such that Ed(g) lies in A(G), and ~ can be realized as the degree function
of a stable G-homotopy equivalence h: YnZ + X*Z. We construct a G-homotopy equiva-
lence f: Y + X by induction over the orbit types.
Start with a map ~: yG + X G with degree one. Assume that we have already
found f: GY >H + X with deg ~K = il for all (K) > (H). Extend ~H: y>H ÷ X H
to a WH-map f: yH + X H. If deg f £ ±I mod IWHI, it can be modified to a map with
degree ±i. Now both f*id and h H have invertible degree functions as WH-maps.
Hence d(f)/d(h H) belongs to A(WH) x by Lemma 4.2. Since deg fK/deg h K = ±i
for each K > H, it suffices to prove the following algebraic lemma
Lemma 4.10. Let G be a finite nilpotent group with abel±an Sylow 2-subgroup. If
the element x of A(G) has XK(x) = ±i for each K + i, then ×e(X) = ±i mod IGI.
Proof. Assume first that G is abel±an. We may clearly multiply x by
a unit ~ of A(G) without changing the assertion. If XG(X) = -I, we first
multiply x with -i. Let H be a subgroup of G of index 2. Then the unit
E H = i - G/H in A(G) has XH(e H) = -I and XK(¢ H) = 1 for all K not contained
in H. Hence by using units e H we may assume that XH(X) = 1 for each H J G of
index at most 2. It follows that XH(X) = I for each H ~ i. Indeed, if this is
always proved for K > H, the congruences (i.i) imply that XH(X) ~ 1 mod IG/HI.
But IG/H I is at least 3 and XH(X) = ±I, so we have an equality XH(X) = i.
Finally the congruences (I.i) once again show that Xe(X) ~ 1 mod IGI. 2
Let now G be a general nilpotent group. The square x in A(G) satisfies
XK(X 2) = 1 for each K + 1 so that Xe (x)2 = Xe(X2) ~ 1 mod IGI. Especially
245
Xe(X)2 n 1 mod pn if the Sylow p-subgroup G has order p . When p is odd, this P
implies that Xe(X) ~ ±I mod p since (z/Pn) × is cyclic. Hence, for odd p there is
a sign gp = ±i such that
Xe(X) ~ £p mod IGpl. (*)
By the abelian case, this holds also for p = 2. We are ready if we can show that
= c for all p and q. P q
Choose a central subgroup H of order p in G for each odd prime divisor P P
p of IGI. Since G is a direct product of its Sylow subgroups Gp, Hp is also
central in G and the subgroup H = G 2 H Hp of G is abelian. Then we know that
P
Xe(X) ~ ~ mod IHl (**)
where e = il. Comparing (~) and (**) we see that e = e when p is odd and P
e2 = e when the order of G 2 is at least 4. If G 2 = Z2, E 2 can be arbitrary x
since ZIG 1 = m O1/2. o
Remark. The lemma fails for dihedral and semidihedral 2-groups G. Both groups
contain a noncentral subgroup H of order 2 with IWHI = 2 and then x =
1 - G/H ~ A(G) has characters Xe(X) = 1 - IGI/2, XH(X) = -I and XK(X) = I for
other subgroups K j G. Conversely, using the multiplicative congruences of
tom Dieck [7] one can extend lemma 4.10 to all nilpotent groups G such that the
Sylow 2-subgroup is not dihedral or semidihedral. From the point of view of theorem 6
such a generalization is useless since one must be able to apply the lemma to all
quotient groups of subgroups of G.
We conclude with an example which shows that stable G-homotopy equivalence does
not imply G-homotopy equivalence in general even for smooth free actions on spheres.
Example 4.11. Let G be a metacyclic group of order pq where p and q are odd
primes, i.e. G is an extension of a cyclic group H of order q by a cyclic group
K of order p such that K embeds into Aut (H) = Z x. The cohomology of G is q
periodic with period 2p and it follows from Swan [19] that there exists a free G-
complex X of dimension 2p - 1 homotopy equivalent to S 2p-I. The oriented G-
homotopy type of X is determined by the k-invariant e(X), a generator of H2P(G;Z) =
ZIG I. All generators occur as k-invariants and X Canv be ~ch°sen finite if and only if
the image of e(X) under the Swan homomorphism 8: Z~G ] + ~o(ZG) vanishes. In this
case the kernel of 8 consists of d which are p'th powers mod q. By using surgery
Madsen, Thomas and Wall show that each finite X is G-homotoDy equivalent to a free
smooth action of G on S 2p-I [22, Th. i, Th. 3].
Let X and Y be smooth free G-spheres diffeomorphic to S 2p-I and let
246
g: Y + X be a G-map, The conjugacy classes of subgroups of G are {I,H,K,G}. If
L < G is nontrivial then yL is empty and deg gL = i. By theorem 2 the degree
d = deg g is determined mod pq. If we define the k-invariants e(X) and e(Y) as
the classes of the angmented cellular chain complexes
C2p_l ~ Z ~ 0 0~Z÷ ~...÷ C °
2p in EXtzG(Z,Z) = H2P(G;Z) then it is immediate that de(Y) = e(X). We may choose
X and Y so that d -z i mod q and d ~ -i mod p, since 6(d) = 0. Then X and Y
are not G-homotopy equivalent. However, the Burnside ring A(G) consists of x such
that
XH(X) ~ XG(X) mod p, Xl(X) ~ XH(X) mod q, Xl(X) ~ XK(X) mod p.
If E ~ C x is the unit with e K = -I and E K = 1 otherwise, then x = ed(g) belongs
to A(G) and X and Y are stably G-homotopy equivalent by theorem 5.
We can realize this geometrically as follows. Induce a faithful representation
of H on C up to a representation V of G. Then V is irreducible and has com-
plex degree p. The subgroup H acts freely on V but dim V V K = i. Hence S(V)
has isotropy groups 1 and K. By theorem 2 we can find a G-homotopy equivalence
f: X*S(V) + Y*S(V) with degree function E. Note that X,Y and S(V) have dimension
2p - i. The lowest dimension 5 occurs for metacyclic groups of order 21.
Remark. Theorem 6 was proved Jn the special case of unit spheres of linear represen-
tations of abelian groups by Rothenberg [16, Cor. 4.10]. In [17] he considers finite
homotopy representations X such that X H is a PL-homeomorphic to S n(H) and WH
acts trivially on H,(X H) for each subgroup H of G. Example 4.11 contradicts the
destabilization theorems 1.8 and 5.7 of [17], which claim that stable G-homotopy equi-
valence implies G-homotopy equivalence if the Sylow 2-subgroup G 2 is "very nice".
Indeed, the groups of 4.11 have odd order and a trivial group should certainly be
very nice. Note that lemma 4.10 is an algebraic version of the basic result [17,
Prop. 2.2], which fails for metacyclic groups of odd order and dihedral groups as
pointed out by Oliver [MR 81c: 57044].
247
References
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298 (1978), 182-195.
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math. 47 (1978), 273-287.
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Ph.D. Thesis, Warsaw 1977.
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Aarhus 1978, Lecture Notes in Mathematics 763, Springer-Verlag, Berlin
Heidelberg New York, 1979.
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267-291.
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pp. 275-301 in Current Trends in Algebraic Topology, CMS Conference
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Department of Mathematics University of Helsinki Hallituskatu 15 00100 Helsinki, Finland
DUALITY IN ORBIT SPACES
Arunas Liulevicius* and Murad Ozaydln
Our aim in this paper is to present a new technique for studying symmetric
products of G-sets. The motivation for this work originally came from the study of
exterior powers in the Burnside ring of a finite group motivated by the work of Doid
[2] which presents a new model for the universal h-ring [i], [3] on one generator.
Some of the results mentioned here will only have sketch proofs - for more detail
the reader can consult [5].
Let G be a group and X a finite G-set. The k-fold s)nnmetric product SkX is de-
fined as follows. The symmetric group S k operates on the k-fold Cartesian product
X k by permutation of coordinates, and we define SkX = xk/s k . The diagonal
action of G on X k commutes with the action of S k , so this means that SkX inherits
an action of G.
The key idea in our approach to the study of SkX is that it is convenient to
study all of them at the same time, We define the graded set S.X,~ = { SkX~k~ N ~ ,
where N is the set of natural numbers.
PROPOSITION I. Suppose X is a finite set. Then S~X = Map(X,N) .
Proof. If z ~S,X, let <z, > : X ~ N be the counting function determined
by z, that is <z,x> is the number of times the element x occurs in z. Even more
precisely, if z = (Zl, ... , Zk).S k , then <z,x> is the number of i such that x =
z.. Notice that k is recaptured from the counting function for z by the identity I
k = ~ <z,x> x£X
* Research partially supported by NSF grant DMS 8303251.
250
Conversely, given a function
z ~ $k X such that e = <z, >
k =
sum,
c : X - ~ N, there exists a unique element
Indeed, here k is given by
c(x) x 6 X
COROLLARY 2. If X and Y are finite sets and X U Y denotes their disjoint
then S,(XL~Y) = S,X x S,Y .
Proof. A function C : X U~Y ~ N is completely determined by the
restrictions C 1 : X ~ N and C 2 : Y - ---~ N.
Notice that if X is a G-set with G acting on the right, then under the corres-
pondence S.X = Map(X,N) the action of G on S.X inherited from the diagonal
action on X k corresponds to the right G-action on Map(X,N) defined by (c.g)(x) =
c(x.g-l). This allows us to prove
COROLLARY 3. If X is a finite right G-set and H~G is a subgroup, then
(s.x) H = S,(X/H).
Proof. To say that a counting function c : X ~ N is in the fixed point
set (S.X) H is the same as saying that c is constant on the orbits of H in X, that
is, it corresponds to a function c : X/H ~ N .
For both the statement and the proof of the statement above it is essential to
use S.X . Without it the statement becomes more complicated, since the orbits of H 4¢
need not have the same number of elements.
COROLLARY 4 (Duality). If X is a finite G-set and G is a finite group, then
S,X = MaPG(X,S,G) . Here the usual right action on S,G = Map(G,N) is used in
defining the set MaPG(X,S,G). There is a second (commuting) right action of G on
S,G = Map(G,N) defined by (c,g)(y) = c(gy), and this action corresponds to the
standard action of G on S,X .
251
Proof. It is enough to check this on orbits G/H. We have just seen that
S,(G/H) = (S,G) H = MaPG(G/H,S,G),
Our second duality result involves the infinite group G = (Z,+), the additive
group of the integers. We wish to determine the structure of the finite Z-sets SeX,
and according to Corollary 2 it is enough to do this for the cycles Z/(n).
PROPOSITION 5. The multiplicity of the cycle Z/(r) in Sk(Z/(rs) ) is zero
if s does not divide k. If k = ms , then the multiplicity of Z/(r) in Sk(Z/(rs) )
is the same as the multiplicity A(m,r) of Z/(r) in S (Z/(r) ). m
Proof. This is a consequence of Corollary 3. See [4] for an alternative argu-
ment and [5] for a more detailed discussion.
COROLLARY 6.
if ~ is the MSbius function,
A(k,n) =
Proof.
Let A(k,n) be the multiplicity of Z/(n) in Sk(Z/(n) ). Then
we have
/ j (~(s)/s).(n/s+k/s-l)!(n/s)!(k/s)! .
(n,k)~'(s)
Use the M~hius inversion formula to solve the recursion relations for
A(k,n) coming from Proposition 5.
COROLLARY 7 (Reciprocity Law). If A(k,n) is the multiplicity of the cycle
Z/(n) in Sk(Z/(n) ) , then A(k,n) = A(n,k) for all k,n.
Proof. Notice that the formula for A(k,n) in Corollary 6 is s~etric in k and
n.
This is not entirely satisfactory, since the reciprocity seems to be an acciden-
tal result of a complicated number-theoretical formula. The key which explains the
reciprocity law is the following duality map of orbit spaces:
252
THEOREM 8 (Duality map). There exists a one-to-one isotropy preserving corres-
pondence
D : ( Sk(Z/(n) )/ Z/(n> -- ~ (Sn (Z/(k) )/ Z/(k) .
That is, for each (k,n) ~" (s) the multiplicity of the cycle Z(n/s) in Sk(Z/(n) )
is the same as the multiplici%y of the cycle Z/(k/s) in S (Z/(k) ). n
Proof. The key point in the proof [5] is to identify the orbit space
(Sk(Z/(n)) )/ Z/(n) as the set of all circular Lazy Susans having n walls and k
balls distributed in the n chambers. The duality map interchanges the roles of the
walls with that of the balls.
REFERENCES
[I] M.F.Atiyah and D.0.TalI, Group representations, h-rings and the J-homomor-
phism, Topology 8 (1969), 253-297.
[2] A.Dold, Fixed point indices of iterated maps. Preprint, Forschungsinstitut
fur Mathematik ETH Zurich, February 1983.
[3] D.Knutson, h-rings and the Representation Theory of the Symmetric Group,
Springer LNM 308 (1973).
[4] A.Liulevieius, Symmetric products of cycles, Max Planck Institut fur Mathema-
tik, Bonn, 1983.
[5] A.Liulevicius and M.Ozaydln, Duality in Symmetric Products of Cycles, Preprint,
University of Chicago, June 1985.
Department of Mathematics
The University of Chicago
Eckhart Hall
5734 University Avenue
Chicago, IL 60637 U S A
and Department of Mathematics
University of Wisconsin
Van Vleck Hall
480 Lincoln Drive
Madison, WI 53706 U S A
CYCLIC HOMOLOGY AND
IDEMPOTENTS IN GROUP RINGS
Zbigniew Marciniak
Warsaw, Poland
We present here an algebraic approach to the Burghelea Theorem on
cyclic homology of group rings. The original proof involves arguments
from the theory of bundles with St-action and it is not easilyaccesible
to algebraists. As an application we offer a new criterion for non-
existence of idempotents in a group ring. In particular, we give a
completely different proof of Formanek's Theorem on polycyclic-by-
finite groups.
Cyclic homology
Let k be a commutative ring with I . For an associative k-algebra
A with I one can consider the Hochschild homology of A which is,
by definition, the homology of the chain complex:
bl A 2 b2 A 3 b3 : O ~ A . . . . . . . ,
where A n = A ®k "'" ~k A (n times) and b n : An+1-~An is given by
n-1
b n ( a o ®...® a n ) = ~ ( - 1 ) i a o ® . . . ® a i a i + l ® . . . ® a n + ~ l ) n a n a o ® a 1 @. . .®an_ l . i=o
A. Cormes o b s e r v e d t h a t t h e a b o v e c o n s t r u c t i o n , when s u i t a b l y m o d i f i e d ,
l e a d s t o i n t e r e s t i n g a p p l i c a t i o n s . The r e s u l t i n g homology i s c a l l e d
" c y c l i c homology" and t h e mos t u s e f u l d e f i n i t i o n seems t o be t h e
following [6~.
In addition to the chain complex J~ we consider its modified ver-
sion
wi th
n-1
bn(a o ®...® a n ) = ~ (-1) i a o ®...® aiai+ I ®...® a n i=o
This complex can be contracted via s : A n - ~ A n+l s ( a o ®. .® an_ 1)
I ® a o ®...® an, I. We put the complexes ~ and ~' together to form
a double complex
254
1 -T N 1 -T N
where T and N are chain maps defined as follows:
T n : A n+S-*A n+1 is the cyclic permutation of coordinates:
® a o ®...® Tn(a o ®...® a n ) = (-S) n a n an_ S
Nn = ~nm(°) + ~nm(1) +...+ ~nm(n) , Tn(k) = T n o ... oT n (k times) .
The cyclic homology of A is just the homology of the total complex
of D(A) :
HC.(A) = H.(Tot D(A))
We notice for further reference that the double complex D(A) has
a shift map S : D(A)---~ D(A) which sends the first two columns
~, ~' of D(A) to zero and shifts the other columns two places
to the left. Consequently, we obtain shift maps
S : HCn(A)~-. HCn_2(A) for all n >_ 2 .
Group rings
Among the algebras which are of interest for topologists we have
group algebras kG , defined for any group G . D. Burghelea skil-
fully used in [2] the theory of circle bundles over the classifying
space of G to determine the groups HCn(kG) . To present his result
we need some notation.
For a group G let TG denote the set of its conjugacy classes.
Let TG be the subset of those classes, which consist of elements
of infinite order. Let c E TG and z E c . We denote by G c the
quotient group CG(Z)/(z> where CG(z) is the centralizer of z in
G . We need the following weak form of Burghelea's result.
Burghelea Theorem
Let G be a group and let k be any commutative ring with unity.
Then
HC.(kG) ~ ® H . (Gc ) @ T . . c E T G
Here H . (Gc ) s tands f o r t h e homology o f g roups w i t h t r i v i a l c o e f f i -
c i e n t s k . The summand T. can be completely described in terms of
homology of some nice fibrations associated with G . However, for our
purposes it is not necessary to go deeper into the structure of T.
We gave a purely algebraic proof of the precise formulation of the
255
Burghelea Theorem in the case when k is a field of char 0 in C7].
In this paper we offer an application.
I dempotents
One way of studing a k-algebra A is to investigate its idempo-
tents: e = e 2 E A . If e % 0,1 then it splits A into a direct
sum A = Ae ® A(1-e) of left A-modules.
Any idempotent e E A generates a sequence of special elements
e n E HC2n(A) for all n ~ 0 (see [&, Prop. 14, Ch. II]). They can
be defined in the following way.
Let e (i) = e @...® e (i times) belong to A i . Set ~I = I
i>I m
(-1) i-1 (2 i ) , ~2i = 2 * i! " '
All these numbers are integers. Consider e n =
and for
~2i+I ~-lji' i! @i
2n+I ~ie (i) E Tot D(A)2n .
i=I
A straightforward calculation shows that e n are cycles of the chain
complex Tot D(A) It is also clear from the definition of the shift
S that we have S(en+ I) = e n for all n ~ 0 .
From now on we assume that k is a field of characteristic 0 o
Let A be a group algebra kG . It is easy to produce an idempotent
e E kG once you have an element g ~ G of finite order n : we set
e = I/n(l+g+...+g n-l) E kG . Another method of producing idempotents
is described in [5] but it still requires the existence of torsion in
G . Moreover, we have the following long-standing.
The Idempotent ConOecture
If a group G is torsion free then its group algebra kG has no
idempotents different from 0 and I . We will prove the following
result.
Main Theorem
Let G be a torsion free group and let k be a field of characte-
ristic zero. If for every conjugacy class c E TG\Ill there exists a
number n c > 0 such that H2nc(Gc;k ) = 0 , then the group algebra kG
contains only two idempotents: 0 and I
The basic tool in the work with idempotents in kG is the trace
function tr : kG--* k given by tr(~ e(g)g) = e(1) . It is very
256
efficient because of the following result.
Kaplansky Theorem [8, Thm. 2.1.8]
Let e = e 2 E kG . Then tr(e) = 0 implies e = 0 and tr(e) = I
implies e = I
We have also other trace functions on kG . For any c E TG we
have a function t c : kG--~ k defined as tc(e ) = [{e(g) Ig E cl.
In particular tll I = tr . These functions are substitutes for charac-
ters from finite group theory and they indeed share some of their
properties.
As the augmentation homomorphism ¢ : kG--+ k is a ring homo-
morphism, we have
Z tc(e) = e(e) = 0 or 1 c E TG
Thus, by the Kaplansky Theorem, the Idempotent Conjecture is equiva-
lent to saying that if G is torsion free and e = e 2 E kG then
tc(e) = 0 for all c E TG\{I~
Proof of the Main Theorem:
Let G be a torsion free group and let e be an idempotent in
kG . As remarked earlier, e generates a sequence fen} of elements
lying in HC2n(kG) for n = 0,1,..., such that S(en+l) = e n .
By the Burghelea Theorem we have
HCo(kG) ~ ® Ho(G c) ® T o • c~TG
From the explicit description of the above isomorphism given in [7]
it is easy to see that the element e o E HCo(kG ) corresponds to the
vector of its traces tc(e) . Further, from the proof of the Burghe-
lea Theorem presented there it is clear that the shift S respects
the direct sum decomposition
HC.(kG) ~ ® H.(G c) ® T.. c E T G
Thus, for any c E TG\{I~ = T G and for any n ~ I we have a homo-
morphism S c : H2n(Gc)--~H2n_2(Gc) .
Fix now a conjugacy class c E TG\II~ - For any n ~ 0 let
x n E H2n(Gc) be the coordinate of e n corresponding to c . Then
we have Sc(Xn+ I) = x n and x o = tc(e ) .
257
Suppose there is an integer n c > 0 such that H2nc(Gc) = O .
Then Xnc = 0 and hence tc(e) = 0. If the same holds for all
c E TG\~ll then all traces tc(e) vanish and e must be O or 1
Corollary: (Compare with Thm. 2.3.10 in [8])
If G is a torsion free polycyclic-by-finite group and k is a
field of char 0 then kG has no idempotents different from O and
I
Proof:
Let h be the Hirsch number of G It is well known that the co-
homological dimension of G is equal to h ~13. Now, for any c E TG
the group G c is also polycyclic-by-finite and its Hirsch number
does not exceed h . Consequently, for 2n > h we have H2n(Gc) = O
(we have coefficients from a field of characteristic zero!) and so
the Main Theorem can be applied. •
Remark :
Whatever we have said about idempotents holds as well for finitely
generated projective modules, as cyclic homology is Morita invariant.
The obvious generalization of the Main Theorem is left to the reader.
References
~1~ K. Brown: Cohomology of Groups, Springer 1982, New York
E2] D. Burghelea: The cyclic homology of the group rings, Comm. Math. Helv. 60 (1985), 354-365
~3~ H. Cartan, S. Eilenberg: Homological Algebra, Princeton 1956
~4~ A. Connes: Non Commutative Differential Geometry, Publ. Math. IHES 62 (1986), 257-360
~5] D. Farkas, Z. Marciniak: Idempotents in group rings - a surprise, J. Algebra 81, No. I (1983), 266-267
E6~ J.-L. Loday: Cyclic homology, a survey, to appear in Banach Center Publications
~7] Z. Marciniak: Cyclic homology of group rings, to appear in Banach Center Publications
C8~ D.S. Passman: The Algebraic Structure of Group Rings, Wiley 1977
~2 surgery theory and smooth involutions
on homotopy complex projective spaces
Mikiya Masuda
Department of Mathematics, Osaka City University, Osaka 558, Japan
§0. Introduction
Let a group act smoothly on a manifold M. One of the
fundamental problems in transformation groups is to study relations
between the global invariants of M (e.g. Pontrjagin classes) and
invariants of the fixed point set. The Atiyah-Singer index theorem
gives profound answers to this problem, which are necessary
conditions of the action. Conversely it is interesting to ask if
those are sufficient conditions. In other words, to what extent are
there actions realizing such relations ? In this paper we deal with
the realization problem of this kind for smooth involutions on
homotopy complex projective spaces.
Let X be a 2(N-l)-dimensional closed smooth manifold homotopy
equivalent to the complex projective space p(~N). We call such X
a homotopy P(C N) briefly. Suppose that X supports a smooth
involution, that is to say, an order two group (denoted by G
throughout this paper) acts on X. Then Bredon-Su's Fixed Point
Theorem (see p.382 of [B]) describes the oohomologieal nature of
the fixed point set X G of X. It depends on the number of
connected components of X G :
Type 0. X G is empty,
T[pe I. X G is connected and has the same cohomology ring as
the real projective space p(~N) of dimension N-I with X 2
coefficients,
259
T z p e ! I . X G c o n s i s t s o f t w o c o n n e c t e d c o m p o n e n t s F 1 , F 2 a n d N.
each F i has the same cohomology ring as p(~ i) with 2 2
c o e f f i c i e n t s . H e r e NI+N 2 = N. M o r e o v e r t h e r e s t r i c t i o n map f r o m
H*(X;Z2) to H~(Fi;Z2 ) is surjective. When the minimum of Ni-i
(: dim F./2) is £, we say more specifically that the involution is I
o f T y p e I I g .
Type I involutions are fairly well understood due to studies of
Kakutani [K], Dovermann-Masuda-Schultz [DMSc], and Stolz [S]. In a
way made precise in [DMSc] we may say that almost all homotopy p(~N)
admit Type I involutions. As a matter of fact no homotopy p(~N) has
been discovered which does not admit a Type I involution.
In this paper we are concerned with Type II involutions. To
illustrate our results we pose
Definition.
component F. 1
Let x be a generator of H2(X;~). For a fixed
N.-I 1
(i = I, 2) of dimension 2(Ni-1), we restrict x
to F. and evaluate it on a fundamental class of F . . We denote 1 1
the value by D(F i) and call it the defect of F i. Due to choices
of a genarator x and an orientation of F i, D(F i) is defined
only up to sign. The defects D(F i) are odd because the
restriction map from H (X;X 2) to H (Fi;Z 2) is surjective.
Clearly the set {D(FI) , D(F2)} is an invariant of the G
action. It is a G homotopy invariant. For instance, if X is G
homotopy equivalent to p(~N) with a linear Type II involution,
then D(F i) = ±I. Therefore one may regard defects as invariants
which measure the exoticness of actions. The concept of defect is
relevant for general ~ actions with the same definition. The m
reader is referred to [HS], [DM], [DMSu], [D2], [M3], [We] in this
direction.
260
The Atiyah-Singer index theorem for Dirac operators associated
with Spin c structures implies that the defects are related to the
characteristic classes of X, F. and those of the normal bundles 1
of F.. It gives many rather complicated integrality conditions, l
from which we deduce a neat congruence between the defects and the
first Pontrjagin class P1(X) of X. In fact Theorem 4.3 says that
~f we choose suitable signs of D(Fi), then the following congruence
(*) holds :
(*)
where k(X)
D ( F t ) + D ( F 2) ~ 4 k ( X ) {mod 8 ) ,
i s t h e i n t e g e r d e t e r m i n e d b y P l ( X ) = ( N + 2 4 k ( X } ) x 2
( s e e Lemma 4 . 1 ) . As a c o n s e q u e n c e { C o r o l l a r y 4 . 4 ) o n e c a n c o n c l u d e
that k(X) must be even if X is G homotopy equivalent to p{~N)
with a linear Type II involution (remember that D(F i) = ±I under
this assumption).
We regard (*) as a guidepost for our construction of Type II
involutions. One of our main results (Theorem 5.1) says that (*) is
also a sufficient condition for Type IIN/2_ 1 involutions in case N
= 4 or 8. The diffeomorphism types of homotopy P(~4)'s are
classified by their first Pontrjagin classes (equivalently, the
integer k(X)) and there are infinitely many sets {D(FI), D(F2)}
satisfying the congruence (*) for each k(X). Hence Theorem 5.1
implies
Corollar Z 5.3. Every homotopy p(~4) admits infinitelz man Z
Type II 1 involutions distinguished by the defects. In particular
they are not G homotopy equivalent to each other.
This is an improvement of Theorem B (I) of [MI]. For a general
N dvisible by 4, a rather weaker result than that of Theorem 5,1 is
261
obtained (Theorem 5.4). For the other values of N we only see
that infinitely many non-standard homotopy p(~N) admit Type II
involutions with non-standard fixed point sets (Theorems 5.6, 5.7).
As for the method, we apply G surgery theory developed by
Petrie and Dovermann. It is a useful tool to construct G manifolds
in the same homotopy (or G homotopy) type as a given G manifold
Z. In fact we take p(~N) with a linear Type II involution as Z.
When we apply G surgery theory, we must work out two things. One
is to produce a G normal map. We construct a nice G
quasi-equivalence in §3, which together with the G transversality
theorem produces a G normal map. The other is to analyse G
surgery obstructions. In all but one case, we can compute those
obstructions by using G signature and Sullivan's Characteritic
Variety Formula. If dim Z ~ 2 (mod 4), then the obstruction in an
L group LdimZ(~[G]'l) ~ ~2 is treated differently. We show the
existence of a framed G manifold with the Kervaire invariant one
in LdimZ(~[G],l), which serves to kill the obstruction.
This paper is organized as follows. In §l we review G
surgery theory and in §2
the Kervaire invariant one.
Petrie is exhibited in §3.
we construct framed G manifolds with
A nice G quasi-equivalence due to
In §4 we apply the Atiyah-Singer
index theorem to deduce congruence (*). Type II involutions are
constructed in §5. In Appendix we apply the ordinary surgery
theory to produce Type II involutions, where the gap hypothesis (see
§i) is unnecessary but the fixed point sets are standard ones.
Throughout this paper we always work in the C ~ category ; so
the word "smooth" will be omitted.
Notations. Here are some conventions used in this paper :
G : an order two group.
262
~2 : the ring ~/2~ : {0, I}.
~m,n (resp. ~m,n) : ~m+n (resp. ~m+n) with the involution
defined by
(z I, .. , Zm+ n) * (z I .... z m, -Zm+ I, . . , -Zm+ n
Such ~m,n is sometimes denoted by ~m,n to distinguish it from +
the space ~m+n with the involution defined by
(z], .. , Zm+ n) , (-z] , . . , -z m, Zm+ I, .. , Zm+n).
The latter G space is denoted by ~m,n
For a complex (or real) representation V (with a metric)
S(V) (resp. D(V)) : the unit sphere (resp. disk) of V,
P(V) : the spaee consisting of complex (or real) lines through
the origin in V.
In concluding this introduction I would like to express my
hearty thanks to Professor T. Petrie for suggesting this problem to
me and for valuable long discussions during his visit to Japan in
the summer of 1983. This paper is an outcome of discussions with
him.
§i. Review of G surgery theory
G surgery theory is a tool to construct a G manifold in the
same homotopy (or G homotopy) type as a given (connected) G
manifold Z. For a general finite group G, we must impose
complicated technical conditions on Z so that G surgery theory
is applicable. But in our case G is of order two; so those
conditions are simplified as follows. Let dim Z G denote each
dimension of connected components of Z G. Then
263
( I . I ) dim Z ~ 5
(1 .2 ) dim Z G ~ 0, 3, 4
(1.3) (Gap hypothesis) 2dim Z G < dim Z.
For simplicity we require in addition :
(1.4) Z and each component of Z G are simply connected,
(1.5) the action of G preserves an orientation on Z.
Throughout this section and the next section the G manifold Z
will be assumed to satisfy these five conditions unless otherwise
stated.
Roughly speaking G surgery theory consists of three concepts
in our construction :
I. G quasi-equivalences or G fiber homotopy equivalences
II. G transversality
III. G normal maps and G surgery.
Here the meaning of these terms will be clarified below little by
little. According to these concepts G surgery theory is divided
into three steps. In the following the (fiber) degree of a map has
a sense up to sign.
First we set up a G
equivalence ~ : V ~ U
quasi-equivalence or a G fiber homopoty
between G vector bundles over Z. Here
a G quasi-equivalence means that e is a proper fiber preserving
G map of degree one on each fiber, and a G fiber homotopy
equivalence is a G quasi-equivalence such that the restricted map
e : -~ to the fixed point sets is also of degree one on each
fiber (note that this implies the existence of a G fiber homotopy
inverse in a stable sense, see §13, Chapter I of [PR]). A G
quasi-equivalence (rasp. a G fiber homotpoy equivalence) is used
to produce a G manifold in the same homotopy (rasp. G homotopy)
264
type as the given G manifold Z.
Next we convert ~ into a G map h A
section Z c U via a proper G homotopy°
encounter obstructions to finding it at this stage. In our case,
however, it is always possible because those obstructions
identically vanish under the gap hypothesis (see Corollary 4.17 of
[P2]). The G transverse map h produces a triple K = (W,f,b)
where W = h l(z), f = hlW : W ~ Z and b : TW ~ f (TZ+V-U) S
notation ~ denotes that b is a stable G vector bundle
G isomorphism). Here we may assume f, : Ho(W G) ~ Ko(Z G) is
bijective, if necessary, by doing O-surgery. Moreover we should
notice that
transverse to the zero
In a general setting we
(the
the degree of
the degree of
f : the fiber degree of ~ : I,
fG : the fiber degree of ~G
Z G (by Smith theory).
(1 .6 )
: an odd integer at each component of
With these observations
Definition. A G normal map is a triple K = (W,f,b) such
that
(i) f : W ~ Z is a G map of degree one,
G (ii) f, : ~o(W G) 4 Ho(Z G) is bijective,
(iii) fG : W G . 4 Z G is of odd degree at each component of Z G,
(iv) b : TW ~ f (TZ+E) for some E e KOG(Z). S
At a final step we perform G surgery on the G normal map K
via a G normal cobordism to produce a new G normal map K' =
(W',f',b') with f' : W'---~ Z a homotopy (or a G homotopy)
equivalence.
To achieve the final step we first do surgery on the G fixed
point set W G and then on the G free part W-W G. Unfortunately
265
we encounter an obstruction at each procedure. The primary one is
the surgery obstruction to coverting fG : W G ~ Z G into a Z 2
homology (resp. a homotopy, if fG is of degree one) equivalence.
This is denoted by oG(f). Since Z G may be disconnected, it lies
in a sum of L groups :
aG(f ) E LdimZG(X(2)[I]) (resp. LdimzG(~[l]))
where Z(2 ) denotes the localized ring of ~ by the ideal
generated by 2 and the orientation homomorphisms from ~I(Z G) to
X2 are omitted in the notation of L groups because they are
trivial by (1.5). The reader should note that we must check the
vanishing of aG(f) for each component of Z G.
When dim Z G m 2 (mod 4), the above L groups are isomorphic to
X2 componentwise. The values of aG(f) via the isomorphisms are
called the Kervaire invariants and denoted by c(fG). The
computation of c(f G) is done in [M2] for G normal maps treated
later.
When dim Z G m 0 (mod 4) and fG is of degree one,
LdimZG(X[1]) is isomorphic to Z componentwise. The values of
aG(f) via the isomorphisms are componentwise differences Sign W G -
Sign Z G of signatures of W G and Z G.
Suppose aG(f ) identically vanishes; so we may assume fG is a
~2 homology (resp. a homotopy, if fG is of degree one)
equivalence. Then we do surgery on W-W G equivariantly to convert
f into a homotopy (rasp. a G homotopy) equivalence. We again
encounter an obstuction. In fact, the vanishing of ~G(f) allows
us to define the obstruction
a f} E LdimZ(~[G]).
When dim Z m 2 (mod 4), LdimZ(~[G]) is isomorphic to Z 2 (see
266
§I3A of [Wl]).
estimate a(f)
is devoted to this problem.
Summing up the content of this section, we have
Proposition 1.7. Let Z be a connected G manifold
satisfying (I.I) - (1.5) and let K = (W,f,b) f : W 4 Z
normal map with b : TW ~ f (TZ+E) for some E E KOG(Z).
(i) dim Z m dim Z G ~ 2 (mod 4),
(ii) e(f G) = 0 (eomponentwise),
(iii) a(f) = 0 in LdimZ(2[G]) ~ 22.
Then there is a G normal map K' = (W',f',b')
that
(I) f' is a homotopy (a G homotopy, if
one) equivalence,
(2) b' : TW' a f' (TZ+E). s
Proposition 1.8. Let Z, K and E be the same as in
Proposition 1.7. Suppose
(i) dim Z ~ 2 (mod 4) and dim Z G ~ 0 (mod 4),
(ii) Sign W G - Sign Z G = 0 (componentwise),
(iii) o(f) = 0 in LdimZ(2[G]) ~ 22 .
Then the same conclusion as Proposition 1.7 holds.
But this time there is no helpful formula to
in terms of K = (w,f,b and Z. The next section
b e a G
S u p p o s e
f': W' ~ Z such
fG i s o f d e g r e e
§ 2 . F r a m e d G m a n i f o l d s w i t h t h e K e r v a i r e i n v a r i a n t o n e
I n t h i s s e c t i o n we w i l l s h o w t h e e x i s t e n c e o f f r a m e d G
m a n i f o l d s w i t h t h e K e r v a i r e i n v a r i a n t o n e . T h i s e n a b l e s u s t o k i l l
267
a ( f ) ( o r c ( f G ) ) i n P r o p o s i t i o n s 1 . 7 , 1 . 8 , i f n e c e s s a r y , by d o i n g
equivariant conneeted sum.
A framed G manifold can be naturally regarded as a G normal
map with a sphere as the target manifold; so we state our results in
terms of a G normal map. We first treat low dimensional cases.
T h e o r e m 2 . 1 . F o r m : 2 o r 4 t h e r e i s a G n o r m a l map
= ( W m ' f m ' b m ) fm : Wm ~ S ( ~ 2 m - l ' 2 m ) s u c h t h a t
(I) W G = s(~m)xs(~m), m
(2) C{fmG) = I in LZm_2(Z[I]) ~ Z 2 ,
(3) TW is a trivial G vector bundle. m
E m
This theorem is obtained by making the following well known fact
equivariant.
P r o p o s i t i o n 2 . 2 . F o r m = 1, 2 , 4 t h e r e i s a n o r m a l map
(WmO f m 0 , b m 0 0 : W 0 ~ S ( ~ 4 m - 1 ) s u c h t h a t = ) fm m
(1) W 0 = S ( ~ 2 m ) × s ( ~ 2 m ) ; h e n c e TW 0 i s t r i v i a l , m m
(2 ) C( fmO) = 1 i n L 4 m _ 2 ( Z [ 1 ] ) ~ Z 2 .
0 K
m
0 0 The map f We s h a l l r e c a l l t h e e x p l i c t c o n s t r u c t i o n o f K m m
0 i s d e f i n e d b y c o l l a p s i n g t h e e x t e r i o r o f a n o p e n b a l l i n W m t o a
p o i n t , a n d b 0 i s t h e t r i v i a l i z a t i o n o f T ( S ( ~ 2 m ) × s ( ~ 2 m ) ) d e f i n e d m
as follows. Remember that ~2m admits a mutiplicative structure
d e f i n e d b y
( q l ' q 2 ) ( q l ' ' q2 ' ) = ( q l q l ' - q 2 ' q 2 ' q 2 ' q l + q 2 q l ' )
w h e r e ( q l ' q2 ) a n d ( q l ' ' q2 ' ) a r e o r d e r e d p a i r s o f r e a l n u m b e r s
i f m = 1 { c o m p l e x n u m b e r s i f m = 2 o r q u a t e r n i o n n u m b e r s i f m =
4) a n d - d e n o t e s t h e u s u a l c o n j u g a t i o n . T h i s e q u i p s
S ( ~ 2 m ) × s ( ~ 2m) w i t h a m u l t i p l i c a t i v e s t r u c t u r e . T a k e a f r a m i n g on
268
S ( ~ 2 m ) x s ( ~ 2m) at a point and transmit it to the other points using
the multiplication. This defines the desired trivialization.
need to take a G
definition of b m
construction that
with Proposition 2.2 proves the theorem.
Proof of Theorem 2.1. Define an involution by (ql' q2 ) '
( q l ' - q 2 )" T h i s p r e s e r v e s t h e m u l t i p l i c a t i o n and t h e l e n g t h o f
(ql' q2 )" Hence S(~ 2m) inherits the involution and so does Wm0
via the diagonal action. This is the required G manifold W . m
0 The G map f is defined similarly to f . But this time we
m m
invariant open ball around a point of W G. The m
is the same as b O. It is immediate from our m
G 0 for 2 4 This = m = or 1:oge~:ner K m Km/2
Q.E.D.
0 One can also use the normal map K to kill the secondary
m
surgery obstruction o(f). In fact, given a G normal map K :
(W,f,b) f : W 4 Z with dim Z : 4m-2 and a(f) : I, then we do
0 connected sum of K and (two copies of) K equivariantly away
m
from W G to obtain a new G normal map K : (W',f',b') f' : W'
Z. Here recall that the inclusion map : 1 , G induces an
isomorphism L4m_2(X[I]) ~ L4m_2(~[G]). This and the additivity
of the Kervaire invariant under connected sum mean that
a ( f ' ) = ~ ( f ) + C(fmO) = 1 + 1 = O.
Now we are in a position to prove
Theorem 2.3.
map K = (W,f,b)
(i)
(ii)
(iii)
then there is a
Let m = 2 or 4. If we are given a G
f : W ~ Z such that
dim Z : 4m-2,
dim Z G = 2m-2 for each component of Z G,
b : TW ~ f (TZ+E) for some E E KOG(Z) ,
G normal map K' = (W',f',b') f' : W' ~ Z
normal
such
269
that
( 1 ) f' i s a h o m o t o p y ( o r a
o n e ) e q u i v a l e n c e ,
(2) b' : TW' ~ f' (TZ+E). s
G homotopy, if fG is of d e g r e e
Remark. K' is not necessarily G normally cobordant to K.
Proof . Since L2m-2(~(2) [ l ] ) ~ L2m-2(~[l]) ~ ~2 and the
degree of fG is odd, the primary obstruction aG(f) : e(f G)
be killed, if necessary, by doing equivariant connected sum with
at fixed points of W and Z. As for the secondary surgery
obstruction, the observation preceding this theorem shows how to
kill it. Finally we note that E in (iii) is unchanged through
these connected sum operations because
G vector bundles. Q.E.D.
can
K m
0 TW and TW are trivial
m m
Now we proceed to higher dimensional case. It is known that
there is no closed framed manifold with the Kervaire invaiant one
except dimensions 2n-2 ; so we are obliged to weaken the results.
Theorem 2.4.
G normal map
(I)
(2)
(3)
(4)
For a positive integer m ~ I, 2, 4
Km = (Wm'fm'bm) fm : W ~ S(~ 2m-l'2m) such that m
there is a
W G is diffeomorphic to S(~2m-1), m
TW is a stably trivial G vector bundle, m G
fm is a homotopy equivalence (hence aG(fm ) = O) ,
~(fm ) = 1 in L4m_2(X[G] ) ~ Z 2.
This time we use the following fact in place of Proposition 2 . 2 .
0 Proposition 2.5. For m ~ I, 2, 4 there is a normal map K
m
Wm0'fm0'bm0) fm0 : (Wm0,0Wm0) ~ (D(~4m-2),S(~4m-2)) such that (
0 (I) 8f is a homeomorphism,
m
270
(2) C(fm0) = 1 in L4m-2(~[l]) ~ ~2'
0 (3) TW is trivial.
m
0 R e m a r k . A c h o i c e o f b
m
provided m ~ 1, 2 , 4 .
does not effect the value of O(fm 0)
0 . An explict construction of K is as follows. Let 6 be a
m
0 small real number. Then W is defined by
m
0 (2.6) W
m : {(Zl, .. , Z2m ) e ~2m I z13+z22+ .. +Z2m 2 = 6} n D(~2m).
0 Pinch the complement of a collar boundary in W to a point.
m 0
Since the boundary of W is known to be homeomorphic to m
S(~4m-2), this defines the desired map f 0. b 0 is defined as a m m
0 trivialization of TW
m
Proof of Theorem 2.4. Since 5 is real, the complex
~I - 0 conjugation map : (zl, .. ,Z2m) ~ ( , .. ,Z2m) preserves W m ,
0 so this defines an involution r on 8W One can easily see that
m
r reverses an orientation on 0W 0 and (aWm0)r is diffeomorphic m
to S(~2m-l).
0* 0 * Now prepare a copy W of W and denote by z the
m m 0* 0 0
corresponding point of W to z E W We glue W and m m m
0* * W along the boundary by identifying z with zz for all z E
m
0 OW The resulting space is a closed and orientable manifold. We
m
define an involution on it by sending z to z and z to z,
which is compatible with the identification because z is of order
two. This is the required G manifold W . This construction is m
due to Lopez de Medrano [L] p.28. The action of G is orientation
G preserving as • is orientation reversing, and W coincides with
m
(SWm0)r, which verifies (I).
0 The proof of (2) is as follows. Since W is a submanifold of
m
271
D(~ 2m) and the involution on 0W 0 comes from the complex m
conjugatin map on S(~2m), we can regard W as a closed G m
submanifold of a G sphere D(~ 2m) U D(~ 2m) : S. Then it is easy T
t o s e e t h a t t h e G n o r m a l b u n d l e o f W i n S i s i s o m o r p h i c t o m
W x ~I,i and that TS is a stably trivial G vector bundle. m
This verifies (2).
We define the G map f by collapsing the exterior of an open m
G invariant ball around a point of W to a point. Then (3) is
m
clear.
b m is defined as a stable equivariant trivialization of TW m.
By Proposition 2.5 C(fm 0) = 1 provided m ~ I, 2, 4. On the other
hand, as indicated before, the inclusion map : 1 ~ G induces an
isomorphism : L4m_2(Z[I] ) ~ L4m_2(~[G]). The above geometric
construction exactly corresponds to this algebraic isomorphism ; so
(4) follows. Q.E.D.
As a consequemce of Theorem 2.4 we have
Corollary 2.7.
Z be a G normal map such that
(i) dim Z = 4m-2,
(ii) dim K = 2m-2 for some connected component K of Z G,
(iii) b : TW s ~ f (TZ+E) for some E E KOG(Z).
If a(f G) = 0, then there is a G normal map K' = (W',f',b')
W' ---* Z such that
(I) f' is a homotopy (a G homotopy, if fG is of degree
one) equivalence,
(2) b' : TW' m f' (TZ+E). s
Let m # 1, 2, 4. Let K = (W,f,b) f : W ....
f' :
272
§3. Construction of G quasi-equivalences
In this section we use the idea of Petrie (see §12, Chapter 3 of
[PR] or §2 of [MAP]) to construct explict and nice G
quasi-equivalences (or G fiber homotopy equivalences) over
p(~m,n). A general method to produce G fiber homotopy
equivalences by means of Adams operations is discussed in [P3].
1 Suppose that we are given a proper S × G map ~ : V 4 U of
degree one between S l × G representations. Then we associate a
proper fiber preserving G map with a principal S l x G bundle
s(~m,n) ~ p(~m,n)
~8 : V = S(~I~ 'n) x V ~' U : S(C~ 'n) x I U S ] ~ S
p(~m,n
where 8 denotes + or -.
on each fiber, i.e. ~ is a 8
desired construction.
Since ~ is of degree one, so is E
G quasi-equivalence. This is the
Forgetting the G action, it is a fiber homotopy equivalence.
We shall denote it by m : V ~ U by dropping the suffix 8.
Here are two interesting examples used later. We refer the
reader to [MeP] for a general construction of ~.
Example 3.1. Let t denote the standard complex
l-dimensional representation of S 1 and t k the k fold tensor
product of t over C. Let p and q be relatively prime
integers greater than one. We set
U p'q = t + t pq, V p'q = t p + t q
Choosing positive integers a and b such that -ap+bq = 1, we
273
define a proper S 1 x G map ~P'q : V p'q ~ U p'q by
o P ' q ( z 1, z 2) = ( z l a z 2 b z lq+z2P) *
One can check that ~P'q
example).
Putting the trivial
be regarded as an S 1 x G map. Since e p'q
induced map ~P 'q B
necessarily a G
values of p and
+. For the case
P(0X~ n )
is of degree one (see §2 of [MAP] f o r
G actions on U p'q and V p'q, ~P'q can
is of degree one, the
is a G quasi-equivalence. However it is not
fiber homotopy equivalence. It depends on the
q. Let us observe the effect for the case 8 =
8 = - the role of the components P(~mxo) and
of p(~m,n)G is nothing but interchanged.
Case I. The case where p and q are both odd, In this case
one can see
(~p,q)G +
UP'q) G : UP'qlp(~mx0) u P(0x~ n) + +
T T r (V,P'q) G = vP'qlP(ll;mxo) u PlOxenl ÷ ~
where the symbol
fiber degree of
^p,q degree of (~p,q~G+ . is also one. Hence ~+
homotopy equivalence.
Case 2. The case where p is even and q
case we have
I denotes the restriction. We know that the
~P'q is one ; so this diagram shows that the fiber
is a G fiber
is odd. In this
(~p,q)G
Ap,q G ^p,q tp q (U+ ) = (U+ )[P(~m×o) u S(O×~ n) ×
^p ,q (v+~P'q)G = (V+ )~P(~mxo) u S(Ox~ n) slX t p.
.^p,q G The fiber degree of ~m+ ) over P(~mx0) is one as before, but
274
that over P(0x~ n) is q as is easily seen from the definition of
~P'q. Therefore ~P'q is not a G fiber homotopy equivalence in w+
this case.
The same argument as in Case 2 works for the remaining case
where p is odd and q is even.
ExamPle 3.2. We take the double of mP'q and define an action
of G by permuting them :
~P'q = oP'q • aP'q : V p'q • V p'q .... ~ U p'q • U p'q.
(~p,q)G u s over P(~mx0) (resp. P(0x~n)) is isomorphic to ~P'q
over P(~mx0) (resp. P(0x~n)). In particular ~'q is necessarily
a G fiber homotopy equivalence independent of values of p and
q; so it has a G fiber homotopy inverse (in a stable sense). We
shall denote it by -^P'q Hence Whitney sum of h copies of
^P'q denoted by h ~p'q has a sense for every integer h
~4. First Pontrjagin classes and defects
In this section we apply the Atiyah-Singer index theorem for
Dirae operators to a homotopy p(~N) with a Type II involution and
deduce some interesting congruences between the first Pontrjagin
classes and the defects defined in the Introduction. This section
is independent of G surgery part, so the reader may take a glance
at the results (Lemma 4.1, Theorems 4.3, 4.5 and Corollaries 4.4,
4.6) and skip their proofs. The following lemma will be established
in the course of the proof of Theorem 4.3.
Lemma 4.1. Let X be a homotopy p(~N). Then the first
275
P o n t r j a g i n c l a s s P l ( X ) o f X i s o f t h e f o r m
P l ( X ) = (N + 2 4 k ( X ) ) x 2
with some integer k(X), where x is a generator of H2(X;~).
Remark 4.2. On the dimensions, where framed closed manifolds
with the Kervaire invariant one exist, the function k(X) takes any
integer ( N = 2, 4, 8, 16, 32 are the cases at presnt). For the
other even values of N one can see that k(X) can take any even
integer. Conversely the recent result of Stolz [S] (together with
(4.4) of [DMSc]) implies that k(X) must be even if N is an even
integer except powers of 2 (note that k(X) modulo 2 agrees with
the ~ invariant in [DMSc]). For an odd integer N the value of
k(X) is more restrictive and complicated.
Our main results of this section are as follows.
T h e o r e m 4 . 3 . I , e t X b e a h o m o t o p y p ( $ N ) w i t h a T y p e I I G
action. Let F i (i = I, 2) be connected components of X G of
dimension 2(Ni-I ) . Then, choosing suitable signs of the defects
D(Fi), we have
D(F]) + D(F2) m 4k(X) (mod 8).
Corollary 4.4. If X is G homotopy equivalent to P(~ NI'N2
then k(X) m 0 (mod 2).
) ,
Proof of Corollary 4.4. The assumption means D(F i) = ±I as
remarked in the Introduction. This together with Theorem 4.3 proves
the corollary. Q.E.D.
Theorem 4.5. If X is G homotopy equivalent to P(~ NI'N2
then 2k(F i) m k(X) (mod 4) provided N i > 2.
,
Corollary 4.6. Let X be the same as in Theorem 4.5. If N. 1
276
is an even integer except powers of 2 for either i, then k(X) m 0
mod 4).
P r o o f o f C o r o l l a r 7 4 . 6 . By t h e a s s u m p t i o n a n d R e m a r k 4 . 2 , k ( F i )
i s e v e n . T h i s a n d T h e o r e m 4 . 5 p r o v e t h e c o r o l l a r y . Q . E . D .
Theorems 4.3 and 4.5 are proved in a similar fashion to each
other. The tools used in the proofs are based on [PI]. We shall
review them briefly. See [PI] for the details.
Since H3(X;~) vanishes, X admits a SpinC(2N-2) structure,
i.e. there is a principal SpinC(2N-2) bundle over X with total
space P such that
P x ~2N-2 ~ TX. SpinC(2N_2) =
By [PI] the G action on X lifts to an action on P which covers
the canonical G action on TX defined by the differential. Then
the half SpinC(2N-2) modules A and A give G vector bundles +
E+ and E_ over TX
E± = P x (~2N-2 x A±) SpinC(2N-2)
and there is a G complex over TX ; E --~ E which defines an +
element 6G E KG(TX ).
Let Id~ : KG(TX ) 4 R(G) denote the Atiyah-Singer index
homomorphism to the complex representation ring R(G) of G. For
V E KG(TX), Id~(V)(g) is the value of the character Ida(V) at g
E G. An element E of KG(X) yields an element E6 G of KG(TX)
through the natural KG(X) module structure on KG(TX). The
following lemma is stated in the proof of Theorem 3.1 of [P]].
Lemma 4.7. Let g be the generator of G. Then the values of
v A
IdA(E6G ) u at 1 and g are as follows :
277
(i) Id~(E6G)(1) : <ch(E)eNX/2A(X), [ X ] >
^ N x , / 2 ~ (ii) Id~(ESG)(g) = Z 8i<Chg(E~Fi)e 1 A(Fi)/chA(vi), [Fi] >
where
(a) E is the element of K(X) obtained from E by forgetting
the action,
(b) x i denotes the restricted element of x to H 2(Fi;Z),
H* (c) Chg : KG(Fi) : R(G)®K(F i) ~ (Fi;~) is defined by
Chg(V®~) = V(g)ch(~) where V(g) is the value of the character V
at g,
( d ) ~. : ± 1 , 1
( e ) c h A ( v i ) i s t h e u n i t o f H ( F i ; Q ) d e f i n e d b y t h e f o r m a l
power series
N-N. N-N. 2 z I I Zcosh(~j/2)
j=l
2 where the elementary symmetric functions of the m. give the
J
Pontrjagin classes of the normal bundle v. of F. to X, + 1 1
(f) A(Y) is the A class of Y and the lower terms are
expressed by A(Y) = 1 - pl(Y)/24 + ...
y ~
Since Id~(E6G) is an element of R(G), the evaluated values
at 1 and g are both integers and their difference must be even.
This fact will give an integrality condition on the Pontrjagin A
classes of X and F i if there is an element E of KG(X). The
following lemma provides such an element.
Lemma 4.8 (Corollary 1.3 of [PI]). Any complex line bundle
over X comes from an element of KG(X).
Lifting of the G action on X to ~ is not unique. There
are exactly two kinds of liftings. The resulting two complex G
278
line bundles are related to each other through the tensor product by
the non-trivial one dimensional complex representation t of G.
Therefore a complex G line bundle, whose underlying bundle is
and the action on a fiber over a point of F 1 is trivial, is
^
unique. We shall denote such a G bundle by ~. Under these
preparations
Proof of Theorem 4.3. Let n be a complex line bundle over X
whose first Chern class is a generator x of H2(X;Z). By Lemma
^ ^ N 1 - 1 N 2 - 2 ^ 4.8 E r = (~-1) (t~-l) n r is an element of KG(X) for any
integer r. As is well known R(G) = ~[t]/(t2-1) ; so one can
express
A
IdX(ErSG ) = ar(l-t) + b r
with integers a r and b r. This means that
IdX(ErSG)(1) = b r
A
X + b . Id (ErSG)(g) = 2ar r
Now we shall apply Lemma 4.7 to compute these values. Remember
that
E = ( n - 1 ) N - 3 ~ r r
A(X) : I - Pl(X)/24 + .. = I - (N/24+k(X))x 2 + ..
N-3 Since the lowest term in ch(E r) is x , one can easily deduce
( 4 . 9 ) b = ( r + N - 2 ) ( r + N - 1 ) / 2 - k ( X ) r
from (i) of Lemma 4.7. This shows the integrality of k(X) ; so
Lemma 4.1 is established,
The computation of (ii) of Lemma 4.7 is as follows. The point
279
is that the cohomological degree of the lowest term in Chg(Er)F I)
(resp. Chg(ErlF2) ) is 2(NI-I) (resp. 2(N2-2)) and both A(F i)
and chA(vi) have values of cohomological degrees divisible by 4.
This means that only the constant terms in A(F i) and chA(Pi),
N-N. which are respectively ] and 2 i, contribute to the
computation. Thus, by an elementary calculation, (ii) reduces to
(4.10) : {D(F ) + ( 2 r + 2 N - 3 ) D ( F 2 ) } / 4 2 a r + b r 1
( r emember t h a t D ( F i ) a r e d e f i n e d up t o s i g n ) .
Eliminate b in (4.10) using 4.9) and multiply the resulting r
identity by 4. Then we get
2(r+N-l ) ( r+N-2) - 4k(X) m D(FI) + (2r+2N-3)D(F 2) (mod 8)
because a is an integer. This congruence holds for every integer r
r ; so take r = 2-N for instance. Then it turns into
4k(X) i D(FI) + D(F2) (mod :B)
which verifies Theorem 4.3. Q.E.D.
Proof of Theorem 4.5. The idea is the same as in the proof of ^ ^ NI-I N2-3^
Theorem 4.3. This time we make use of E' = (~-I) (t~-l) r
instead of E r. Then one can deduce the desired congruenee for F 2.
We omit the details because the computation is similar to the
before. The parallel argument works for F I. Q.E.D.
§5. Construction of Type II involutions
In this section we apply the preceding results to construct
28O
homotopy p(~N) 's with Type II involutions. The gap hypothesis then
restricts our object to Type IIN/2_ 1 actions and N =- 0 (mod 2). As
observed in §I, the surgery obstructions which we encounter are
different by the values of N modulo 4.
First we treat the case N = 0 (mod 4). We consider the
realization problem of Theorem 4.3 and Corollaries 4.4, 4.6. The
first main result is Theorem 5. I. The author believes that it is
valid iff N is a power of 2 greater than 2 (of. Remark 4.2).
But it is related to the Kervaire invariant conjecture; so it would
be beyond our scope.
Theorem 5.1. Let N = 4 or 8. Suppose we are given a triple
(k, d], d 2) of integers satisfying these conditions :
(I) d. are odd, 1
(2) d I + d 2 - 4k (mod 8) or d I - d 2 m 4k (mod 8).
Then there is a homotopy p(~N) X with a Type IIN/2_ 1 G action
such that
(k, {dl{ , {d2{ ) = (k(X), {D(FI){, {D(F2) {)
where F. are connected components of X G. In addition there is a 1
G map f : X --~ p(~N/2,N/2) giving a homotopy (or a G homotopy,
if d. = +I equivalence. 1
Proof. Since (q2-I)/8 - 0 or 1 (mod 2) according as q -
or + 3 (sod 8), the assumption means that k+(d2-1)/8+(d2-1)/8 ±I
is an even integer.
quasi-equivalence
We denote it by 2h and consider a G
^ ^2'd2 ^2 'd l ^2,3
over p(~N/2,N/2)
yields a G normal map
(see Examples 3.1 and 3.2). By Theorem 2.3
(X,f,b) with a homotopy equivalence f.
281
This is the desired one.
definition of the above
In fact it easily follows from the A
that
D(F 1) = the fiber degree of
D(F 2) = the fiber degree of
~GIp(~N/2×0 ) = d 1
~GIp{o×~N/2 ) = d 2
^ 2 , d 1 _ ~ 2 , 24k(X)x 2 : P l ( V d l + v 2 ' d 2 _ u 2 d 2 _ 2 h ( ~ 2 , 3 _ U 2 , 3 ) )
: { - 3 ( d ~ - l ) - 3 ( d ~ - l ) + 48h}x 2 2
: 24k x Q.E.D.
Corollary 5.2 (cf. Corollary 4.4). Let N = 4 or 8 and k be
even. Then there is a G homotopy p(~N/2,N/2) X with k(X) : k.
Proof. Apply Theorem 5.] to {k, I~ I). Q.E.D.
Corollary 5.3. Every homotopy p($4) admits infinitely many
Type II 1 involutions distinguished by the defects. In particular
they are not G homotopy equivalent to each other.
Proof. By [W2] the set of homotopy P(¢4)'s bijectively
corresponds to • via the function k(X). For a fixed integer k
there are infinitely many triples (k, dl, d2) satisfying the
conditions of Theorem 5.1. This verifies the corollary. Q.E.D.
For higher dimensional cases we use Corollary 2.7 instead of
Theorem 2.3. There it must be arranged that the Kervaire invariant
on the fixed point set vanishes. This forces us to put a constraint
that k is even, but it is essential unless N is a power of 2
(see Remark 4,2).
Theorem 5.4. Let N m 0 (mod 4). Suppose we are given a
triple (k, d I, d 2) of integers satisfying these conditions :
(I) d. are odd and k is even, i
(ii) d I + d 2 m 4k (mod 16) or d I - d 2 m 4k (mod 16).
282
Then the same conclusion as in Theorem 5.1 holds.
Proof. The proof is similar to that of Theorem 5.1. The
assumption means that k+(d~-l)/8+(d~-l)/8 m 0 or 2 (mod 4)
according as d. m ±1 (mod 8) or d. m ±3 (mod 8). We denote it by 1 1
2h and consider a G quasi-equivalence e defined in the proof of
Theorem 5.1. Observe that
~GIp(~NI2x0 ) = (¢2,2d I ~ 2 , d 2
• e • (-h)~2'a)Ip(~N/2x0)
^ 2 , d 1 ~ G I p ( 0 x ~ N / 2 ) : (~ • ¢2,2d 2
e (-h)~2'3)IP(0x~N/2)
where ¢ is the v times map from u to uv and where UjV
denotes the canonical line bundle over p(¢N/2). The following
a s s e r t i o n i s p r o v e d i n Lemma 3 .11 and Theorem 3 .1 o f [M2].
A s s e r t i o n . (1) C ( ¢ u , v ) = 0 i f u i s e v e n ,
(2) c ( ~ p ' q ) m ( p 2 - 1 ) q 2 - 1 ) / 2 4 (mod 2 ) .
Since the Kervaire invariant is additive with respect to
Whitney sum of odd degree fiber preserving proper maps (see [BM]), A
the above assertion implies c(~ G) = 0. Therefore it follows from
Corollary 2.7 that m yields a G normal map (X,f,b) with a
homotopy equivalence f. In a similar way to the proof of Theorem
5.1 one can see that this is the desired one. Q.E.D.
As a consequence of Theorem 5.4, if we weaken the dimensional
assumption N : 4 or 8 in Corollary 5.2 to N m 0 (mod 4), then
we get
Corollary 5.5 (cf. Corollary 4.6). Let N ~ k m 0 (mod 4).
Then there is a G homotopy p(~N/2,N/2) X with k(X) : k.
Proof. Apply Theorem 5.4 to (k, i, i). Q.E.D.
283
For the case N m 2 (mod 4) we again apply Corollary 2.7. This
time the surgery obstructin aG(f) is detected by the signature
(Propsition 1.8). We shall outline the proof of Theorem 5.6 stated
below.
First recall that ~'q is a G fiber homotopy equivalence
over p(~N/2,N/2) if p and q are both odd (see Example 3.1).
Consider an abelian group Q generated by all such ~P'q. We want 8
to find an element e of ~ such that the surgery obstruction of
^G vanishes. By Proposition 1.8 the obstruction is detected by the
componentwise differences Sign wG-sign p(~N/2,N/2)G where W is a
G manifold obtained from e. Since the fixed point set consists of
two connected components, we get a map
Sign : ~ ~ ~ •
given by ~ 4 Sign wG-sign p(~N/2,N/2)G.
Unfortunately this is not a homomorphism. However, if we restrict
it to a certain subgroup of ~, then it turns out to be a
homomorphism and hence its kernel would contain infinitely many
elements provided that the rank of the subgroup is greater than two.
This is the case if N > 6. The trick to make the map Sign a
homomorphism in this way is due to W.C. Hsiang [H].
Consequently we have
Theorem 5.6. Let N ~ 2 (mod 4) and N ~ 10. Then there are
infinitely many G homotopy p(~N/2,N/2) X such that the total
Pontrjagin classes of X and F. are not of the same form as the 1
standard ones, where F. are components of X G as before. 1
Remark. The reason why we exclude the case N = 6 is to avoid
4-dimensional surgery on the fixed point set.
284
For the remaining case N m I (mod 2) a Type II involution on
a homotopy p(~N) X has a fixed point component of dimension at
least N-I. Hence the gap hypothesis is never satisfied and hence
we cannot apply the preceding G surgery theory. But, for a Type
II(N_I)/2 involution, one of the fixed point components is of
dimension equal to I/2dim X = N-I and the other is of dimension
less than i/2dim X. The G surgery obstruction under these
situations is analyzed by Dovermann. We quote it in our setting.
Proposition ([DI]). Let K = (W,f,b) f : W ~ P =
p(~(N+I)/2,(N-I)/2) be a G normal map such that fG is of degree
one. Then if the following conditions are satisfied, then one can
convert f into a G homotopy equivalence via a G normal
cobordism :
(I) c(f G) : 0
(2) Sign W G = S i g n pG
(componentwise)
(componentwise)
Sign(G,W) = Sign(G,P).
As before we can produce many G normal maps from G fiber
homotopy equivalences over P because the G transversality still
holds ([P2]). We must carefully choose a fiber homotopy equivalence
so that the associated G normal map satisfies the above (I) - (3).
We may neglect (I) by virtue of additivity of the Kervaire invariant
with respect to Whitney sum of fiber homotopy equivalences. We
apply the Hsiang's trick to adjust (2). For (3) we again apply the
Hsiang's trick. However at this last step we must evaluate
Sign(G,W), which consists of two elements : one is the ordinary
signature of W and the other is the equivariant signature of W
at the generator of G. Here the later causes a problem. Namely,
in order to compute it using G signature theorem, we need to know
285
the Euler class of the normal bundle w of the fixed point
component of dimension equal to I/2dim W. However the stable G
isomorphism b does not provide us with any information for it
because the Euler class is not a stable invariant. To solve this
problem we consider a semi-free Z 4 action extending the G
action. Namely we consider a ~4 fiber homotopy equivalence. It
then equips the normal bundle v with a complex structure induced
from the Z 4 action. Since the Chern classes are stable invariants
and the top Chern class agrees with the Euler class up to sign, this
method enables us to evaluate the Euler class of v through the
stable ~4 isomorphism b.
Consequently we use the Hsiang's trick twice to obtain the
following result similar to Theorem 5.6. The details are omitted.
T h e o r e m 5 . 7 . L e t N m I ( m o d 2) a n d N > 1 1 . T h e n t h e r e a r e
infinitely many G homotopy p(~(N+I)/2,(N-I)/2) X such that the
total Pontrjagin classes of X and F. are not of the same form as 1
the standard ones.
Appendix
In this appendix we apply the o r d i n a r y s u r g e r y t h e o r y to exhibit
infinitely many non-standard G homotopy P(cm'n). Here the gap
hypothesis is unnecessary, but the fixed point sets and their
equivariant tubular neighborhoods are equivariantly diffeomorphic to
those of p(¢m,n). The following lemma is easy.
Lemma A.I. Let P0 be the exterior of an equivariant open
tubular neighborhood of p(¢m,n)G in p(~m,n). Then P0 is a free
286
G space and equivariantly diffeomorphic to the product of
(s(~m'0)xs(~n))/Sl and the unit interval, where the S l action is
the diagonal one induced from the complex multiplication.
We shall denote the G orbit space of PO by T 0. Suppose we
are given a manifold XO together with a homotopy equivalence T 0 :
X0 -~ P0 which restricts to a diffeomorphism on the boundary. Then
we lift T 0 to the double coverings and glue the equivariant
tubular neighborhood of p(~m,n)G in p(~m,n) to X 0 (the double
cover of ~0 ) and P0 respectively along their boundaries via the
lifted map. This yields a G homotopy p(~m,n) together with a G
homotopy equivalence.
In order to produce such a pair (X0,f 0) we use the ordinary
surgery theory (relative boundary). The surgery exact sequence
yields
0 : L2N_I(G) --~ hS(P0,eP O) ~ [P0/OP0,F/O] ....... a ~ L2N_2(G)
where N = m+n and hS(P0,SP 0) denotes the set of such pairs
(Xo,f 0) identified by a natural equivalence relation (see §I0 of
[WI]). By Lemma A.I T 0 is diffeomorphic to the product of a
closed manifold with the unit interval; so the above surgery
obstruction o turns out to be a homomorphism (see p.lll of [WI]).
As is easily seen, the rank of the abelian group [Po/OPo,F/O] is
[(N-1)/2]-[(max(m,n)-1)/2] (see [DMSu] for the details). Moreover
L2N-2(G) = ~2 or ~e~ according as N is even or odd (see p.162
of [Wl]).
elements
if either
(I) N = m+n
or (2) N = m+n
These mean that hS(~0,0~0) contains infinitely many
(X0,f 0) distinguished by the Pontrjagin classes of ~0
is even and max(m,n) _< N-2,
is odd and max(m,n) _< N-5.
287
Thus we have established
Theorem A.2. Suppose m and n satisfy either of the above
(1) or (2). Then there are infinitely many G homotopy p(~m,n)
such that the fixed point sets and their equivariant tubular
neighborhoods are equivariantly diffeomorphic to those of p(~m,n).
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PROPER SUBANALYTIC TRANSFORMATION GROUPS AND
UNIQUE TRIANGULATION OF THE ORBIT SPACES
Takao Matumoto
Department of Mathematics
Faculty of Science
Hiroshima University
Hiroshima 730, Japan
Masahiro Shiota
Department of Mathematics
Faculty of General Education
Nagoya University
Nagoya 464, Japan
§ i. Introduction
Let G be a transformation group of a topological space X,
Triangulation of the orbit space X/G was treated by several people
(e. g. [5], [12] and [13]) in some cases of compact differentiable
transformation groups. The authors showed in [7] a unique triangula-
tion of X/G, provided that G is a compact Lie group, X is a real
analytic manifold and the action is analytic. Moreover, the uniqueness
was extended to the case of differentiable G-manifolds and played an
important role in defining the equivariant simple homotopy type of
compact differentiable G-manifolds when G is a compact Lie group.
Let us explain what the uniqueness means here. Under the above condi-
tions we can give naturally X/G a subanalytic structure. On the
other hand we know a combinatorially unique subanalytic triangulation
of a locally compact subanalytic set ([3] and [ii]). Hence X/G comes
to admit a unique subanalytic triangulation.
Now we consider a problem under what weaker condition X/G has a
natural subanalytic structure. Of course we may assume that X, G and
the action are subanalytic; as a subanalytic set is Hausdorff, it is
natural to assume a condition that the action is proper in the sense of
[6] and [9] (see §2); moreover, in order to simplify the description we
assume that X is locally compact. In this paper we shall show that
these conditions are sufficient (Corollary 3.4) and hence we obtain a
unique subanalytic triangulation of the orbit space of a proper subana-
lytic triangulation of the orbit space of a proper subanalytic trans-
formation group of a locally compact subanalytic set (Corollary 3.5).
291
We shall see that a subanalytic group is homeomorphic to a Lie
group. But we shall not use properties of Lie group except for the
Montgomery-Zippin neighboring subgroups theorem [8].
See [7] for more references and our terminology.
§ 2. Subanalytic transformation groups
Let G be a topological group contained in a real analytic mani-
fold M. If G is subanalytic in M then we call G a subanalytic
group in M.
Remark 2.1. A subanalytic group in an analytic manifold is homeo-
morphic to a Lie group. It seems that G may be subanalytically
homeomorphic to a Lie group.
Proof. As the Hilbert's fifth problem is affirmative [8] it
suffices to see that G is locally Euclidean at some point of G.
But this is clear by the fact that a subanalytic set admits a subana-
lytic stratification (see Lemma 2.2, [7]).
Let G be a subanalytic group in M 1 and X a subanalytic set
in M 2. If G is a topological transformation group of X and the
action G × X 9 (g, x) ~ gx 6 X is subanalytic (i~e. the graph is subana-
lytic in M 1 × M2) then we call (G, M I) a subanalyitc transformation
~roup of (X, M2).
A transformation group G of a topological space X is called
proper if for any x~ y 6 X, there exist neighborhoods U of x and V
of y such that {hE G: hUN V~ ~} is relatively compact in G ([6]
and [9]). This is equivalent to say that G × X9 (g, x) ~ (gx t x) 6 X × X
is proper when G is locally compact and X is Hausdorff.
Remark 2.2. Let G be a locally compact proper transformation
group of a completely regular space X. Then X/G is completely
regular [9].
Lemma 2.3, Let (G, M I) be a subanalytic proper transformation
group of a subanalytic set (X, M 2) and {X i} be the decomposition of X
by orbit types. Then {X i} is locally finite in U of X in M 2,
Proof. For each x 6 X let G denote the isotropy subgroup of x
G at x. Put
292
A = U G x x x = {(g, x) E G x X: gx =x} xEX
and let ~:M 1 × M 2 ~M 2 be the projection~ Then A is subanalytic in
M 1 x M 2. Moreover, we can choose an open neighborhood U of X in M 2
so that ZIA,:A' ~ U is proper from the fact that a subanalytic set is
o-compact and the assumption that ~[A:A~ X is proper, where A' is
the closure of A in G x U. We may consider the problem in U and an
open neighborhood of G in M 1 in place of M 2 and M 1 respectively,
and this U will satisfy the requirements in the lemma. Hence we can
assume from the beginning that G is closed in M 1 and the map
zI~:A~M 2 is proper where A is the closure of A in M 1 x M2. Let
also denote the closure of X in M 2. We remark A N G × X =A
because A is closed in G × X.
Now we note that the following assertion is obtained from
Hironaka's theorem [4, p.215] since zl~:A~ X is a proper map.
Assertion: A and X have subanalytic stratification A = {A i}
and V= {Yj} respectively such that zI~:A~ V is a stratified map
compatible with X: i.e.,
(i) For each stratum A i of A, ~(Ai) is contained in some Yj.
:A ~ Y is a C ~ submersion. (ii) For such i and J' ~IA i l 3
(iii) For each j, Aj = {A i 6 A: ~(Ai) cYj} is a Whitney stratification
([2] or [I0]).
(iv) X is a union of some strata of V.
Apply the Thom's first isotopy lemma to ~I~!A ~ Y (e,g. 5.2,
Chapter_l II. [i]). Then for each Y.3 and xl, x 2 6 Yj, -i (x I) n A and
(x 2) N A are homeomorphic. Here it is important that Yj are con-
nected. Now if x 6 X then
-I (x) N A = -l(x) N A = G × x.
X
Hence for x I, x 2 E Yj c Z, Gxl and Gx2 are homeomorphic. Furthermore,
for such x I and x2, Gxl and Gx2 will be conjugate. To see this recall
the Montgomery-Zippin neighboring subgroups theorem [8~ p,216], which
states that each compact subgroup H of G has a neighborhood O in
G such that any compact subgroup of G included in O is conjugate
to a subgroup of H. Hence, by the properness assumption~ each x 6 X
has a neighborhood V in X such that G is conjugate to a subgroup Y
293
of G x for any Y 6V- But a proper subgroup of G is never homeo- x
morphic to G x as G x is compact. Therefore if y 6 V is located in
the same stratum as x then G is conjugate to G . Thus we have y x
proved for x I, x 2 6 Yj cX, Gxl and Gx2 are conjugate. Hence each of
cX. Therefore {X i} satisfies X l in the lemma is a union of some Y3
the requirements in the lemma, which completes the proof.
Remark 2.4. In Lemma 2.3 and Lemma 3.1 below we can replace the
properness condition by a weaker condition that X is a Cartan G-space
in the sense of [9], which is clear by their proofs.
In Lemma 2,3 if X is closed in M 2 we can put U=M 2 for the
following reason (Lemma 2.1, [7]). A subset Y of an analytic mani-
fold M is subanalytic in M if each x £ M has an open neighborhood
W in M such that Y N W is subanalytic in W.
3. Subanalytic structure on an orbit space and its triangulation
Let X be a topological space. A subanalytic structure on X is
a proper continuous map ~: X~M to an analytic manifold such that
~(X) is subanalytic in M and ~: X~(X) is a homeomorphism. Let
XI, X 2 be topological spaces with subanalytic structures (~I ~ M I) and
(~2' M2) respectively. A subanalytic map f: X 1 ~ X 2 is a continuous -I
map such that the graph of ~2 o f 0 ~i : ~l(Xl ) ~2(x2 ) is subanalytic
in M 1 x M 2. Subanalytic structures (~i' MI) and (~2' M2) on X are
equivalent if the identity map of X is subanalytic with respect to
the structures (~i' MI) on the domain and (~2' M2) on the target. We
shall regard equivalent subanalytic structures as the same.
If X is a locally compact subanalytic set in an analytic mani-
fold M from the outset, then X is regarded as equipped with the sub-
anaytic structure given by the inclusion ~ X~ U where U is some open
neighborhood of X in M such that X is closed in U. We give
every polyhedron a subanalytic structure by PL embedding it in a
Euclidean space so that the image is closed in the space, Then a PL
map between polyhedra with such subanalytic structures is subanalytic
and hence the subanalytic structure on a polyhedron is unique,
Let X be a subanalytic set or a topological space with a subana-
lytic structure, Then a subanalytic triangulati0n of x is a pair
consisting of a simplicial complex K and a subanalytic homeomorphism
294
• :[K[ ~X. For a family {Xi} of subsets of X, a triangulation (K, T)
of X is compatible with {X i} if each X i is a union of some
T(Int ~), o 6 K.
We remark that when we consider a subanalytic structure on a
topological space or a subanalytic triangulation of the space we shall
treat only a locally compact space, Of course we can define a subana-
lytic structure and a subanalytic 'triangulation' (in this case a sub-
analytic 'triangulation' consists of open subanalytic simplices and may
not contain the boundary of the simplices) without the locally compact
assumption. But the description, e.g, the definition of equivalence
relation of subanalytic structures, will be complicated~ because the
composition of two subanalytic maps is not necessarily subanalytic in
the usual sense (but always "locally subanalytic" [Ii]) ; and to make
matters worse a subanalytic finite 'triangulation' (= a decomposition
into finite open subanalytic simplices) of a subanalytic set is not
unique in general.
Let q:X~ X/G be the natural quotient map for a transformation
group G of a topological space X. The following is the key lemma to
the main theorems.
Lemma 3.1~ Let (G, M I) be a subanalytic proper transformation
group of a subanalytic set (X, M 2) and x 0 a point of X. Assume that
X is locally compact. Then there exist a neighborhood U of x 0 in
X and a G-invariant subanalytic map f:GU~2k+l, k = dim X, such that
the induced map f:GU/G~ f(U) is a homeomorphism.
Proof. By properly embedding M 2 in a Euclidean space we can
assume M2 =~n and x 0 = 0. It is sufficient to define a G-invariant
subanalytic map f:GU~2k+l so that f:GU/G~2k+I is one-to-one,
because GU/G is locally compact. Put
Z = { (x, y)< X × X: q(x) = q(y)).
Then Z is the image of the projection on X × X of the graph of the
action G x X~ X. As the problem is local at 0 we can assume by the
properness condition that the projection on ~n ×~n of the closure of
the above graph is proper and hence by (2.6), [i0] Z is subanalytic in
~n x~n. Let B(s, a) and S(s, a) for ~ > 0 and a E~ n or 6 ~n x~n
denote the open s-ball and s-sphere with center at a respectively.
We shall construct open neighborhoods V 0m ~. mV2k+l of 0 in
295
X and G-invariant bounded subanalytic maps f. :V. ~z, i = 0, .'-, 2k+l, 1 1
such that
fi+l = (f I , gi+l ) V : X N B 0) i Vi+ I ' i (~i'
for some subanalytic function gi+l and some si > 0, and
= . × V.- Z : f. (x) = fi(y)} Z i {(x, y) 6 V l z l
is of dimension ~ 2k- i, If we construct these and put U=V2k+I and
f = the extension of f2k+l to GU then f:GU/G~2k+I will be
one-to-one, because dim Z2k+l = -i means that if x r y £ U belong to
the distinct orbits then f(x) ~ f(y).
We carry out the above construction by induction on i. For i = 0
we put trivially V 0 =XN B(I, 0) and f0 = 0. So assume that we have
already constructed V i and fi" Clearly Z i is subanalytic in
~n ×~n. Assume that dim Z = 2k - i, otherwise it suffices to put 1
Vi+ 1 =V i and gi+l = 0. Let Yi+l be the union of all strata of
dimension < 2k- i in a subanalytic stratification of Z i, Then
× V - Z and Yi+l ( c Z i) is a subanalytic set in ~n ×~n t closed in V i i
of dimension $ 2k- i - 1 such that Zi - Yi+l is an analytic manifold of
dimension 2k - i. For every large integer m we put
= - is an analytic manifold of dimen- W m (Z i Yi+l ) N S(i/m, 0) . Then W m
sion 2k- i - 1 since (Zi- Yi+l' 0) satisfies the Whiteny condition
(Prof. 4.7, [8]). Choose a sequence of points {aj}j:l,2,... in U W m
so that for any large m and x 6 W m, B(exp(-m), x) contains at least
f one aj. Write aj = (aj, a'~) .3. Then Ga i N Ga~ = %, Put
G O = {g 6 G: gV0 N V0 ~ ~}
where V0 denotes the closure of V 0, Then we have G01 = GOt G O is
compact by the properness condition, and hence X 0 = G0V 0 is compact.
Let {P } be the decomposition of X 0 such that x and y in X 0 are
contained in the same P if and only if there exists a finite sequence e
x = x0, Xl, ..., x Z = y in X 0 with gixi = xi+ 1 for some gi of G O ,
Here Z = 3 is sufficient for the following reason. Let x0t ..o x i
be a sequence in X 0 chained by go' "''' gi-i in G O as above,
Then by definition of X 0 there are Y0' "''' YZ in V0 and h0, ,..,
h Z in G O such that x i = hiY i. Hence we have
yZ = h~Ig~_l.~.glg0h0Y0 ,
296
Therefore, by definition of G0, hzlg~_l ...g]g0h 0 c~ G011 Hence the se-
quence x0' Y0' Yi' xi is chained by the elements h 0 , -i
h i gZ_l---glg0h0 , h i of GO, which proves that ~ = 3 is sufficient.
The above proof shows also that (i) for each ~ and x 6 P N V0'
= N V0 = Gx N V0 (i.e. {P~ N V0 } is the family of P G0(G0x n V0 ) and P
intersections of G-orbits with V0 ) , From the first equality it
follows that each P is compact and subanalytic, because G0x N V0 is
compact and subanalytic. Moreover Z = 3 shows the following. (ii)
let el' ~2' "'" be a sequence such that there exist b I 6 P~I' b2 6 P~2'
• converging to a point b Then N ~ ' ._ .. . r=iU~_rW is identical with
P which contains b. ±
Define a map A:C 0(X 0) ~C 0(V0 ) by
Ah(x) = sup{h(y) : y6 P for ~ with xC P } for x6V 0,
Then, by (ii) and by the fact that X 0 is compact, (iii) A is well-
defined (i.e. Ah 6 C0(V0 ) for h 6 C0(X0 )) and continuous with respect to
the uniform C O topology on C0(X0 ) and C O - (V0); (iv) by (i) Ah are
G-invariant for h E C0(X0) ; and (v) if h is subanalytic then Ah is
subanalytic for the following reason. Let h be subanalytic. By (i)
the set
D = { (x, Y) 6 X 0 × X0: x, y 6 P for some ~}
2 × -2 3 2 of the sub- is the image under the proper projection X 0 V 0 × G O ~ X 0
analytic set
2 -2 ~3 { (xl,Yl,x2,Y2,gl,g2,g) 6 X 0 × V 0 × ~0: Xl = glx2 ' Yl = g2Y2 ~ x2 = gY2 }'
Hence D is subanalytic. Now by definition Ah(x) = sup{h(y) : (x, y) 6 D},
and the graph of Ah is the boundary of the image by the proper pro-
jection V0 × X0 ×~9 (x, y, t) ~ (x, t) 6 V0 ×~ of the subanalytic set
{ (x, y, t)6 Q0 × V0 ×~: (x, y)6 D, t ~ h(y) }.
Therefore, Ah is subanalytic.
Assertion: Let <0j EC0(X0) , j : I, 2 ..... be a sequence satisgying
(a") Let also b. > 0. Then there exist c. > 0 e j = I, 2, A~j (a i) ~A~j J " 3 3 = cO .... such that cj < bj, [c <0 uniformly converges to some <0 6 (X 0)
j 3 3 and A~(a{) ~ A~(a 3) for all j.
Proof of Assertion: We define c inductively as follows, Put ]
297
c I =b I. Assume we have already defined c I .... , cj so that if we put
9i =Clel + "'" + czei for £ & j then
(I) i A~i(a ~) =A~i(a [) and
(2)ip cz(IA~z(a ~) +iA@z(a~)]) ! ]A@p(a~) -A@_p.(a")p. I/2 Z-p+1 for p< £
We want cj+ 1 satisfying (1)j+ 1 and (2)j+ip, p ~ j. If A@j(a~+ I)
A@j(a~+I) , it suffices to put ej+ 1 = 0. If A@j(ai+ I) =A~j(aS+I),
then we choose positive cj+ 1 so that (2)j+ip , p ~ j, hold. In this
case
A~j+l(a~+l) - A@j+I (a~+l) = cj+ l(A~j+l (a3+l) - A~j+l(a~+l)) ; 0,
hence (1)j+ 1 holds.
and (2)£p for p < £.
Thus we obtain a sequence Cl, c2~ ..., with (I)£
Then for any integer p > p' > 0
(3) -A " > (ap,) - (a" ~ I12, IA@p(ap,) ~p(ap,) I = [A@p, A~p, p,,
Furthermore, diminishing cj if necessary we can assume @j uniformly
converges to some ~. Then it follows from (3) that
A~(ai) ~ A~(a'~) for all j, 3
which proves Assertion.
For every a. the polynomial approximation theorem assures the 3
existence of a polynomial ~j on ~n such that
(ai) ~A(~jIX 0) a") ~ A(~j IX0) ( j
Let bl, b2, ... be small positive numbers such that the power series
Zb~ is of convergence radius ~ where ~9(x) means ~Id Ix e when jJJ we write ~j (x) = ~d x e,
Apply Assertion~ to these ~jI~0 and bj. Then we obtain cj ~ 0
such that Zj=ICj~ j converges to an analytic function ~ on ~n and
A(~IX0) (ai) ~A(~ IX0) (aS) for all j.
!
Put gi+l =A(~Ix 0) on V i. Then we have already seen that gi+l is
subanalytic. Hence we only need to see that
Z'i+l = { (x, y) 6 Z i : gi+l' (x) = gi+l' (Y) }
298
is of dimension ~ 2k - i - i in some small neighborhood Vi+ 1 × Vi+ 1 of
' is what we wanted. 0. In fact gi+l =gi+llVi+ 1
! Assume the dimension of Zi+ 1 at 0 is 2k- i. Then there is a
subanalytic analytic manifold Ni(c Z'i+l n (Z i - Yi+l)) of dimension 2k-i
whose closure in ~n contains 0. Recall the subanalytic version
(Prop. 3.9, [2]) of a theorem of Bruhat-Whitney which states that there
exists a real analytic map p : [0, I] ~N i U {0] such that p(0) = 0 and
p((0, I]) cN~. Define a continuous function X on [0, i] by 1
x(t) =dist(p(t) , Z i -N i) .
Then it is easy to see that X is subanalytic and positive outside 0
and hence that
x(t) _-> Cltl d, t 6 [0, I]
for some C, d > 0 (the Lojasiewicz' inequality).
B(CItl d, p(t)) n ZioN i
These imply
in other words
gi+l(X) =g~+l(y) for (x, y) 6 B(CIt] d, p(t)) n Z i.
On the other hand, by definition of gi+l
g'i+l (a~) ~ gi+l (' a")j for all j.
Hence
(4) a. ~ B(Cltl d, p(t)) for all j. ]
consider now the Zojasiewicz' inequality to the inverse function of
Ip(t) I =dist(0, p(t)) . Then, we have
IpIt) I < c"ItT d'' c" d" = for some and > 0.
Hence it follows from (4) that for some C' and d' > 0
a. ~ B(C'Ip(t) Id', p(t)) for all j. ]
But this contradicts the fact that for any large m and x 6 Wm, w B(exp(-m), x) contains at least one aj. Hence Zi+ 1 is of dimension
2k- i- 1 in some neighborhood of 0. Thus we have proved that
is one-to-one.
299
Remark 3.2 In Lemma 3.1 we can choose f to be extensible on X
as a G-invariant subanalytic map by retaking U =V2k+2 =x 0 B(S2k+2, 0)
sgb ~. Moreover, we have a G-invariant subanalytic map with C2k+2 < ~t± + 2k 2
F= (f, ~2k+2 ) :X~ with the properties (3.2.1) and (3.2.2) below.
Indeed let 8 be a subanalytic function on X with support in
such that 0 ~ @ ! 1 and 0-1(1) is a neighborhood of U. Put V 0
AISIx ) (y) on GV 0 h(x)
0 on X - G~ 0 , ~
where y6 V 0 N G x. Then hf is extensible on X so that the extension
vanishes on X- GU. We denote the extension by f for simplicity.
Let ~2k+2 : X~ be defined by
inf{lyl : (x, y) 6 Z} for x 6 GU
~2k+2 (x) = { S2k+2 otherwise.
Then M2k+ 2 is a G-invariant subana!ytic function, and X ~2k+2 F= (f, M2k+2 ) : satisfies moreover
(3.2.1) F(GU) N F(X- GU) = ~.
For such F it follows from (2.6), [I0] that
(3.2.2) F(X) is subanalytic in ~2k+2
because of F(X) = F(X N B(I, 0)) and because the closure of graph
FIxNB(I, 0) is bounded and subanalytic.
Theorem 3.3, Let (G~ M I) be a Subanalytic proper transformation
group of a locally compact subanalytic set (X~ M2) ~ Then there exist
an open neighborhood M½ of X in M 2 and a G-invariant subanalytic
map M : X~2k+l with respect to subanalytic structures (inclusion~ M~)
and (identity, ~2k+l) such that ~(X) is closed and subanalytic in
]R 2k+l and that the induced map ~ : X/G~ ~(X) is a homeomorphism r
where k = dim X.
Proof. For each point x of X let U x be an open neighborhood
of x in M 2 such U x A X is contained in a neighborhood of x in X !
which satisfies the requirements in Lemma 3.1 and Remark 3.2, Let M 2
be the union of all U x. By properly embedding Mi in a Euclidean
space, we can assume Mi =~n and we give always X a subanalytic
300
structure (inclusion, ~n).
The case where X= G(K N X) for some compact set K in ~n As K
is covered by a finite number (say s) of Ux, there exists a G-invari-
ant subanalytic map ~ : X~2s(k+l) by Lemma 3.1 and Remark 3.2 such
that the induced map ~ : X/G~ ~(X) is a homeomorphism. Here we use
(3.2.1) for the existence of ~-i and we see that ~(X) is subanalytic i
in ~2s(k+l) for the same reason as in (3.2.2), because we can choose
K subanalytic, e.g. B(s, 0) for some large E, so that ~(X)
= ~(K nx) . we note also that ~(x) is closed in ~2s(k+l) by the com-
pactness of K n X. Let (K, ~) be a subanalytic triangulation of
~2s(k+l) compatible with ~(X) (see Lemma 2.3, [7]), K' the family of
o 6 K whose interior is mapped by < into ~(X) and ~ : IK, I ~2k+l
be a PL embedding. Then ~= ~ o -I o 7 : X~2k+l is what we want.
The case where there is no compact set K in ~n such that X
= G(K N X) : Let 8 be a G-invariant subanalytic function on X such
that for any compact set H in ~ there exists a compact K in ~n
with 8-1(H) = G(K N X)
(e.g. 8(x) = inf{[gx I : g6 G}),
and let e be a subanalytic function on ~ such that for each integer
i
S 1
~= ~0
on [2i, 2i + I]
on [2i- 2/3, 2i- 1/3].
For each i consider the G-invariant subspace
-i X =0
l ([2i - i/3, 2i + 4/3])
of X. By the property of 8, (Xi, G) corresponds to the first case.
Hence there exists a G-invariant subanalytic map ~i : Xi~2k+l such
that Mi : Xi/G ~ Mi(Xi) is a homeomorphism. Define ~ : X~2k+2 by
(x) = S (~ o O(x)<~ i(x) , O(x))
I(0, 0(x))
for x 6 X. 3-
for x ~ iUiXi
Then % is G-invariant and subanalytic, ~I (y8-I((2i_i/3, 2i+4/3)))/G 1
is a h o m e o m o r p h i s m o n t o t h e i m a g e , and f o r a n y i n t e g e r s j ;~ j '
dist(%(0-1([j+I/3, j+2/3])), %(0-I([j'+I/3. j'+2/3])))> 0.
301
In the same way we obtain a G-invariant subanalytic map ¢' : X~2k+2
such that ~'I (U8-i((2i_4/3, 2i+I/3)))/G is a homeomorphism onto the 1
image. Hence 9 = (%, %') : X~4k+4 is G-invariant subanalytic map
whose induced map ~ : X/G~ ~(X) is a homeomorphism. Recalling the pro-
perty of 8, we have a closed neighborhood U of x and a compact set
K in ~n such that ~(K N X) = ~(X) D U for any point x of ~4k+4.
From this it follows that ~(X) is closed and subanalytic in ~4k+4,
since we can choose a subanalytic K. Moreover we can diminish 4k + 4
to 2k + 1 in the same way as the first case. Therefore the theorem is
proved.
Corollary 3.4. Let (G, M I) be a subanalytic proper transformation
group of a subanalytic set (X, M2). Assume X is locally compact,
Then X/G admits a unique subanalytic structure such that q : X~ X/G
is subanalytic.
Proof. Trivial by Theorem 3,3.
Corollary 3.5. Let (G, M I) and (X, M 2) be as above and give X/G
the above subanalytic structure. Then there exists a subanalytic tri-
angulation of X/G compatible with the orbit type stratification and
uniquely in the following sense. If there are two subanalytic triangu-
lations (K, ~) and (K', T'), we have subanalytic triangulation isotopies
(K, T t) and (K', Ti) of X/G such that T O = T, ~ = T' and
(~{)-I o T1 : IK I ~ IK, I is a PL map (see [7] for the definition of sub-
analytic triangulation isotopy).
Proof. Follows immediately from Lemma 2.4 in [6], Corollary 3.4
and the next fact. Let {X.} be the decomposition of X by orbit types. l
Then Lemma 2.3 tells us that {q(Xi)} is a locally finite family of sub-
analytic subsets of X/G.
References
[i] C. G. Gibson et al. Topological stability of smooth mappings,
Lecture Notes in Math., Springer, Berlin and New York~ 552 (1976).
[2] H. Hironaka, Subanalytic set, in Number theory, algebraic geome-
try and commutative algebra, in honor of Y. Akizuki, Kinokuniya,
Tokyo (1973), 453-493.
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[3] , Triangulations of algebraic sets, Proc. Symp. in
Pure Math., Amer. Math. Soc., 29 (1975), 165-185.
[4] , Stratification and flatness, in Real and complex
singularities, Oslo 1976, edited by Holm, Sijthoff & Noordhoff,
Alphen aan den Rijn (1977), 199-265.
[5] S. Illman, Smooth equivariant triangulations of G-manifold for
G a finite group, Math. Ann., 233 (1978), 199-220.
[6] J. L. Koszul, Lectures on groups of transformations, Tata Inst.,
Bombay (1965).
[7] T. Matumoto-M. Shiota, Unique triangulation of the orbit space
of a differentiable transformation group and its application r
(to appear in Advanced Studies in Pure Math. 9)
[8] D. Montgomery-L. Zippin, Topological transformation groups r
Wiley (Interscience), New York (1955).
[9] R. S. Palais, On the existence of slices for actions of non-
compact Lie groups, Ann. of Math., 73 (1961), 295-323.
[i0] M. Shiota, Piecewise linearization of real analytic functions,
Publ. Math. RIMS, Kyoto Univ., 20 (1984), 727-792.
[ii] M. Shiota-M. Yokoi, Triangulations of subanalytic sets and local-
ly subanalytic manifolds, Trans. Amer. Math. Soc., 286 (1984),
727-750.
[12] A. Verona, Stratified mappings-structure and triangulability,
Lecture Notes in Math., Springer, Berlin-Heiderberg, 1102 (1984).
[13] C. T. Yang, The triangulability of the orbit space of a differ-
entiable transformation group, Bull. Amer. Math. Soc., 69 (1963),
405-408.
A remark on duality and the Segal conjecture
by J. P. May
The Segal conjecture, in its nonequivariant form, provides a spectacular
example of the failure of duality for infinite complexes. The purpose of this note
is to point out that the Segal conjecture, in its equivariant form, implies the
validity of duality for certain infinite G-complexes in theories, such as
equivariant K-theory, which enjoy the same kind of invarianee property that
cohomotopy enjoys.
To establish context, we give a quick review of duality theory. For based
spaces X, Y, and Z, there is an evident natural map
~: F(X,Y) A Z ) F(X,Y Z).
Here F(X,Y) is the function space of based maps X + Y and v is specified by
v(f^z)(x) = f(x)^z. Any up-to-date construction of the stable category comes
equipped with an analogous function spectrum functor F and an analogous natural
map ~ defined for spectra X, Y, and Z. If either X or Z is a finite CW-
spectrum, then ~ is an equivalence. The dual of X is DX = F(X,S), where S
denotes the sphere spectrum. Replacing Z by the representing spectrum k of some
theory of interest, we obtain ~: DX^k + F(X,k). On passage to ~q, this gives
~.: kq(DX) + k-q(x), and ~. is an isomorphism if X is finite. Classical
Spanler-Whitehead duality amounts to an identification of the homotopy type of
DZ~X when X is a polyhedron embedded in a sphere, where Z ~ denotes the
suspension spectrum functor. This outline applies equally well equivariantly, with
spectra replaced by G-spectra for a compact Lie group G. We need only remark that
a map of G-spectra is an equivalence if and only it induces an isomorphism on
passage to ~q(?) = [G/H+ Asq,?] G for all integers q and all closed subgroups
H of G (where the + denotes addition of a disjoint basepoint) and that homology
and cohomology are specified by
k (Xl : and :
for any G-spectra X and k G. See [6] for details on all of this.
304
We restrict our discussion of the Segal conjecture to finite p-groups for a
fixed prime p, and we agree once and for all to complete all spectra at p
without change of notation. See [4] for a good discussion of completions of
spectra. Completions of G-spectra work the same way (and have properties analogous
to completions of G-spaces [7]). The nonequivariant formulation of the Segal
conjecture [1,5,8] asserts that a certain map
m: V:E°~BWH+ > DBG+
is an equivalence, where B denotes the classifying space functor and the wedge
runs over the conjugacy classes of subgroups H of G. Since both the mod p
homology of DBG+ and the mod p cohomology of BG+ are concentrated in non-
negative degrees, we see that the duality map v,: H,(DBG+) + H-*(BG+) cannot
possibly be an isomorphism. It is not much harder to see that the corresponding
duality map in p-adic K-theory also fails to be an isomorphism.
As explained in [5], the map a above is obtained by passage to G-fixed point
spectra from the map of G-spectra
B: S ~ F(sO,s) > F(EG+,S)
induced by the projection EG + pt, where EG is a free contractible G-space. The
equivariant form of the Segal conjecture asserts that 8 is an equivalence. More
generally, the analogous map with EG+ replaced by its smash product with any based
finite G-CW complex X is an equivalence. The crux of our observation is just
the following naturality diagram, where k G is any G-spectrum.
DX^kG--SAI ;D(EG+^X)^k G
i F(X,kG ) 8 ; F(EG+ AX, k G)
The left map ~ is an equivalence since X is finite. The top map 6, hence
also ~^i, is an equivalence by the Segal conjecture. If ~ carries G-maps which
are nonequivariant homotopy equivalences to isomorphisms, then 6 on the bottom is
an equivalence (as we see by replacing X with G/H+^X for all H C G) and we can
conclude that v on the right is an equivalence. In particular, duality holds in
d -theory for the infinite G-complex EG+~X; that is,
v,: k~(D(EG+ ^X)) --~ k~q(EG+ ~ X)
305
is an isomorphism. Of course, equivariant K-theory has the specified invariance
property by the Atiyah-Segal completion theorem [3]. Equivariant cohomotopy with
coefficients in any equivariant classifying space also has this property [5,8,9]°
In the examples just mentioned, k G and its underlying non-equivariant
spectrum k (which represents ordinary K-theory or ordinary cohomotopy with
coefficients in the relevant nonequivariant classifying space) are sufficiently
nicely related that, for any free G-CW spectrum X,
k~(X) ~ k*(X/G) and k~(X) ~ k.(X/G).
(See [6,II].) With X replaced by EG+~X for a finite G-CW complex X, this
may appear to be suspiciously close to a contradiction to the failure of duality in
non-equivariant K-theory cited above. The point is that the dual of a free finite
G-CW spectrum is equivalent to a free finite G-CW spectrum [2,8.4; 5,III.2.12],
but the dual of a free infinite G-CW spectrum need not be equivalent to a free G-
CW spectrum, and in fact Z~EG+^X provides a counterexample.
Bibliography
i. J.
2. J,
3. M.
4. A.
5- L.
6. L.
F. Adams. Grame Segal's Burnside ring conjecture. Bull. Amer. Math. Soc. 6(1982), 201-210.
F. Adams. Prerequisites (on equivariant theory) for Carlsson's lecture. Springer Lecture Notes in Mathematics Vol. 1051, 1986, 483-532.
F. Atiyah and G. B. Segal. Equivariant K-theory and completion. J. Diff. Geometry 3(1969), 1-18.
K. Bousfield. The localization of spectra with respect to homology. Topology 18(1979), 257-281.
G. Lewis, J. P. May, and J. E. McClure. Classifying G-spaces and the Segal conjecture. Canadian Math. Soc. Conf. Proc. Vol. 2, Part 2, 1982, 165-179.
G. Lewis, J. P. May, and Mark Steinberger (with contributions by J. E. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics. To appear.
7. J. P. May. Equivariant completion. Bull. London Math. Soc. 14(1982), 231-237.
8. J. P. May. The completion conjecture in equivariant cohomolog<f. Springer Lecture Notes in Mathematics Vol. 1051, 1984, 620-637.
9. J. P. May. A further generalization of the Segal conjecture. To appear.
On the hounded and thin h-cobordism theorem parameterized by R k
by
Erik Kjaer Pedersen
0. Introduction
In this paper we consider bounded and thin h-cobordisms
parameterized by ~k. We obtain results similar to those obtained by
Quinn [QI,Q2] and Chapman [C] , but in a much more restricted
situation. The point of the exercise is to give a self contained
proof, based on the algebra developed in [PI,P2] in the important
special case, where the parameter space ~s euclidean space. We also
get a nice explanation as to why the thin and bounded h-cobordism
theorems have the same obstruction groups. Unlike the general version
being developed by D.R,Anderson and H.J.Munkholm [A-M], we only
consider h-cobordisms with constant (uniformly bounded) fundamental
group.
In case of the bounded h-cobordism theorem+ it is however clear,
that the discussion we carry through will generalize to more general
metric spaces than ~k namely to proper metric spaces (every ball
compact). We mention this because in this case, we have computed KI of
some of the relevant categories i. e. the obstruction groups, in joint
work with C. Weibel.
This work was completed while the author spent a most enjoyable
year at the Sonderforschungsbereich f~r Geometrie und Analysis at
G~ttingen University. The author wants to thank for support and
hospitality. The author also wants to acknowledge useful conversations
with D.R.Anderson and H.J.Munkholm.
I. Definitions. Statements of results.
D e f i n i t i o n I . I A m ~ n l y o l d ff p a r a m e t e r l z e d bV ~k c o n s i s t s o f a
m a n i Y o l d W t o ~ e t h e r w i t h a p r o p e r map W ~ : ~ k w h i c h i s o n t o .
307
We use the map p to give a pseudo metric on W by which we measure
size. This is distilled in the following definition.
D e f i n i t i o n 1 . 2 G i v e n K ~ W, ff p a r a m e t e r i z e d by ~k by ~ : W
d e f i n e t h e s i z e o f ~ , S ( K ) t o be
S ( E ) i n f ( r l 3 y E ~k: p ( K ) ~ B ( y , r / 2 ) )
w h e r e B ( y , r / 2 ) i s t h e c l o s e d b a l l i n ~k w i t h r a d i u s r / 2 .
, ~k, w e
S(K) is thus the diameter of the smallest ball containing p(K).
We shall now introduce uniformly bounded and locally constant
fundamental groups. Given t 6 ~+~ we shall define t-bounded
fundamental group as follows:
D e f i n i t i o n 1 . 3 T h e f u n d a m e n t a l g r o u p o f W i s t - b o u n d e d i f t h e
f o l l o w i n g 2 c o n d i t i o n s h o l d :
i ) F o r e v e r y ( x , y ) 6 W and f o r e v e r y h o m o t o p y c l a s s oJ p a t h s f r o m x t o
y , t h e r e i s a r e p r e s e n t a t i v e a : ( I , 0 , 1 ) ~ , ( W , x , y ) so t h a t
$(a(1)) < t + $ ( ( x , y ) ) .
2 ) For e v e r y n u l l h o m o t o p i c map a: S 1 , W, t h e r e i s a n u l l h o m o t o p y
A: D 2 , W so t h a t $ ( A ( D 2 ) ) < S(a(sl)) + t .
In other words, generators and relations of KI(W) are everywhere
representable by something universally bounded. We say the fundamental
group is bounded, if for some t it is t-bounded, and we say it is
locally constant if it is t-bounded for all t.
We shall now consider h-cobordisms in the category of manifolds
parameterized by ~k.
D e f i n i t i o n 1 . 4 The t r i p l e ( W , ~ o W , ~ I W ) p a r a m e t e r i z e d by ~ k i s a
b o u n d e d h - c o b o r d l s m ( b o u n d e d by t ) i f t h e b o u n d a r y o f W, 8W, i s t h e
d i s j o i n t u n i o n o f ~0 W and ~1 W, and t h e r e a r e d e f o r m a t i o n s
D i : W×t , W o f W i n OiW, so t h a t S ( D i ( w × I ) ) < t f o r a l l w E W . .
Given an h-cobordism of this kind, it is natural to ask for a
product structure:
308
Definition 1.5 A bounded product structure (bounded by t ) on
(W,~OW,D1W) i s a h o m e o m o r p h i s m
h : ( ~ o W X I , ~ o Y X O , ~ o W X l ) , (W,DoW,DIW)
w h i c h i s t h e i d e n t i t y on D0W a n d s a t i s Y i e s t h a t S ( H ( w X I ) ) < t Yor a ~ l
w 6 DoW.
We are now able to formulate the thin and bounded h-cobordism
theorems.
Bounded h-cobordism theorem. Let (W,OoW,~IW) be e bounded h-cobordism
o f dimension a t ~east 6, parameterized by IRk w ~ t h bounded ]undamenta~
g r o u p ~ . T h e n t h e r e i S an i n u a r i a n t i n g _ k + l ( E n ) , w h i c h u a n i s h e s i Y
and o n l y i f P] a d m i t s a b o u n d e d p r o d u c t s t r u c t u r e . A l l s u c h i n v a r i a n t s
a r e r e a l i z e d b y b o u n d e d h - c o b o r d i s m s .
This bounded h-cobordism theorem is a formal consequence of the
thin b-cobordism theorem, which we proceed to formulate. However it is
much easier to prove the bounded h-cobordism theorem. In the above
statement, one could replace ~k by any other metric space X, which is
proper in the sense that every ball is compact, at the price of
the obstruction group by KI(Cx(Z~)). (see section 5 for replacing
definition and discussion of this).
We now formulate the thin h-cobordism theorem:
Thin h - c o b o r d i s m t h e o r e m : T h e r e i s a ] u n c t i o n ] : N × N ", ~ s o t h a t
i f (W,DoW,~IW) i s an h - c o b o r d i s m o f d i m e n s i o n n b i g g e r t h a t 6 ,
p a r a m e t e r i z e d b v ~ k b o u n d e d b y t , w i t h f u n d a m e n t a l g r o u p b o u n d e d by
t , t h e n t h e r e i 3 a p r o d u c t s t r u c t u r e on W b o u n d e d b y f ( n , k ) . t , i f a n d
onZ~ i~ t h e o b s t r u c t i o n t o a b o u n d e d p r o d u c t s t r u c t u r e , i n E R + I ( Z ~ ) ,
v a n i s h e s .
Remark 1.6 The difference between the thin and bounded h-cobordism
theorems parameterized by ~k thus lies in the predictability of the
bound of the product structure. This of course implies that one may
let t go to 0, whereas in the bounded h-cobordism theorem, that has no
309
effect.
It iS natural to relate bounded h-cobordism theorems to classical
compact h-cobordism theorems. This is done in the following:
T h e o r e m 1 .7 Le t (M,~0M,OlM) be a c o m p a c t h - c o b o r d i s m w i t h y u n d a m e n t a ~
g r o u p rXZ k, and Se t M , T k ~ n d u c e t h e p r o j e c t i o n ~XZ k , Z k on
] u n d a m e a t a l g r o u p s . T h e n t h e p u l l b a c k o ~ e r ~k ...... , T k d e y i n e s a b o u n d e d
h - c o b o r d i s m (W,DoW,D1 W) ( t h e Z k - c o v e r i n g ) a n d t h e t o r s i o n ~ n v a r i a n t s
a r e r e l a t e d b y t h e B a s s - H e l l e r - S w a n e p i m o r p h i s m
Wh(KXZ k) ~ ~_k+ l (Z~ ) .
Remark 1.8 K k+l(Z~) means Wh(~) for k = 0, K0(Z~) for k = 1 and
K k+l(Z~) for k > i.
2:. Reviewin 8 the alBebra.
In this section, we review some of the algebra from [PI,P2]. We
also develop the algebra needed to make it possible to treat not only
the bounded h-cobordism theorem, but also the thin h-cobordism
theorem. This amounts to a discussion of the "size" of the "reason"
for the vanishing of an invariant, which is known to vanish. A reader
familiar with [PI,P2] and only interested in the bounded h-cobordism
theorem, may thus skip this section.
Given a ring R we define the category ~k(R) to be zk-graded, free,
finitely generated, based R-modules and bounded homomorphisms. That
means an object A is a collection of finitely generated, free, based
R-modules A(J), J 6 zk and a morphism #: A , B is a collection
I :A(I) : B(J) of R-module morphisms with the property that there ~j
I is a r = r(~) so that #j = 0 when HI-JH > r. Here it is convenient to
use the max norm on ~k. A morphism ~ will be called degree preseruing
or homogeneous if ~ = 0 for I different from J.
Another way of thinking of ~k(R) is to think of A as ~A(J). Then
the condition on ~ is that # : A , B is a usual R-module morphism
310
satisfying that ~(A(J)) ~ ~ B(1). I I I - J I l ~ r
The description given here differs from the one given in [PI] in
that we take based R-modules. This however does not change anything
and makes applications to geometry easier. In [PI] we proved that
KI(~k(R)) ~ K_k+I(R). The definition of KI(~k(R)) is, that as
generators we take [A,ff] where A is an object and ~ an automorphism
and as relations [A,~B] [A,~] - [A,B] and A~B A~B . The reason
it does not make a difference whether we consider based or unbased
R-modules, is that [A,aB~ -| ] = [A,B]. Thus a basis change will have no
effect on the invariant.
Given an object A of ~k+l(R) there is an obvious object A[t,t -I ]
-! of ~k+l(R[t,t ]). This object has a homogeneous automorphism ~t which
is the identity on homogeneous elements, whose last coordinate is
negative, and multiplication by t when the last coordinate is
positive. If ~ is an automorphism of A bounded by r, then the
commutator [~,Bt] is the identity on any element whose last coordinate
is numerically bigger than r, since ~ both commutes with
multiplication by t and with the identity. This means that [~,~t] only
does something interesting in a certain band. If we then restrict to
that band, and forget the last coordinate in the grading ( by taking
sum), then we get a Z k graded automorphism in ~k(R[t,t-l]). direct
This is the B~ss-Heller-Swan monomorphism
K_k(R) = KI(~k+I(R)) , Kl(~k(R[t,t-I ]) ) = K_k+l(R[t,t-I ]).
The details are given in [PI]. Here we want to use this for some
simple observations:
Let K be a fixed integral k-tuple. We may then regrade Z k by
vector addition of K. This will clearly induce a functor of ~k(R).
Lemma 2.1 The map on K_k+I(R ) induced by the regrading given by vector
addition of K is the identity.
Proof The map A ~ (regraded A) induced by the identity is bounded,
and the map on K_k+I(R ) is thus given by conjugation by this map.
This lemma is used to prove the more interesting
Lemma 2.2 Let A be an object of Ok(R) and ~ and B two automorphisms of
311
A bounded by r. Suppose there is a K 6 Z k so that ~ and ~ agree on all
A(J) with HJ-KH ~ r, i.e. on some box with sides 2r, ~ and B agree.
Then [A,~] = [A,B] in K_k+l(R).
-i Proof Using Lemma 2.1 we may assume K = 0. Now conside T = ~B We
have T = id on a box with side length 2r~ and after application of the
Bass-Heller-Swan monomorphism this is still the case. After k
applications of the B-H-S monomorphism , we thus have the identity.
The above lemma is used to show that parameterized torsion is
well defined under subdivision.
Now consider the map r : Z k , zk multiplying by r > 0. This
induces a functor r, : ~k(R) ~ ~k(R) sending A to r,A with
r,A(J) = A(rJ) and 0 otherwise, morphism induced by the identity.
Lemma 2.3 The map induced by multiplication by r > 0 is the identity
on K k+I(R ) .
Proof After k applications of the Bass-Heller-Swan monomorphism, we
clearly have the identity.
Finally we have to do the algebra needed to get the thin
h-cobordism theorem, rather than just the bounded h-cobordism theorem.
At this point we need to remind the reader as to what we mean by an
elementary automorphism ~ of A. By this we mean there is a direct sum
decomposition A = A I ~ A 2 of based submodulesp so that ~ may be given
the matrix presentation 0 . We also need to remind the reader that
there is an alternative description of K_k+I(R) as the Grothendieck
construction of zk-l-graded projections. We call a projection
Qeometr~c when it sends any basis element either to itself or to 0.
Lemma 2.4 There is a function f : ~ ~ ~ so that the following is
true:
I) If A E ~k_[(R) and p : A ~ A is a projection bounded by 1 and so
that [A,p] = 0 in K_k+I(R). Then after stabilization there is an
automorphism B bounded by f(k)or-24 so that BpB -I is geometric.
2) If A E ~k(R) and ~ : A ~ A is an automorphism bounded by r, so
that [A,=] s 0 E K_k+I(R)- Then stably ~ may be written as a product
312
Of 24 elementary automorphisms, each of which is bounded by f(k)-r.
Proof is by induction on k on the statements i) and 2) for any ring.
We will show, that if the ring is of the form R = S[t,t -1 ] and the
given automorphism (projection) only ~nvolves finitely many t-powers,
then the automorphisms produced have the same property. We shall allow
ourselves to refer freely to [PI ]. To facilitate the reading, we do
the first two steps rather than the general step. For k = 1 statement
1 disappears, so consider statement 2. The map P0 : A , A is the
identity in positive gradings and the 0-map in negative gradings. The r
map ap0 Q-I restricted to ~ A(i) is conjugate to P0 at least after i=-r
r stabilization of say A(0)~ so there is an automorphism B of ~ A(i)
i=-r
so that B~p0~-iB -I = P0 or B~P0 = P0 B~" Extending B to all of A by the
identity, we have an automorphism B bounded by 2r so that B~P0 = P0 ~"
We thus get ~ = B-I(B~) where B -I and Ba both are bounded by 2r. Since
is the identity away from the interval -r to r~ £ preserves the two
halves when we split up A say at r. Denote B~ or B -I by T. The trick
used in [PI ] is the equation
( Y ~ I ~ I ~ . . . ) = ( T ~ y - I ~ y . . . ) ( I ~ y ~ y - I ~ . . . )
each term on the right side may be written as a product of 6
elementary isomorphisms each of which is bounded by 4r, so f(1) may be
taken to be 4. If the ring R is of the form S[t,t-I |, and the
automorphism a only involves finitely many t-powers, then clearly all
the elementary automorphisms produced have that same property.
For k = 2 consider a Z-graded projection p of A as in statement
I). Then pt + (I-p) is a Z-graded automorphism of R[t,t-i | modules
involving only finitely many t-powers and bounded by r. By what we
just proved pt + (l-p) may be written as a product of 24 elementary
matrices, each only involving finitely many t-powers and each bounded 24
by 4r, i. e°, pt + (l-p) = H E.. Turning t-powers into a grading, and i=l x
conjugating the projection PO by this automorphism, delivers back the
projection p at t-degree O, the id in positive t-degrees and the O-map
-I in negative t-degrees. Considering (pt + (l-p))pO(pt + (l-p)) in a
band around t-degree 0 corresponds to stabilization. Using the trick
of lemma i. I0 in [PI ] which turns an elementary matrix into a product
Of one with support in a band around t-degree 0 and one far away, we
313
obtain ~ bounded by 24.4-r so that in a broad band (of t-degrees)
BPB-I = PO" The trick being employed is that it does not matter how
high t-powers get involved, because the grading introduced by the
t-powers will immediately be forgotten.
It is now clear how the induction proceeds) one essentially uses
the same words.
3-- Bounded simple homotopy theory parameterlzed by ~k.
In this section we elaborate a little on the results of (P2]) and
carry these results into the manifold category. First we recall
D e f i n i t i o n 3 . 1 A I i n i t e , b o u n d e d CW c o m p l e x p a r a m e t e r i z e d b y ~k
c o n s i s t s oy t h e I o l l o w i n g : A I i n ~ t e d i m e n s i o n a l CW c o m p Z e x X t o g e t h e r
w i t h a map X , ~ k w h i c h i s o n t o a n d p r o p e r , s o t h a t t h e r e i s a
t E ~+ s o t h a t t h e s i z e , S ( C ) < t y o r e a c h c e l l C.
D e f i n i t i o n 3 . 2 L e t K b e a s p a c e p a r a m e t e r l z e d b y ~ k . A s i m p l e h o m o t o p y
t y p e on g c o n s i s t s oY
1) a b o u n d e d , ] i n i t e CW c o m p l e x E p a r a m e t e r i z e d b y ~ k
2 ) a b o u n d e d h o m o t o p y e q u i v a l e n c e K ) E
Two s u c h a r e s a i d t o be e q u i v a l e n t i ] t h e i n d u c e d b o u n d e d h o m o t o p y
e q u i v a l e n c e oY Y i n i t e b o u n d e d C W - c o m p l e x e s h a s o t o r s i o n i n
K k + l ( Z r l K ) ( s e e [ P 2 ] 2 o r d e l i n t t i o n s )
T h e o r e m 3 . 3 A m a n i f o l d W p a r a m e t e r i z e d b y ~k w i t h b o u n d e d f u n d a m e n t a l
g r o u p , h a s a w e l l d e f i n e d s i m p l e h o m o t o p y t y p e g i v e n b y a
t r i a n g u l a t i o n w i t h b o u n d e d s i m p l i c e s ( i n t h e P t o r DIFF c a t e g o r i e s ) o r
b y a b o u n d e d h a n d t e b o d y s t r u c t u r e i n t h e TOP c a t e g o r y .
Proof We give the argument in the PL category. This extends to the
DIFF category by smooth triangulations. The TOP category requires the
usual modifications in the argument. Given t E ~+, we choose a
triangulation with simpliees of size less than t. This is a bounded
finite CW complex, hence the identity defines a simple homotopy type
314
on W. We have to compare this to another arbitrary triangulation with
simplices of size less that t' The two triangulations have a common
subdivision, so as in compact topology it suffices to show that the
identity is a homotopy equivalence with trivial torsion, when thought
of as a map from W with some triangulation K to a subdivision K. We
pick out one of the coordinates in ~k say the last, and call this x.
Rather than comparing the triangulation and its subdivision directly,
we introduce an intermediate subdivision cell complex K' which is a
subdivision of K and has K as a subdivision. Furthermore if a simplex
of K' has barycenter with x-value bigger than 3t the simplex is also a
simplex of K, whereas if the x-value is smaller than -3t, the simplex
is also a simplex of K. In other words the cell decomposition agrees
with K for large positive values of x and with K for large negative
values of x. It is not possible to have K' be a triangulation, because
we have to subdivide a face of a simplex without subdividing the
simplex itself. This however is no problem when we only want a cell
complex. We now compare K and K' . At the level of chain complexes the
identity induces a map sending a generator corresponding to a cell to
the sum of the simplices it is being divided into~ and the homotopy
inverse sends one of these back to the generator and the rest to 0.
For large positive x-values there is no subdivision, so the map is the
identity. By Lemma 2.2, it suffices to know the map on a big chunk, so
we are done. Comparing K' and K is treated similarly, but now using
the fact that the cell decompositions agree for large negative
x-values.
Note that the reason we can not simply refer to the usual compact
proof is, that we may not subdivide equally much everywhere, so there
may be more than finitely many steps in the subdivision procedure.
We are now ready to define the obstruction and prove the theorems.
4. Proof of thin and bounded h-cobordism theorem parameterized by ~k.
Consider an h-cobordism (W,~oW,~IW) parameterized by ~k and
bounded by t, with fundamental group ~ bounded by t. For the purposes
315
of the bounded h-cobordism, these can be taken to be the same number
by taking the bigger, while for the thin h-cobordlsm theorem it is
part of the assumption. By assumption the inclusion ~0 W ~ W is a
bounded homotopy equivalence. Since ~0 W as well as W have well defined
simple homotopy types by theorem 3.3, this homotopy equivalence has a
well defined torsion in K_k+I(ZK). If (W,~0W,~IW) is boundedly
equivalent to (~0WXI,~0W,~0WXI) then W is obtained from ~0 W by
attaching no handles, and it is clear that this torsion must vanish.
Assuming the invariant vanishes~ we give W a filtration as
~0WXlV0-handlesVl-handlesV...Vn+l-handlesVOlWXI in such a way that the
size o f each handle is bounded b y t, and the size of each wXl in ~0WXI
or ~IWXI is bounded by t. The aim now is to get rid of all the handles
in between, without changing the size of the product structure lines
too badly. The procedure is the usual handlebody theory, with
attention paid to size, and the arguments are very similar to those
applied by Quinn in [Q|], but of course with different algebra.
Cancelling 0-handles is done in standard fashion, but one has to
worry that one does not get too long a sequence of 0 and 1 handles~
letting the size get out of control. We have a t-bounded deformation
retraction of W to ~0 w . The restriction to 0-handles defines a map
(0-handles)XI : W
defining a path from the core of each 0-handle to ~0 W . Using (very
small) general position, one may assume this path runs in the
l-skeleton of W, relative to ~0 W , so from the core of every 0-handle,
there is a path through cores of 1 and O-handles to ~0 W , bounded by t
when measured in ~k. If this path has any loop, we may simply discard
the loop. That does not increase the size. Also if the path from one
0-handle is a part of a longer path from another 0-handle, we may
forget the shorter path. In the end we would llke to have an embedding
(cores of some 0-handles)XI : W
which goes through all 0-handles and retaining the control of size.
This is done by subdividing every 0-handle with more than I path going
through into so many 0 and l-handles, that they have been made
disjoint. We now have a disjoint embedding of paths from ~0 W going
through 0 and I handles and with size being bounded by t. Cancelling
these 0-handles accordingly will change the boundedness of the collar
316
structure on the boundary to a controlled multiple of t.
The cancelling of l-handles is now done in standard fashion by
introducing 2 and 3 handles, and using the 2 handles to cancel the
l-handles. Having done this from both ends of the handlebody~ we have
a handlebody without any 0,l,n and n+l handles, and the product
structure on the collars of the boundary is bounded by a constant
times t. All 2 and n-I handles must be attached to the boundary by
homotopically trivial maps (otherwise they would change the
fundamental group)~ so we now have the same fundamental group ~ at all
levels of the decomposition.
The cellular ~ chain complex of (W,~0W) may he tho'ught of as a
chain complex in ~k(~) by associating to each cell an integral
lattice point in ~k near the points in ~k over which the cell sits. As
elaborated in [P2], this cellular chain complex
0 , c n _ , ~ , Cn_ 2 ~ . . . . ~ , C 3 ~ , C 2 , 0
will be contractible in ~k(~ff), with a contraction s whose bound is
directly related to the bound of the deformation of W in ~0 W. We now
proceed to cancel handles following the scheme indicated by the
algebra: we introduce cancelling 3 and 4 handles corresponding to all
the 2-handles, and sitting over the points in ~k where the 2-handles
sit, to obtain a chain complex in ~k(~) which in low dimensions is
C2eC 4 ~ C2eC 3 ~OjD~ C2 - 0.
At the level of 3-handles we now perform handle additions, so that to
each handle x in C 2 we add s(x) in C 3 . Since s is bounded, this will
increase the cell size by a controllable amount. Since in dim 2 we
have ~s = I, the chain complex, after having performed this handle
addition, now has the form
C2~C 4 j C2~C 3 (I.*~ C2 .......... , 0.
We are now in a situation to cancel the 3-handles we introduced
against the 2-handles, since we have obtained algebraic intersection
I, and after some small Whitney isotopies we will have geometric
intersection ] and can cancel handles. After the cancellation the
chain complex has the form C2~C 4 ~ C 3 b 0, and is of course still
contractible in ~k(Z~).
Continuing this procedure, we get into a two-lndex situation
317
0 ~ Cr+l ~s C r ....... ~ 0
and the collars have bounded product structures, bounded by some
predictable (even computable as a function of dim(W)) constant times
t. The invariant in K_k+I(Z~) = Klek(Z~) is given by the torsion of
this chain comp]ex, which is exactly the isomorphism 8. Of course ~ is
not an automorphism but an isomorphism. The point is that if ~ is of
the type sending a generator to a generator, then we may cancel
handles. It is however easy to see that, at least stably ( see e.g.
[Pl]) C r and Cr+ 1 are isomorphic by an isomorphism sending generators
to generators~ hence composing ~ w2th such an isomorphism, we obtain
an automorphism. At this point there is a choice involved, but for
k > ] the torsion of an automorphism sending generator to generator is
0. This is Lemma 1.5 of [PI]. When k=l this is not true, and what
Quinn calls a flux phenomen occurs. The invariant thus only becomes
well defined after dividing out by automorphisms that send generators
to generators, which amounts to saying the Invarlant lives in reduced
K-groups. At this point one might mention that the choices involved in
finding representing cells of the Z~ modules have no effect since an
automorphism multiplying generators by elements of ~ will have 0
torsion~ because it is homogeneous.
Since we have assumed the invariant is 0 in K_k+l(Z~)~ the
automorphism can be written as a product of elementary automorphisms
after stabilization. After stabilizing geometrically by introducing
cancelling handles, we may then change ~ to cancel one of these
elementary automorphisms at a time, at the expense of letting the
handles grow bigger. At this point, as in all handle addition
arguments, we of course use the boundedness of the fundamental group,
to be able to judge how much bigger the handles get. In the end ~ will
be equal to the isomorphism from Cr+ 1 to C r chosen, that sends
generators to generators. We now cancel handles and are done.
To prove the thin h-cobordism theorem, we have to worry about how
many handle additions we perform, but by lemma 2.4 this is controlled.
To sum up the difference between the thin and the bounded h-cobordism
theorem, to do the thin version one needs to do the following: First
multiply the reference map in ~k by I/K so the h-cobordism will be
bounded by I. Here we use lemma 2.3 to show this does not change the
obstruction. To get into the 2 index situation~ there is no difference
318
between the two proofs. In the 2-index situation, we need lemma 2.4 to
see that we can control how many handle additions we need to perform,
and how far away the handles that have to be added can sit.
Proof of Theorem 1.7
Consider a compact h-cobordism (M,~0M,~IM) with fundamental group
~XZ k. The torsion of this h-cobordism will be represented by the
torsion of the based chain complex of the universal cover of (M,~oM)
as Z[ffXZ k] modules. This is exactly the same chain complex as that of
the zk-covering, but now the Z k has been turned into a zk-grading. On
the other hand, the description of the Bass-Heller-Swan epimorphlsm
given in [PI] is exactly that.
Realizability o_~f obstructions
Given a manifold ~0 W ~ ~k with unifomly bounded fundamental
group ~ and an element ~ E K k+l(Z~), we wish to construct an
h-cobordlsm (W,OoW,~IW) with obstruction oo However ~ is represented
by a zk-graded bounded automorphism ~ : C ..... ~ C, where C is some
object of ~k(Z~). We start out with ~0WXI. Then we attach infinitely
many trivial handles of the same dimension r corresponding to the
generators of C, and each placed at a point which in ~k is near by the
integral lattice point of the generator in C. As in the standard
realizability theorem we now attach r+l-handles by maps given by
above. It is easy to extend the reference map to ~k and we get a
manifold (W,O0W,~IW) with the chain complex 0 ~ C aj C ~ 0 and
will thus have torsion given by the class of ~ which is o. To prove it
is a bounded h-cobordism, we do however need to invoke the Whitehead
theorem type results of Anderson and Munkholm [A-M].
5. Parameterlzln~ by other metric spaces.
In the proof of the bounded h-cobordism theorem (not the thin
h-cobordism theorem) we have nowhere used that the metric space we
parameterized by is ~k Any other metric space X will do, as long as X
satisfies that every ball in X is compact (A proper metric space in
319
the sense of [A-M]). The groups in which the obstructions will then
take values will then be KI(~X(Z~) where Cx(R) is an additive category
described as based, finitely generated, free R-modules parameterized
by X and bounded homomorphisms. That means an object A is a set of
based, finitely generated, free R-modules A(x), one for each x E X
with the property, t h a t for any ball B C X, A(x) = 0 for all but
finitely many x E B. A morphism ~ : A J B is a set of R-module
morphisms #x : A(x) * B(y), so that there exists k = k(@) with the Y
property that ~x = 0 for d(x,y) > k. The study of this sort of Y
category is the object of forthcoming joint work with C.Weibel, in
which we obtain results about the K-theory of such categories. In the
case of X = ~k we have preferred to have the modules sitting at the
integral lattice points, but this is not an important difference. In
general when the fundamental group is uniformally hounded with respect
to the metric space X, the proof of the bounded h-cobordism theorem
will go through word for word. The obstructions will be elements of
KI(~X(Z~) , where stands for the reduction by automorphlsms sending
generators to generators. The case where the fundamental group is not
necessarily being assumed to be uniformally bounded is presently being
studied by D.R.Anderson and H.J.Munkholm.
320
References
[A-M]
[ e l
[PII
[P21
IOl]
I021
D.R.Anderson and H.J.Munkholm: The simple homotopy theory of
controlled spaces, an announcement. Odense university preprint
series no7,1984.
Chapman: Controlled Simple Homotopy Theory and
Applications~Springer Lecture notes 1009.
E.K.Pedersen: On the K i-functors + Journ. of Algebra,90, (1984)
461-475.
E.K.Pedersen: K_i-invariants of chain complexes. Proceedings of
Leningrad topological conference, Springer Lecture Notes in
Mathematics 1060, 174-186.
F.Quinn: Ends of maps I~ Ann. of Math. Ii0 (1979) 275-331.
F.Quinn: Ends of Maps II, Invent Math. 68 (1982) 353-424.
Sonderforschungsbereich Geometrie und Analysis
Matematisches Institut der Georg August Universitat
Bunsenstra~e 3-5
D-3400 G~ttingen BRD
and
Matematisk Institut
Odense Universltet
DK-5230 Odense M
Danmark
ALGEBRAIC AND GEOMETRIC SPLITTINGS OF THE K- AND L-GROUPS
OF POLYNOMIAL EXTENSIONS
Andrew Ranicki
Introduction
This paper is an account of assorted results concerning the
algebraic and geometric splittings of the Whitehead group of a
polynomial extension as a direct sum
Wh (~x~) = Wh (~) ~0 (~ [~ ] ) ~N~(~ [~ ] ) ~Ni'-~(~ [~] )
and the analogous splittings of the Wall surgery obstruction groups
{ L~(~×~) s h = L. (~)@L._I(~)
h h p L. (~x~) = L. (z)@L._l (~)
Such a splitting of Wh(~x~) was first obtained by Bass, Heller and
Swan [ 2 ]. Shaneson [29] obtained such a splitting of Pedersen and Ranicki [18]
[ L~(~x~) geometrically~ Novikov [17] and Ranicki [20] obtained such L~{~×~)
L-theory splittings algebraically.
The main object of this paper is to point out that the geometric
L-theory splittings of [29] and [18] are not in fact the same as the
algebraic L-theory splittings of [17] and [20] (contrary to the claims
put forward in [18], [20], [23] and [24] that they coincided), and to
express the difference between them in terms of algebra. The splitting
s ), ,,>L~ (~×~) /LS(~xm) > rh [L.(~ * > ~*-i (~)
maps I h ' m are the same in algebra
and geometry, the split injections being the ones induced functorially
from the split injection of groups ~:~ ~ ~×~ . However, the splitting
s {L h >L~ fL~(~x~) ~L.(~) ._i(~)> (~x~) are in general
maps L~(~x~) ~>L.(g)h ' L~_l(~)> >L~(~x~)
di*fferent in algebra and geometry. In particular, the geometric split
split surjections are not the algebraic split surjections induced
functorially from the split surjection of groups c : ~ x ~ >~ [
This may be seen by consfdering the composite eB' of the geometric
split injection
322
I ~. : L h (~)> ~ LS(~x~) ; n-i n h s ~×x S 1 ) a.((f,b):M, >X)~ ,~ 0.((f,b) xl:M × S 1
~, : L p (~1> eLh(~x~) n-i n '
a~((f,b) :M " ~X)1 ~o~((f.b)xl:M xS 1 )X x S I)
(denoted B' to distinguish from the algebraic split injection B of [20])
and the algebraic split surjection
I e : Ls(~x~)--------~L (~) ; n
S s a.((g,c) ~N >Y)~ ~[~]®~[~x~]O.(g,c)
6 : Lh(~x~) ~Lh(~) ; n n
h h a.((g,c) :N ~Y), ~[~]®~[~x~]a.(g,c)
i f in i t e NOW ~B' need not be zero: if X is a (n-1)-dimensional
(finitely dominated
~ simple n-dimensional geometric Poincar6 complex then X × S 1 is a ~homotopy finite
geometric Poincare complex, the boundary of the - 1 finite (finitely dominated
(n+l)-dimensional geometric Poincar6 pair (X x D2,X × SI), but not in
tsimple pair (W,X × S I) with general the boundary of a (homotopy finite
~I(W) = nl(X) , so that E and B' do not belong to the same direct sum
system.
The geometrically significant splittings of L.(~×~) obtained
in ~6 are compatible with the geometrically significant variant in ~3
of the splitting of Wh(~x~)due to Bass, Heller and Swan [2 ]. In both
K- and L-theozy the algebraic and geometric splitting maps differ in
2-torsion only, there being no difference if wh{~) = O.
I am grateful to Hans Munkholm for our collaboration on [16].
It is the considerations of the appendix of [16] which led to the
discovery that the algebraic and geometric L-theory splittings are not
the same.
This is a revised version of a paper first written in 1982 at the
Institute for Advanced Study, Princeton. I should like to thank the
Institute and the National Science Foundation for their support in that
year. Thanks also to the G~ttingen SFB for a visit in June 1985.
Detailed proofs of the results announced here will be found in
Ranicki [26], J27], [28].
323
§i. Absolute K-theory invariants
The definitions of the Wall finiteness obstruction [X] £ ~O(~[~i(×)])
of a finitely dominated CW complex X and the Whitehead torsion
T(f) eWh(~l(X) ) of a homotopy equivalence f:X ~Y of finite CW
complexes are too well known to bear repeating here. The reduced
algebraic K-groups ~0' Wh are not as well-behaved with respect to
products as the absolute K-groups Ko,K I. Accordingly it is necessary
to deal with absolute versions of the invariants. The projective class
of a finitely dominated CW complex X
[X] = (k(X),[X]) C KO(~[~I(X) ] = KO(~)¢~O(~[~I(X)])
is well-known, with ×(X) C KO(~) = ~ the Euler characteristic.
It is harder to come by an absolute torsion invariant.
Let A be an associative r~ng wlth 1 such that the rank of f.g.
free A-modules is well-defined, e.g. a group ring A = ~[~]. An A-module
chain complex C is finite if it is a bounded positive complex of based
f.g. free A-modules d d
C : ... ) O ~C n ~Cn_ 1 ~ ... ~ C 1 ~ C O ~ O ~ ....
in which case the Euler characteristic of C is defined in the usual
manner by n
x(C) : ~ {-)rrankA(C r) C ~ + r=0
A finite A-module chain complex C is round if
x ( C ) = 0 ~
The absolute torsion of a chain equivalence f:C >D of round finite
A-module chain complexes is defined in Ranicki [25] to be an element
7(f) ~ KI(A)
which is a chain homotopy invariant of ~ such that
i) if f is an isomorphism 7(f) = [ (-)r~(f:Cr---~Dr) . r=0
ii) ~(gf) = 7(f) + 7(g) for f:C----eD, g:D ~E.
iii) The reduction of T(f) in KI(A) = KI(A)/{~(-I:A ~A)} is the
usual reduced torsion invariant of f, defined for a chain equivalence
f:C >D of finite A-module chain complexes to be the reduction of the
torsion T(C(f)) ~ KI(A) of the algebraic mapping cone C(f). Thus for
A = ~[~] the reduction of y(f) @ KI(~[~]) in the Whitehead group
Wh(~) = KI(~[~])/ {n} is the usual Whitehead torsion of f.
fv) T (f) = • (D) - ~ (C) ~ KI(A) for contractible finite C,D.
v) In general T(f) M T(C(f)) ~ KI(A ) , and T{f@f') ~ ~(f) + T(f')
{although the differences are at most ~(-I:A ---+A) ~ KI(A))-
324
vi) The absolute torsion T(f) ~ KI(A) of a self chain equivalence
f:C ~ D = C agrees with the absolute torsion invariant T(f) ~ KI(A)
defined by Gersten [i0] for a self chain equivalence f : C ~ C of a
finitely dominated A-module chain complex C.
I round
A finite structure on an A-module chain complex C is an
r o u n d equivalence class of pairs (F,$) with F a finite A-module chain
complex and ¢:F ~C a chain equivalence, subject to the equivalence
relation
(F,$) - (F',@') if ~({'-I$:F---~C ---~F') = O ~ [ KI(A)
£~I(A)
In the topoloqical applications A = ~[~] , and KI(A) is replaced
by Wh(~).
Proposition i.i A finitely dominated A-module chain complex C admits a
~round (absolute finite structure if and only if it has ]reduced projective
mo(i) class [C] = O ~ , in which case the set of such structures on C
Ko(A)
I KI(A)- carries an affine _ structure.
KI(A) []
Let X be a (connected) CW complex with universal cover X and
fundamental group ~I(X) = ~. The cellular chain complex C(X) is
as usual, with C(X) r = Hr (x(r) ,X([-I)) (r ~ O) defined the free
[~]-module generated by the r-cells of X. The cell structure of X
determines for each C(X) a ~[~]-module base up to the multiplication r
of each element by ±g (g ~ ~ ) . Thus for a finite CW complex X the
cellular ~[~]-module chain complex C(X) has a canonical finite structuse.
A CW complex X is round finite if it is finite, X(X) = 0 8 ~ ,
and there is given a choice of actual base for each C(~) r (r ~ O) in
the class of bases determined by the cell structure of X.
fabsolute The ~ torsion of a homotopy equivalence f;X-----~¥ of
L Whitehead
I round finite CW complexes is defined by
T(f) = r(f:C(X) ~C(Y)) e I KI(~[~I(X)])
{
tWh(~i(X))
325
I round A finite structure on a CW complex X is an equivalence
r o u n d claSS of pairs (F,~', with F a finite CW complex and ~:F ~ X a
homotopy equivalence, subject to the equivalence relation
(F,¢) -- (F' ¢') if ~(~-i - IKI(~[~I (X)] , ' :F----~X---~F') = O
L Wh(~l(X) ) •
The finiteness obstruction theory of Wall [34] gives:
round Proposition 1.2 The finite structures on a finitely dominated
CW complex X are in a natural one-one correspondence with the t i round
L
finite structures on the ZZ[~l(X)]-module chain complex C(X).
[]
The mapping torus of a self map f:X- ;X is defined as usual by
T(f) = X × [O,l]/{(x,O)= (f(x),l) Ix6 X}
Proposition 1.3 (Ranicki [26]) The mapping torus T(f) of a self map
f:X-----~X of a finitely dominated CW complex X has a canonical round
finite structure.
[]
The circle S 1 = [0,i]/(0= i) has universal cover SI= ]R and
fundamental group ~I (SI) = 2Z. Let z ~ ~i (SI) = 2Z denote the generator
such that
z : ]R >]R ; x ~-------->x+l .
The canonical round finite structure on the circle
S 1 = eO~2 e I = T(id.:{pt.} ~ {pt.}) is represented by the bases
~r eC(~ 1 ) = 2Z[z,z -I] (r : 0,i) with [
(ZI) = - c(gl) = ,z-i ~i -O -O d = l-z : C 1 2Z[z,z i] ~ O 2Z[z ] ; ~e - ze ,
= -i O i corresponding to the lifts ~O {O}, e = [O,I] C]R of e ,e .
In particular, Proposition 1.3 applies to the product
X × S 1 = T(id.:x ~X) , in which case the canonical round finite
structure is a refinement of the finite structure defined geometrically
by Mather [14] and Ferry [ 8 ] , using the homotopy equivalent finite
CW complex T(fg:Y------~Y) for any domination of X
(Y , f : X >Y , g : Y >X , h : gf -- I ; X-- ~X )
by a finite CW complex Y.
326
Given a ring morphism e:A ~ B let
~! : (A-modules) ~ (B-modules) ; M , ; B®AM
be the functor inducing morphisms in the algebraic K-groups
~ : Ki(A) ~ Ki(B) (i =0,i) ,
which we shall usually abbreviate to ~. Given a ring automorphism
a:A -~A let KI(A,~) be the relative K-group in the exact sequence
l-e j 8 l-e
Kl(A) ~ KI(A) ~ KI(A,~) ~ Ko(A) >Ko(A) ,
as originally defined by Siebenmann [33] in connection with the
KI(A [z,z-l]) recalled in 53 below. By definition splitting theorem for
KI(A,e) is the exotic group of pairs (P,f) with P a f.g. projective
A-module and f C HomA(e!P,P) an isomorphism. The mixed invariant of a
finitely dominated A-module chain complex C and a chain equivalence
f:~,C----+C was defined in Ranicki [26] to be an element
[C,f] ~ Kl(A,e)
such that e([C,f]) = [C] ~ Ko(A), and such that [C,f] = O £ Kl(A,e) if
and only if C admits a round finite structure (F,~:F )C) with
r(~-if(~,~) : e!F ~ ~!C ~ C > F) = 0 6 Kl(A)
The inva[iant is a mixture of projective class and torsion, and
indeed for e = 1 : A ,A
[C,f] = (T(f) , [C]) e KI(A,I) = KI(A)@Ko(A)
The absolute torsion invariant defined by Gersten [10] for a
self homotopy equivalence f:X ~X of a finitely dominated CW complex X
inducing f, = 1 : ~l(X) = n ~.
T(f) = T(f:C(X) )C(g)) e Kl(Z~[n])
was 9eneralized in Ranicki [26]: the mixed invariant of a self homotopy
equivalence f:X -~X of a finitely dominated CW complex X inducing any
automorphism f, = 0~ : ~l(X) = n , ~ is defined by
[X,f] = [C(X),f:e!C(X) ,C(X)] e KI(2Z[~],~)
This has image 9([X,f]) = [X] ~ KO(2Z[n]) , and is such that [x,f] = 0
if and only if X admits a round finite structure (F,~:F---wX) such that
T(~-lf~ : F ~ X ------~X ~F) = O £ KI(YZ.[~])
If X admits a round finite structure (F,~) then [X,f] = j(T(~-if#))
is the image of T(#-If~:F ~F) eKl(2Z[~]) -
327
§2. Products in K-theory
For any rings A,B and automorphism 8:B ~ B there is defined a
product of algebraic K-groups
: Ko(A)~KI(B,B) ~ KI(A®B,I~B) ;
[P]®[Q,f:B!Q--~Q], ~ [P®Q,I®f: (i~8} ! (P~Q) = P®~!Q ~P®Q] ,
which in the case B = 1 is made up of the products
® : Ko(A)~Ko(B)-------~Ko(A®B) ; [P]®[Q]I ~[P®Q]
® : Ko(A)®KI(B)------+ KI(A®B) ; [P]®~(f:Q--+Q)! ~T(I~f:P®Q---~P~Q).
The product of a finitely dominated A-module chain complex C and a
finitely dominated B-module chain complex D is a finitely dominated
A®B-module chain complex C~D with projective class
[C®D] = [C]®[D] C Ko(A®B) ,
and if f:B!D ~D is a chain equivalence then the product chain
equivalence l®f : C®8!D ~C®D has mixed invariant
[C®D,I~f] = [C]®[D,f] C KI(A®B,I~B)
The following product formula is an immediate consequence.
Proposition 2.1 Let X,F be finitely dominated CW complexes with
~l(X) = 7, ~l(F) = 0, and let f : F. ~ F be a self homotopy equivalence
inducing the automorphism f, = B : 0----+p. The mixed invariant of
the product self homotopy equivalence 1 × f : X × F .... ~ X × F is given by
[X × r,1 × f] = [X]®[F,f] e KI{~[~×p] ,I~B) ,
identifying ~[~×p] = ~[w]~[D].
[]
In the case ~ = 1 : D---~p the result of Proposition 2.1 is made up
of the product formula of Gersten [ 9] and Siebenmann [30] for the
projective class
[x ×r] = [X]®[F] e KO(~[~×~])
and the product formula of Gersten [i0] for torsion
T(I × f:X × F ~X × F) = [X]®T(f:F---*F) e Kl(~[~xp]) .
If also X is finite the product formula T(I x f) = [X]~T(f) is an
absolute version of the special case e = 1 : X ..... ~X' = X , f, =i of the
formula of Kwun and Szczarba [12] for the whitehead torsion of the
product e × f : X × F ............. ~ X' × F' of homotopy equivalences e : X }X',
f : F > F' of finite CW complexes
T(e × f) = x(X)®T(f) + ~(e)~x(F) e Wh(~xp)
328
The product A~B-module chain complex C®D of a finitely dominated
A-module chain complex C and a round finite B-module chain complex D
was shown in Ranicki [26] to have a canonical round finite structure,
with
~(e®f:C®D ~ C'~D') = [C]~(f:D ........ ~D') ~ KI(A®B)
for any chain equivalences e:C ~C',f:D ~ D' of such complexes. The
following product structure theorem of [26] was an immediate
consequence.
Proposition 2.2 The product X × F of a finitely dominated CW complex X
and a round finite CW complex F has a canonical round finite structure,
with
~(e × f:X × F ~X' × F') [X]®T(f:F ~F') e KI(~ [~3 (X)× ~I(F)])
for any homotopy equivalences e:X >X',f:F ~ F' of such complexes.
[)
The canonical round finite structure on X x S 1 = T(id.:X >X)
given by Proposition 1.3 coincides with the canonical round finite
structure given by Proposition 2.2.
The product
KO(~[~])®KI(~[p]) > Kl(~[~xp])
has a reduced version
~O(~[~])®{±p} ~ wh(~xp) ;
[P]®T(±g:~[p] ~ [~]) , ~T(I®~g:P[p] ~ P[P])
with {±p} = {±l}×p ab = ke[(Kl(~[p]) ~Wh(p)) . ~e shall make much
use of this reduced version with p = ~ , for which {±~} = KI(~[~]) .
329
53. The White.head group of a polynomial extension
Im the first instance we recall some of the details of the direct
sum decomposition
Wh(~x2Z) = Wh(~)SKo(2Z[~])SNiI(~[~])SNiI(~Z[~])
obtained by Bass, Heller and Swan [ 2 ] and Bass [ i ,XII] for any group
We shall call this the algebraically significant splitting of Wh(~xgZ.)..
The relevant isomorphism
a n d i t s i n v e r s e
8K 1 (~)@Ko "-'-" = (c B A+ A_) : Wh (2Z[~])eNiI(ZZ[~])@Nil(~[~]) ~ Wh(~×2Z)
I s u r j e c t i o n
involve the split (injection of group rings
ajzJ, , ~ a, I c : 7Z[~x~] = Z~[~] [z,z-I]-------~Z[~] ; J=-~ L 3
: 7z[~]~ }2Z[~] [z,z -I] ; a, ,'a (a,ajeZ~[~]} .
The split in~ection B:Ko(~Z[~]b ;Wh(~×ZZ) is the evaluation of the
product Ko(2Z[~])®KI(~[Z~]) ..... ~Wh(~×2Z) (the reduction of
Ko(~Z[~])~KI(~[2Z]) ~KI(~Z[~×~Z])) on the element 7 (z) £ KI(ZZ[ZZ])
= -®~ (z) : Ko(Z~[~]b ~Wh(~×ZZ) ;
[P], ~? (z:P[z,z -I] .... ~p[z,z-l])
If P = ira(p) is the image of the projection p = p2 : 2Z[~]r ~ZZ[~]r
then
B([P]) = ~ (pz+l-p:Z~[~xZZ] r ~ 2Z[~×ZZ] r) ~ Wh(~x2Z)
By definition, Nil(Tz[~]) is the exotic K-group of pairs (F,~) with F
a f.Q. free ~Z[~]-modu]e and ~£ HOm2Z[~ ] (F,F)a nilpotent endom~)rphism.
The split injections A-+, A_ are defined by
A-± : N'~(2Z[~])~ ..... ,Wh(~x~Z) ;
(F,~)~ *T (l+z-+iv:F[z,z -I] ~.F[z,z-I]) .
330
The precise definitions of the split surjections B,A± need not detain
us here, especially as they are the same for the algebraically and
geometrically significant direct sum decompositions of Wh(~).
The exact sequence
c ~ Wh ( ~ ) .... ) Wh (~X~)
B
I::l was interpreted geometrically Dy Farrell and Hsiang [5 ], [ 7 ] :
if X is a finite n-dimensional geom~tric Poin~ar~_ complex with Zl(X) =
and f : M ~X x S 1 is a homotopy eq0ivalence with H n+l a compact
(n+l)-dimensional manifold then the Whitehead torsion T(f)eWh(~×ZZ)
is such that
T(f) e im([:Wh(~)~------~Wh(~x2Z))
(i:l 0 = ker( : Wh(~x2Z) ,~ ZZ[~])eN'~I(ZZ[~I)SNi~'~I(TZ[n]))
if (and for n >45 only if) f is homotoplc to a map transverse regular
at X x {pt.}C X x S 1 with the restriction
g = fl : Nn = f-l(x × {pt.}) > X
also a homotopy equivalence. Thus T(f)ecoker(~:wh(~)~ ,Wh(#x2Z))
is the codimension 1 splitting obstruction of f along X × {pt.} cX x S I.
For a finitely presented group n every element of Wh(~×ZZ) is the
Whitehead torsion T(f) for a homotopy equivalence of pairs
(f,~f) : (M,~M) >(X,dX) × S 1 with (M,~M) a compact (n+l)-dimensional
manifold with boundary, and (X,~X) a finite n-dimensional geometric
Poincar@ pair with ~l(X)=7, for some n>z 5. In this case
T(f) e coker(~:Wh(~)~ ~Wh(~×2Z)) is the relative codimension 1
splitting obstruction.
The geometrically significant splitting
Wh(~×Z~) = Wh(~)eKo(ZZ[#])@~I(2Z[7])SN~(2Z[#])
is defined by the isomorphism
g, 8~ = : Wh (~×2Z) ~ Wh (~) @Ko(ZZ [~ ] ) @NIl (2Z [n ] ) 8N~ (2Z [z ] )
with inverse
331
8~ -I = (¢ B' A+ A_) : Wh(~)@Ko(~Z[~])@Ni"~(ZZ[~])@Ni'-'~(2Z[~]] ,Wh(~×2~) ,
where
B' = -~ (-z) : ~O(~[~])>
£' = ¢(I-B'B) : Wh(~x~)
x(f:P[z,z -I]
~Wh(~×~) ; [P]~ ~(-z:P[z,z-l]----~P[z,z-l])
(= ~(-pz+l-p) if P = im(p= p2)) ,
>7 Wh(~) ;
~P[z,z-l]) )%(£f:P---~P) + T(-I:Q--~Q)
-i with f an automorphism of the f.g. projective ~[~×~]-module P[z,z ]
induced from a f.g. projective ~[~]-module P, and Q a f.g, projective
[~]-module such that B(T(f)) = [Q] ~ KO(~[~]) -
Ferry [ 8 ] defined a geometric injection for any finitely
presented group
B" : ~0(~[~]b , Wh(~x~) ; -i ~-i ¢ l Ix-i 1 ¢
IX], ~(f= (I×-i)¢ : Y >XxS ~- ~ ~ X×S -------~Y) ,
with [X] C KO(~[~]) the Wall finiteness obstruction of a finitely
dominated CW complex X with HI(X) = ~ and x(f) ¢ Wh(~x~) the Whitehead
torsion of the homotopy equivalence f = ~-l(ix-l)¢:Y >Y defined
using the map -I:S 1 > S 1 reflecting the circle in a diameter and
any homotopy equivalence ¢:Y ~XxS 1 from a finite CW complex Y in the
finite structure on X×S 1 given by the mapping torus construction of
Mather [14].
Proposition 3.1 The geometrically significant injection B' agrees
with the geometric injection B"
B' = ~" : ~0(~[~1)~- ~Wh(~×~)
Proof: By Proposition 2.2
B"([X]) = [X]~(-I:S 1 } S I) £ Wh(~x~) ,
with 7(-I:S 1 ~ S I) ~ Kl(~[z,z-l]) the absolute torsion. Now -I:SI---+S 1
induces the non-trivial automorphism z, > z -I of ~I(S I) = <z>,
and the induced chain equivalence of based f.g. free ~[z,z-l]-module
chain complexes is given by
l_z -I (-l),C(~ I) : ~[z,z -I] ~ ~Z[z,z -I]
(-i) 1 -z
c(~l) : 2Z[z,z-l] l-z ) 2Z[z,z-l] ,
s o t h a t
332
T(-I:S 1 ~S I) = T(-z:ZZ[z,z -I] ~ZZ[z,z-I]) ~ KI(ZZ[z,z-I]) .
Thus
B" = -~(-z) = B' : ~0(~[~]), ~Wh(~×Z~) .
[]
Ferry [ 8 ] characterized im(B') c_Wh(zxZZ) as the subgroup of the
elements T@Wh(nxZZ)such that (pn) " (~) = 7 for some n >. 2, with I
(pn) " : Wh(~×~) ~ Wh(~×Tz) the transfer map associated to the n-fold
covering of the circle by itself
Pn : S1 sl n
See Ranicki [27] for an explicit algebraic verification that
im(B')c Wh(~x2Z) is the subgroup of transfer invariant elements.
The algebraically significant decomposition of Wh(~xZg) also has a
certain measure of geometric significance, in that it is related to the
Bott periodicity theorem in topological K-theory - cf. Bass [ i ,XIV].
More recently, Munkholm [15] identified the infinite structure set
-~(X x ]R 2) = ker(E:Ko(2Z[~xTz]) ...... +Ko(2Z[~])) (X compact, ~I(X) = ~) of
Siebenmann [32] with the lower algebraic K-groups derived from the
algebraically significant splitting of Wh(~xZZ) by Bass [ 1 ,XII] -
to be precise ,~(X × IR 2) = (K_I(~NKo(gNKo) (ZZ[~I) •
Both the injections B,B':Ko(2Z[#])} >Wh(nxZg) can be realized
geometrically for a finitely presented group n, as follows. Given a
2 (2Z[~]r,ZZ[I:] r) be a f.g. projective 2Z[~]-module P let p = p C Homzz[~ ]
projection such that P = im(p) . Let K be a finite CW complex such that
~l(K) = ~. For any integer N>/ 2 define the finite CW complexes
X (Kx S 1 vbIS N) 'J z+l ([jeN+l' =
r P -P r
X' = (K x S 1 V ~/S N) </_pz+l_p(~jeN+i)r ' r
such that the inclusions define homotopy equivalences
K×S 1 )X , K×S 1 ~X'
Proposition 3;2 The injections B,B' are realized geometrically by
: Ko(2Z[~] ~ > Wh(~ × ZZ) ; [P], -~(-)NT(K × S I ..... ~X)
B' : K0(Z~[n] > ~Wh(,~ × ~Z) ; [P] ~ , (-)NT(K × S ± ) X')
[]
Nevertheless, B' Is more geometrically sianificant than B.
333
(Following Siebenmann [31] define a band to be a finite CW complex X
equipped with a map p:X ~S 1 such that the pullback infinite cyclic
cover ~ = p* (JR) of X is finitely dominated. For a connected band X the +
infinite complex X has two ends E , 6 which are contained in finitely
dominated subcomplexes X+ X c X such that X+ , n X- is finite and
.X+uX- = X. The finiteness obstructions are such that
Ix] = [x+] ÷ [x-] e ~0(2z[~]) (~ = ~l(X)) . --+
For a manifold band X the finiteness obstructions [X-] ~Ko(Z~[~]) are
images of the end obstructions [c +-] 8Ko(2Z[~I(C-+)]) of Siebenmann [30].
For any finitely presented group ~ the surjection B:Wh(~x2Z) ~Ko(~Z[~])
is realized geometrically by
B(~(f:X ~Y)) = [Y+]- [X+] ~ Ko(ZZ[~]) ,
with z(f) ~Wh(~x2Z) the Whitehead torsion of a homotopy equivalence of
bands f:X )Y with ~l(X) = ~×2Z , ~I(X) = ~. For the bands used in
Proposition 3.2
[~] =-[X-] = IX'+] =-[X' ] = (_)Nip] ,
[(K x SI) +] = [(K x SI) -] = (K x ]R +] = [K] = O ~ Ko(2Z[z]) ).
We shall now express the difference between the algebraically and
geometrically significant splittings of Wh(~x2Z) using the generator
T(-I:2Z. ~2Z) 8 KI(2Z) (= 2Z2) and the product map
0~ = -®T(-I) : Ko(2Z[~]) ~ Wh(~) ; [p]L ~ 1(-I:P ~ P) .
If P = im(p) for a projection p p2 = : F >F of a f.g. free
2Z[~]-module F then the automorphism I-2p:F ~F is such that
~([m]) = 7(l-2p:F ~F) ~ Wh(~) .
Proposition 3.3 The algebraically and geometrically significant
surjections ~,~':wh(~×2Z) ~>Wh(~) differ by
injections B,B':Ko(2Z[~])~ ) Wh(~x2Z)
B w
g ~w Ko(2Z[~] ) ~ ~Wh (~)~ -~ ~Wh (~xZZ)
334
In particular, the difference between the algebraic and geometric
splittings is 2-torsion only, since 2~ = O.
It is tempting to identify the geometrically significant surjection
[':Wh(~×~) ~Wh(~) with the surjection induced functorially by the
split surjection of rings defined by z, ~-i
]z ] q : ~[~×~] = mrs] [Z,Z -1] ~ ~[~] ; ~ a , ~ ~ a.(-l) j ., j=_~ j=_~ 3
and indeed
e'I = DI : im((£ B) :Wh(~)@Ko(~[~])~ ~Wh(~x~))
= im((e B'):Wh(~)@Ko(~[~])) ~Wh(~×~))- : Wh(~)
However, in general
~'[ / nl : im((A+ & ) :Nil(~[~])~Nil(~[~])~ ...........
SO that e' ~ ~ : Wh(n×~) ~Wh(~) .
> Wh(~xTZ) )
>Wh(~)
For an automorphism <~:~---+~ of a group ~ Farrell and Hsiang [6 ]
and Siebenmann [33] expressed the Whitehead group of the e-twisted
extension ~xe2Z of ~ by ~ = <z> (gz = zS(g) e nxcLZ~ for g~ ~) as a
natural direct sum
Wh(~x ZZ) = Wh(r~,e)@Ni'-~l(2Z[~],e)@Ni'-~(Z~[~],a -I)
with Wh(~,e) the relative group in the exact sequence
i-~ j ~ i-~ Wh(~) > Wh(~) ---~Wh(~,~) >Ro{2Z[~]) .... ) KO{TZ[~])
(the reduced version of the group KI(ZZ[~],e) discussed at the end of ~i)
and N]~(Z~[~],@ +I) the exotic K-group of pairs (F,v) with F a f.g. free
2Z[~]-module and v ~ Hom~{~] ((@±i) !F,F) nilpotent. Given a f.g. projective
~.[~]-module P and an isomorphism f 8 Hom2z[~ ] (eBp,P) there is defined a
mixed invariant [P,f] ~Wh(~,@) with ~([P,f]) = [P] ~ Ko(TZ.[~]).
As in the untwisted case e = i there are defined an algebraically
significant splitting of Wh(~x ZK) , with inverse isomorphisms
~+ \~_/
Wh(~x ?Z)~. ~ Wh(~,~)~Nil(~[~] ,~)~)NiI(Z~[~], ,
(B ~+ ~_)
and a geometrically significant splitting of Wh(~xa?z) with inverse
isomorphisms
335
with
I B' ) Wh(~× 2Z) - -~ Wh(~,{~)(BNi"-~I(2Z[~] ,(~J~Ni"~l(~[~] ,a -1)
(B' ~+ ~_)
: Wh(~,~)} %Wh(~x ~Z) ; [P,f]~-'---~(zf:P [z,z -!] '~Pa[z,z-I])
B' : Wh(~,~)y-------~Wh(~x ]~) ; [P,f]~ ~.T(-zf:P [z,z -l] ~P [z,z-l])
+I A÷ : NiI(2Z[~],~- )) ~Wh(~x ZZ) ; - *I -i -I]
(P,v), >~(l+z- ~:P [Z,Z ] -*P~[z,z ) ,
identifying Zg[~xeZZ] = Zg[~]~[z,z-l]. The automorphism
: Wh(r~,e) . .~ Wh(~,~) ; [P,f]~- ..... } [P,-f]
is such that ~2 = 1 and
B' = B9 : Wh(~,a)> ............ )Wh(~×~ZZ)
B' = ~B : Wh(~×a~Z) .... bWh(~,~)
In the untwisted case s = 1 ~x 2Z is ]ust the product ~ ×.2Z, and there
is defined an isomorphism
Wh (~)S~o(2Z [~] ) ..... ~ Wh(.~,l) ;
(~(f:p ~p),[Q])~ ~[P,f] - [P,l] + [Q,I]
with respect to which
( i w) CKO (Zz )¢Ko (ZZ = : Wh(~) [~]) ...... ,Wh(~ l~]) 0 1
The algebraically (resp. geometrically) significant splitting of
Wh(~×aZZ) for ~ = 1 corresponds under this isomorphism to the
algebraically (resp. geometrically) significant splitting of Wh(~x~Z)
defined previously.
A self homotopy equivalence f:X ~X of a finitely dominated CW
complex X has a mixed invariant
IX,f] e Wh(~,e)
with ~ = f, : ~ = Zl(X) } ~, such that B([X,f]) = IX] ~ Ko(2Z'~n]),
a reduction of the mixed invariant [X,f] ~ K I(ZZ[~] ,e) described at
the end of §I. Let f-l:x '~X be a homotopy inverse, with homotopy
e:f-lf_ - I:X------~X. The mapping tori of f and f-i are related by the
homotopy equivalence
336
U : T(f -I) ,T(f) ; (x,t)" ~ (e(x,t),l-t)
inducing the isomorphism of fundamental groups
-i) = = 2Z ; U, : ~l(T(f ) n × (-i 2Z ~ ~l(T(f)) ~×
-i g (e ~) ~-----+g , z ~ ~ z
The torsion of U with respect to the canonical round finite structures
given by Proposition 1.3 is
T(U) = T(-z~:C(X)a[Z,Z-I] ----+C(X)a[z,z-I]) e KI(2Z[~]~[Z,Z-I]) ,
so that:
Proposition 3.4 The geometrically defined split injection is given
geometrically by
B' : Wh(~,~)~ > Wh(~xeZZ) ; [X,f]~ > ~(U:T(f -I) ~T(f))
[]
Proposition 3.3 is just the untwisted case e= 1 of Proposition 3.4,
with f = 1 : X } X and
U = i x -i : T(I:X--~X) = X × S I- ~ T(1) = X x S I ,
-i : S 1 = ]R/ZZ ~ S 1 ; t ~ > l-t .
The exact sequence
i-~ -6 Wh(~) -~ Wh(~) >Wh(~x ~Z)
KO "~ ~ -i > (ZZ [~] )~Nil (~. In] ,~)eNil (~z [~] ,e )
(i-~ O O)
..... ~ ~o(~[~]) >~O(ZZ[~×~]) (-£ = Bj = B'j , ~B = ~B')
The obstruction theory of Farrell [ 4 ] and Siebenmann [33] for
fibering manifolds over S 1 can be used to give the injection
B':Ko(~[n]), ~ )Wh(nx~) a further degree of geometric significance,
as follows.
has a geometric interpretation in terms of codimension 1 splitting
obstructions for homotopy equivalences f:M n ~X with ~I(X) = ~xa~
(Farrell and Hsiang [ 5 ], [ 7 ]) , as in the untwisted case e = i.
3 3 7
Let p:X .... ,X be the covering projection of a regular infinite
cyclic cover of a connected space X, with X connected also. Let
~:X ~X be a generating covering translation, inducing the
automorphism ~, = s : ~I(X) = ~ ~ ~. The map
T(~) ) X ; (x,t)~ )p(x)
is a homotopy equivalence, inducing an isomorphism of fundamental
groups ~l(T(~)) = ~x ~ ~l(X) . If X is a finite CW complex and
is finitely dominated the canonical (round) finite structure on T(~)
given by Proposition 1.3 can be used to define the fibering obstruction
¢(X) = 7(T(~) ) X) e Wh(~xe~)
This is the invaria~t described (but not defined) by Siebenmann [31].
If X is a compact n-manifold with the finite structure determined by a
hand!ebody decomposition then ¢(X) = 0 if (and for n > 6 only if) X
fibres over S I in a manner compatible with p, by the theory of
Farrell [4 ] and Siebenmann [33].
Given a finitely dominated CW complex X with Zl(X) : ~ let
Y ~X × S 1 be a homotopy equivalence from a finite CW complex Y
in the canonical finite structure. Embed Y CS N (N large) with closed
regular neighbourhood an N-dimensional manifold with boundary (Z,~Z) ,
and let (Z,~Z) be the infinite cyclic cover of (Z,~Z) classified by
the projection
HI(Z ) = ~I(~Z) = ~I(X x S I) = ~x~ - ~ ~ .
Thicken up the self homotopy equivalence transposing the sl-factors
1 x T : X x S ] S 1 ~ X x S 1 S I × ; (x,s,t) I > (x,t,s)
to a self bomotopy equivalence of a pair
(f,~f) : (Z,~Z) ~ S 1 >(Z,%Z) x S 1
inducing on the fundamental group the automorphism
v x ~ x ~ ~ ~ x ~ x ~ ; (x~s,t)~ ~ (x,t,s)
transposing the Z-factors. Thus (f,~f) lifts to a ~-equivariant
homotopy equivalence
(f,~f) : (Z,~Z) x S 1 > (Z,~Z) x ~ .
In particular, this shows that ZZ is a finite CW complex with a
finitely dominated infinite cyclic cover ~Z.
Proposition 3.5 The geometrically significant injection is such that
B' : ~O(~[~])~ >Wh(~x~) ; {xj~ ~¢(~Z)
I]
338
C (S l) i-* !
IS 1]
c (~l)
so that S 1 has torsion
§4. Absolute L-theory invariants
The duality involutions on the algebraic K-groups of a ring A
with involution--:A ~A;a, ,a are defined as usual by
* : Ko(A) >Ko(A) ; [P]! ~[P*] , P* = HomA(P,A)
* : KI(A) >KI(A) ; T(f:P ..... ~P)l >T(f*:P* ~P*) ,
with reduced versions for Ko(A), KI(A). We shall only be concerned
with group rings A = ZZ[~I and the involution g = w(g)g -I (ge~)
determined by a group morphism w : ~ ~ZZ 2 = {+_i} , so that there
is also defined a duality involution *:Wh(~) ,Wh(~) .
projective class ifinitely dominated The of a n-dimensional
LWhitehead torsion finite
geometric Poincare complex X with ~l(X) =
I [x] = [c(~)] e Ko(~[~])
T(X) = T(C(X) n-*- >C(X)) e wh(~)
satisfies the usual duality formula
[x]* = (-)nix] e KO(~[~])
T(X)* = (-)nT(x) e Wh(n)
The torsion of a round finite n-dimensional geometric Poincar$ complex X
T(X) = T(C(X) n-* -~C(~) ) e K I(ZZ[n])
is such that
T(X)* = (-)nT(x) e KI(ZZ[~]) .
The Poincar4 duality chain equivalence for the universal cover
~I = JR of the circle S 1 is given by
l_z -I : Z~[z,z -I] ,~ ZZ[z,z -I]
1 - z
2Z[z,z -I ] 2Z [Z,Z i] ,
T(S I) = T([S I] n-:c(~l) I-* ~c(~l))
= T(-z:2Z[z,z -I] -~2Z[z,z-I])
e KI(ZZ[z,z-I] )
This is the special case f = 1 : X = {pt.} ){pt.} of the following
formula, which is the Poincar6 complex version of Propositions 1.3,3.4.
339
Proposition 4.1 Let f:X )X be a self homotopy equivalence of a
finitely dominated n-dimensional geometric Poincare complex X inducing
the automorphism f, = e : ~i (X) = ~ ~ ~ and the ~[~]-module chain
equivalence f : a,C(X)- ~C(~). The mapping torus T(f) is an
(n+l)-dimensional geometric Poincar6 complex with canonical round
finite structure, with torsion
~(T(f)) = T(-zf:C(X) e[z,z -I] ~C(X) a[z,z-l]) e Kl(~[~]a [z,z-l]) .
[]
For f = 1 : X ~ X the formula of Proposition 4.1 gives
~(X ×S I) = ~(-z:C{~)[z,z -I] ~C(~)[z,z-l])
= [X]®7(S I) = B' ([X]) ~ KI(~[~] [z,z-l])
with IX] £ KO(~[~+]) the projective class and B' the absolute version
B' : Ko(~[z]); ~ ml(~[~] [z,z-l]) ;
[P]I } T(-z:P[z,z -I] >p[z,z-l])
(also a ~plit injection) of B':Ko(~[~])~ ~ Wh(~x~).
For a finitely presented group ~ every element x £ KO(~[~]) is
the finiteness obstruction x = IX] of a finitely dominated CW complex
X with ~I(X) = z, by the realization theorem of Wall [34]. We need
the version for Poincare complexes:
Proposition 4.~ (Pedersen and Ranicki [18]) For a finitely presented
group ~ every element x ~ KO(~ [~]) is the finiteness obstruction
x = [X] for a finitely dominated geometric Poincar6 pair (X,~X)
with ~I(X) = ~.
[]
The method of {18] used the obstruction theory of Siebenmann [30].
The construction of Proposition 3.5 gives a more direct method, since
(Z,~Z) is a finitely dominated (N-l)-dimensional geometric Poincar6
pair with prescribed [Z] ~ Ko(~[z]) . (Moreover, if the evident map
of pairs (e,}e) :(Z,~Z) ~S 1 is made transverse regular at pt. 8 S 1
the inclusion
(M,OM) = (e,$e)
lifts to a normal map
(f,b) : (M,3M)
-I ({pt.]) } (Z,~Z)
~(Z,3Z)
from a compact (N-l)-dimensional manifold with boundary. This gives a
more direct proof of the realization theorem of [18] for the projective
surgery groups L~(~), except pdssibly in the low dimensions).
340
By the relative version of Proposition 4.1 the product of a
finitely dominated n-dimensional geometric Poincare pair (X,~X) and
the circle S 1 is an (n+l)-dimensional geometric Poincare pair
(X,~X) x S I = (X x sl,~x x S I)
with canonical round finite structure, and torsion
T(XxS 1,3xxs I) = <{-z:C(~)[z,z -I]
= [x]®~(s l) = ~'([x]
Combined with Proposition 4.2 this gives:
Proposition 4.3 The geometrically significant
>, C(~)[z,z-1])
e KI{2Z[~] [z,z-l])
njection is such that
: ~O(~[~])~ >Wh(~x~) ; [x]~ ><(X x Sl,~x × S l) ,
for any finitely dominated geometric Poincare pair (X,~X) with ~I(X) = T
[]
In §5 this will be seen to be a special case of the product formula
for the torsion of (finitely dominated) x (round finite) Poincare
complexes.
Given a *-invariant subgroup S~ KO (~[~]) (resp. S g Wh(~)) let
I:!i: finite)
(n ~O) be the cobordism group of finitely dominated (resp.
Isymmetric n-dimensional C quadratic Poincare complexes over ~[~]
(C,$e Qn(C)) ~ with finiteness obstruction [C] e S ~Ko(~[~]) (resp.
(C,~ Qn(C))
{i(C,$) = ~(~o:C n-* ~c) Whitehead torsion £ Sg Wh(~)) .
(C,~) T((I+T)~o:C n-* > C)
A finitely dominated (resp. finite) n-dimensional geometric Poincare
complex X with Zl(X) = ~ and [X] ~ S (resp. T(X} ~ S) has a symmetric
signatu[e invariant
n o~(X) = {C(X),~) e LS(~)
with ¢O [X] m : C(X) n-* . . . . ~C(X) , and a normal map (f,b) :M ~X
of such complexes has a ~uadratic @ignature invariant
L~(~) c,(f,b)
such that (l+T)O S ,(f,b) = O~(M) - o~(X) . See Ranicki [22],[29] for the
details. In the extreme cases S = {O},Ko(~[~]) (resp. {O},Wh(~))
the notation is abbreviated in the usual fashion
341
LKO (2Z Ln i n n [7]) (~] = (7) (~) = Ls(~) p L{ O}-_CWh (7)
LKo (]Z ' [7]) (~)= Lp(~ ) L {O}c~Wh(~) (7) = LS(~) n n n
n n n L{O}C.~o(ZZ[~] ) (7) = Lwh(z ) (z) = Lh(7)
L{ O}C KO (Z~ [11] ) ~"h L h (11) n (7) = L~ (7) (7) = n
s In particular, the simple quadratic L-groups L.(~) are the original
surgery obstruction groups of Wall [35], with aS(f,b) the surgery
obstruction.
The torsion of a round finite n-di.~ensional
complex over 2Z[~] I (C,¢)
L (C,~)
{ ~(C,¢ ~(C,~
and is such that
I symmetric
quadratic
is defined by
= ~(¢o:C n-* ~C) ~ K I(2Z[7])
= T((I+T)~o:Cn-* ~ C) ~- K](2Z[~])
I(C,¢ * = (-)n~(c,¢)
(C,¢)* (-)n~(c,~)
Poincar$
e KI(ZZ [~] )
define the round! symmetric
quadratic Given a *-invariant subgroup S_CKI(2~[~]
L-group) (n>/O) to be the cobordism group of round finite L rS (~)
t n symmetric i (C,~)
n-dimensional Poincar6 complexes over 2Z[7] with quadratic (C,~)
ii (C,,) torsion £ S _CKI(2Z[z]) . See Hambleton, Ranicki and Taylor [ii] (C,9)
for an exposition of round L-theory. We shall only be concerned with
the round symmetric L-groups LrS here, adopting the terminology
n L n L n, (7) = L n (7) Lrs(Z) = (z) rn rKI(2Z[~]) ' r{±7} "
The Rothenberg exact sequence for the quadratic L-groups
... ,LSn(~.) , Lh(~) ,Hn(Tz2;Wh(~)) >LS_I(~) , ...
has versions for the symmetric and round symmetric L-groups which fit
together in a commutative braid of ex_~ct sequences
342
"r
L n /rh\ L n (n) L n rs h (~)
Ln(~) s
I
Hn(~z2;Wh(~)) ~ Ln-l(~)
with the maps 7 (resp. X)
Y Ln-l(~)
rs
(zz2 ;Ko(zz) / LrTIC
defined by the Whitehead torsion (resp. Euler
characteristic). In the case Wh(~) = O the L-groups are
I L*(~) r
L*(~) abbreviated to The L-groups of the trivial group ~ = {i} are
given by
Ln({l}) = ~2
with isomorphisms
L4k({I})
L4k+l({l})
h4k({1}) r
L4k+l({l}) r
Ln({l}) 2Z2OZZ 2
if n - (mod 4) 0
0
~ZZ ; (C,#)~ ~ signature(C,¢)
]l ~ 2 ; ( C , ~ ) ~ ~deRham(C,¢) = x½(C;TZ 2 ) + x½(C;Q)
>~ ; (C,¢) ~ ~ ½(signature(C,¢))
• '2Z2$2Z 2 ; (C,¢)~ ~(x½(C;ZZ2),x½(C;~))
(See Ill] for details. The F-coefficient semicharacteristic of a
(2i+l)-dimensional Z~-module chain complex C is defined by
i X½(C;F) = [ (-) rrankFHr (C) e 2Z ,
r=O
for any field F).
The torsion of a round finite n-dimensional geometric Poincare
complex X with ~l(X) = ~ is the torsion of the associated round finite
n-dimensional symmetric Poincare complex over 2Z[~] (C(X),4p)
~(X) = T(C(X),~) = ~(~O = [x] n- : C(X) n-* ~C(X)) ~ KI(ZZ[~])
If SC_KI(ZZ[~]) is a *-invariant subgroup such that T (X)G S the round
343
symmetric signature of X is defined by
n 0rs(X) = (C(~),~) e LrS(~)
n In the case S = KI(2Z[~]) (resp. {+~]_ ~ ) this is denoted O*rh(X) ~ Lrh(~)
r *(X) ~ Ln(~). (resp. O;S(X) ~L s(~)), and if also Wh(~) = 0 by o r r
We shall be particularly concerned with the round symmetric
s~gnature of the circle S 1
*(S I) = (c(gl),~) ~ LIr(~Z) . o r
The imaoe of the 2Z[z,z-l]-module chain complex
l-z C(g l) : 2z[z,z -l] ~ 2Z[z,z -1]
under the morphism of rings with involution
[? : ~[2Z] = 2Z[z,z -I], ,~Z ; z, ~i (z = z_l )
: ~Z[~Z] = ZZ,z,z -I] ~2Z ; zl ~-i
is the 2Z-module chain complex 0
J e!C(~ I) ~ 2Z . . . . . ZZ
~tn:C(Z 1 ) : 2
~(×½(C;ZZ2),x½(C;@)) = (i,i) with mod2 and rational semicharacteristics
~L(x½(D;ZZ2) ,x½(D;@)) = (i,O)
so that o;(S I) £Ll(2Z)r has images
I~!0r{S I) = (l,1) L I({I}) = ZZ2e2Z 2 .
tnl0r(S I) (i,O) r
The algebraic proof of the splitting theorem for the quadratic L-groups
Ls(~×2Z) = Ls(~)~LB n n n _l(r~) discussed in §6 below can be extended to prove
analogous splitting theorems for the symmetric and round symmetric
L-groups
n n-i , Lrs(Z×2Z ) = L n {zl(~Lh-l(~ ) Ln(z×~Z) = Ls(~)@L h (z) rs
Thus LI(~z)= LI({I})~LO({I}) = ZZ2@ZZ2@Z~, although we do not actually r
need this computation here.
344
55. Products in L-theory
~ s y m m e t r i c The product of an m-dimensional Poincare complex over A
[quadratic
(C,¢) and an n-dimensional symmetric Poincar6 complex over B (D,8) is
symmetric an (m+n)-dimensional Poincar~ complex over A~B
[quadratic
(C,¢)®(D,e) : (C®D,~e) ,
allowing the definition (in Ranicki [22]) of products in L-theory of
the type
Lm(A)®Ln(B) __>Lm÷n(A®B)
Lm(A)®Ln(B ) ~ Lm+n(A®B)
We shall only be concerned with the product L ~Ln ------>L here, with m m+D
A = ~[~], B = ~[p] group rings, so that A®B = ~[[xp].
I f i n i t e l y d o m i n a t e d The product of a ~finite m-dimensional symmetric (reap.
quadratic) Poincar6 complex over ~[~] (C,¢) and a !finitely dominated
finite
n-dimensional symmetric Poincar~ complex over ~[p] (D,e) is a
finitely dominated (m+n)-dimensional symmetric (reap. quadratic)
finite
Pcincar6 complex over ~[~xp] (C®D,¢~e) with I pr°jective class
<Whitehead torsion
I [C®D] = [C]®[D] e KO(~[~×p])
T(C~O,¢®6) = T(C,%)®x(D ) + x(C)~T(D,6) @ Wh(vxp)
The following product formulae for geometric Poincar6 complexes are
immediate consequences.
f f i n i t e l y d o m i n a t e d Proposition 5.1 The product of a m-dimensional
finite
geometric Poincar6 complex X with ~I(X) = ~ and a I finitely dominated
t finite
n-dimensional geometric Poincare complex F with ~I(F) = P is a
Ifinitely geometric complex X × F dominated
(re+n) -d inlensional Poincar6 finite
345
with projective class
Whitehead torsion
i iX ~ F] = [X]®[F] £ Ko(~l~xp])
~(X × F) ~(X)®x(F) + x(X)®~(F) Wh (~xp)
[ ]
¢ Wh (p)
[P] e S, [Q] e T
~(f) e S, ~(g) eT
~I(F) = p and
f i n i t e l y d o m i n a t e d
product normal map of [finite
Poincar6 complexes
(g,e) = (f,b) x 1 ; ~ x F -----~ X x F
is given by
U S(f,b)®o~(F) e L U (~xp) o.(g,c) = O. m+n
*(F) ~ n(0). the product of 0~(f,b) ~ L (~) and 0 T L T
[IF] 8T~_K0(~[O]) { [~(F} ~ T_CWh(0)
, then the quadratic signature of the
(m+n)-dimensional geometric
I]
Given *-invarJant subgroups [SgWh(~)
s u c h t h a t f o r ~Wh(~×p) [T(f)®l,l®~(g) ~ U
there is defined a product in L-theory
® : h~(z)®LSfp) ~m Um+n(~×p) ; (C,~)~(D,8)' ~ (C®D,~®8)
with the following geometric interpretation.
Proposition 5.2 (Ranicki [23]) If (f~b) :M ~ X is a normal map of
IfJnitely m-dimensional geometric Poincare complexes with dominated
finite
{ [M]- IX] eS~o(m[~]) ~l(X) = ~ and , and if F is a
(N) - ~{X) e S @wh(~)
I finitely n-dimensional geometric complex dominated
Poincar6 with finite
346
The methods of Ranicki [26] apply to the products of algebraic
Poincare complexes, giving the following analogues of Propositions 2.2,
5.2:
Proposition 5.3 i) The product of a finitely dominated m-d~mensional
quadratic Poincar6 complex over ~[7] (C,$) and a round finite
n-dimensional symmetric Poincar~ complex over ~[p] (D,e) is an
(m+n)-dimensional quadratic Poincar~ complex over ~[~×p] (C~D,$®@)
with canonical round finite structure, and torsion
T(C®D,~®0} = [C]~T(D,0) e Kl(~[~xp])
the product of [C] e Ko(~[n]) and T(D,O) ~ Kl(~[p]) •
ii) Given *-invariant subgroups S ~Ko(~[7]), TqKl(~[p]) , U~Wh(~xP)
such that S®T¢ U there is defined a product in L-theory
® : nS(~)®L~m T (p) >L~+n(ZX p) ; (C,~)®(D,O)~ > (C~D,o®O)
If (f,b) :M >X is a normal map of finitely dominated n-dimensional
geometric Poincare complexes with ml(X) = ~ and [H] - [X] ~S~Ko(~[n]),
and if F is a round finite n-dimensional geometric Poincare complex
with ~I(F) = p and T(F) e T ~Kl(~[p]) then the product map of
(m+n)-dimensional geometric Poincare complexes with canonical (round)
finite structure
(g,c) = (f,b) × 1 : M x F ~ X x F
has quadratic signature
U = o s ®O~T L U (~xp) O. (g,C) . (f,b) (F) e m+n
s ~ n t he p r o d u c t of o . ( f , b ) ~ L (7) and O*rT(F)~LrT(P) . []
An n - d i m e n s i o n a l g e o m e t r i c P o i n c a r ~ complex F i s roun,d,, simple,
i f i t i s round f i n i t e and
r(F) ~ {±p} ~KI(~[P]) (P = ~I(F)) ,
so that Y(F) = O £ Wh(p) and the round simple symmetric signature
o* (F) eL n (P) is defined. rs rs
f f i n i t e Proposition 5.3 shows in particular that for a round tS imple
n-dimensional geometric Poincar~ complex F product with the round n
finite 1 * (F) eLrh(P) °rh defines a morphism of symmetric signature Ln
simple (O~s(F) e rs(P)
347
< -®a~h(F ) ; LP(z) ~L h (~xp) m m + n
-~°rs(F) : Lh(~)m ~LS" m+n(~XP)
In the simple case these products define a map of generalized
Rothenberg exact sequences
S ('rxO) >Lh+n ( ' n x O ) m ^m+n • .. >Lm+ n >H (2Z~;Wh(Tx~)). >LmS~n_l(~×p)---~.. •
with T(F)~ {-+D] c. KI(ZZ[P]). The map of exact sequences in the appendix
of Munkholm and Ranicki [16] is the special case F = S I. Moreover,
the split injection
~, = -®T(S I) : Hm(2Z2;Ko(~[~])) ; Hm+l(]~2;Wh(~x2Z))
was identified there with the connecting map 6 arising from a short
exact sequenc.- of 2Z[~2]-modules
T
O > Wh(~×~) >Wh(p') > KO(ZZ[~]) ....... >O ,
with Wh(p:) the relative Whitehead group in the exact sequence of
transfer maps
~! ~: Pl = O Po = O
Wh(~). )Wh(~x2Z) . ~Wh(p !) . ~ KO(~[~]) ~.Ko(2Z[~xZZ])
associated to the trivial sl-bundle
p = projection
S 1 ; E = K(~,I) x S 1 > B = K(,~,I)
and 7z 2 acting by duality involutions. The relationship between transfer
maps and duality in algebraic K-theory will be studied in L~ck and
Ranicki [13] for any fibration F >E P ~ B with the fibre F a
finitely dominated n-dimensional geometric Poincar6 complex. In particular, !
there will be defined a duality involution *:KI(P') ...... -~KI(P~) on the I
relative K-group KI(P') in £he transfer exact sequence T
KI(2Z{~I(B)] ) Pi > KI(ZZ[~I(E)] ) ~ Kl(P !) I
P0 >' Ko(2Z[~I{B) ]) > Ko(2Z[~I(E) ]) ,
as we!l as assorted transfer maps p!:Lm(~I(B)) ..... ~ Lm+n(~l(E)) in
alqobraic L-theory. If F is round simple and Zl(B) acts on F by self
348
equivalences F
with a round manifold fibre) then there is also defined a transfer
exact sequence
~F with T = 0 £ Wh(~I(E)) (e.g. if p is a PL Dundle
i > Wh(p')
~! PO
--+K 0(2Z[7 I(B)]) >Ko(2Z[7 I(E)])
Pl Wh(~l(B)) > Wh(,~l (E))
p[ with a duality involution *:Wh( ) ~Wh(p') on the relative
Whitehead group. The connectina maps ~ in Tate ~2-cohomology arising
from the short exact sequence of ~[~2]-modules
O * coker(pi) ~ Wh(p') > ker(p~) > 0
and the transfer maps in L-theory together define a morphism of
exact sequences
>Hm(zz2;ker(PO) ~Lh_l(7)
ira(> i) ~Hm+n(zg2;coker (~i))--+Lm+n_ I (E)
(n = 71(B) , E = ~I(E))
i k e r (~0)
.... ~Lh(~) ~L (~) m m
P P
i
im(#i) __+Lhm+n ... ~ Lm+ n (E) (F.)
In the case of the trivial fibration
p = projection F ~E = BxF ~ B
(with the fibre F a round simple Poincare complex, as before) the
algebraic K-theory transfer maps are zero
= -®[F] : 0 : Ki(~[,~]) ~ Ki(2Z[~x4)]) Pi
(i = O,i 0 = ~I(F))
so that Pi =o. Also, the algebraic L-theory transfer maps are given
by the products with the round symmetric signatures
i P" = -®Crh(F) : LmP(~) > Lhm+n(ZXP)
P! = -®O~s(F). : Lh(~)m ~LSm+n(~×0) ,
and 6 is given by product with the torsion T(F)~ {-+p} C_ KI(TZ[p])
6 = -®T(F) : Hm-(2Z2;Ko(ZZ[#])) ~ Hm+n(zz2;Wh(7×p))
as in the case F = S 1 considered in [16].
)...
349
§6. The L-groups of a polynomial extension
There are 4 ways of extending an involution a~ +a on a ring A
to an involution on the Laurent polynomial extension ring A[z,z-l], -i -i
sending z to one of z,z ,-z,-z In each case it is possible to
express L.(A[z,z-l]) (and indeed L*(A[z,z-l])) in terms of L.(A), and
to relate such an expression to splitting theorems for manifolds
- see Chapter 7 of Ranicki [24] for a general account of algebraic
and geometric splitting theorems in L-theory. Only the case
A = ~[~] , ~ = z -I
is considered here, for which A[z,z -I] = ~[~] [z,z-l].
The geometric splittings of the L-groups L.(~×~) depend on the
I Wall [35]
realization theorem of ~ Shaneson [29] , by which every ! <Pedersen and Ranicki [18]
I LS(~)
n
L h (~) n
L p (-~) n
element of
rel~ surgery obstruction
simple
(n ~5, ~ finitely presented) is the finite
projective
t o , I f , b )
a~(f,b) of a normal map
0~(f,b)
(f,b) : (M,~M) ~ (X,SX)
from a compact n-dimensional manifold with boundary (M,~M) to a
f simple finite n-dimensional geometric Poincar6 pair <X,~X)
finitely dominated k
equipped with a reference map X ....... ,K(~,I), and such that the
restriction Zf = fl : ~M ...... ~'~X is a
simple
homotopy equivalence.
A morphism of groups
induces functorially morphisms in the L-groups, given geometrically by
350
¢, : Lq(*) ~ Lq(H) ; • n n
(f,b) a~((M,~M) ~ (X,~X) ~K(~,I))
(f,b) a~((M,~M) ) (X,~X) ' K(~,I)
¢ *K(]],I))
and algebraically by
(q = s,h,p) ,
¢, : Lq(~) ~ Lq(]]) ; (]q(f,b)~ n n
In general %~ will be written ¢.
~2Z [~] ®2Z [~] a*q(f ,b)
The geometric splitting of Shaneson [29]
was obtained in the form of a split exact sequencc
[ B O ; L:(~) ~ LS(~ x ~} ~L h (7)- > O
n n-I
with ~ the split injection of L-groups induced functorially from the
split injection of groups ~:~> ~ z ~ . The split surjection B was
defined geometrically by
B : LS(~ × ~) ~L h n n-i (~) ;
s (f,b) a, ((M,~M) ) (X,dX) × S 1 ~ K(~,l) x S 1 = K(~×2Z,I))
~ ah (g,c} ,((N,~N) ~ (X,oX) > K([,I))
using the splitting theorem of Farrell and Hsiang [ 5 ] , [ 7 ] to
represent every element of LS(~x2Z) as the rel~ simple surgery n
s obstruction o,(f,b) of an n-dimensional normal map
(f,b) : (M,~M) >(X,~X) x S 1 with (X,~X) a finite (n-l)-dimensional
geometric Poincare pair, such that f is transverse regular at
(X,%X) x {pt.} C(X,dX) × S 1 with the restriction defining an
(n-l)-dimensional normal map
(g,c) = (f,b) I - (N,~N) = f-I((x,~x) x {pt.}) ~ (X,~X)
with ~f:ZM' ~X× S 1 a simple hcmotopy equivalence and ~g:~N ~X a
homotopy equivalence. There was also defined in [29] a splitting map
for B
3 5 1
L h : n-I (~)~
h 0, ((M,%M)
>LS(~ x 2Z) ; n
(f,b) ~' (X,~X) } K(~,I))
S sl (f,b) × 1 ) O.((M,ZM) x ...... (X,~X) × S 1
>K(~,I) × S 1 = K(wx2Z,l))
( = 0~(f,b)~a;(S I) by Proposition 5.3 ii))
''LS(zx~) )~ L:(z) be the geometric split surjection determined Let e " n
by ~,B,B', so that there is defined a direct sum system
~ B LS(~)~< * LS(~ x ZZ)~ "~Lh n i(~) n n -
Although it was claimed in Ranicki [20] that E' coincides with the
split surjection induced functorially from the split surjection of
groups e:~x~ )~ (or equivalently ~[~l[z,z -I] ~[~] ; z ~ ~i)
it does not do so ~n general. This may be seen by considering the
composite
£B' : L~_l(~)~-- - - - - - - - -~LS(~x~)n ~LS(~)n '
which need not be zero. A gene~ic element
h (7) 0.((f b) : (M 8M) . (X,3X)) C L h ' ' n-i
is sent by B' to
B' (o~(f,b)) = o~((g,c) = (f,b) x iS1 : (M,~M) ~ S 1 ~ (X,~X) × S I)
C L h (~ x ~) • n
Now (g,c) is the boundary of the (n+l)-dimensional normal map
(f,b) x I(D2,SI ) : (M
such that the target
(X,~X) x (D 2,S 1
is a finite (n+l)-dimens
boundary and
~((X,~X) x (D2,S 1
3M) × (D 2 , S I) (X,~X) × (D2,S I)
= (X x D2'X x slk] ~X x S I~X x D2)
ional geometric Poincare pair with simple
) = T(X,~X)®x(D 2) + x(X)ST(D2,S I)
= T(X,3X) e Wh(~)
(by the relative verslon of Proposition 5.1). It follows that
352
¢~='a,h'#,~. b) ~ LS(~)n is the image of
T((X,~X) x (D2,SI)) = T(X,}X)
~n-l(2z2;Wh(~)) = ~n+l(Tz 2
under fhe map ~n+l(2z2;Wh(~)) ~ LS(~) n
sequence
... ~Lhn+l(~) ,-~n+l(Tz2;Wh(~)) > LS(~) ) Lh(~) n n
:Wh(~))
in the Rothenberg exact
) ....
The discrepancy between ~ and ¢' will be expressed algebraically in
Proposition 6.2 below; it is at most 2-torsion, and is 0 if Wh(~)= O.
Novikov [17] initiated the development of analogues for algebraic
L-theory of the techniques of Bass, Heller and Swan [2 ] and Bass [ i ]
for the algebraic K-theory of polynomial extensions. In Ranicki [19],[20]
the methods of [17] (which neglected 2-torsion) were refined to obtain
for any group n algebraic isomorphisms
I L =
I BL =
: (~×Tz) ~ LS(r~)$L _ 1 ( 7 ) B n n
L h : (~xZg) ~ (~)$L _i(~) B n n
with inverses
~ , l =
B) : Ls(~)$Lh_I(~) ~LS(~×2Z) n n
(~ B) : Lh(v.)$LP_l(~)n ~Lh(~×~)n
by analogy with the isomorphism of [2 ]
8 K : Wh(~×2Z) • Wh(~)$Ko(2Z[~])(gNi'-~(TZ[~.])$Ni'-'~-(77[~])
recalled in §3 above. The isomorphisms ~L define the algebraically
significant splitting
As already indicated above this does not in general coincide with the
geometric splitting of LS(~xZZ) due to Shaneson [29], although the n
B:LS(zxZZ) split sur jection )>L n n _i(~) of [29] agrees with the
algebraic B of [20].
353
Pedersen and Ranicki [18,~4] claimed to be giving a geometric
interpretation of the algebraically significant splitting
h = h p (z) However the composite L, (~×2Z)L, (w)@L,_ 1 .
£B' : L p (~)~ ." Lh(~xZZ) ~Lh(~) n-i n n
of the geometric split injection
B' : L p n_l(W)> : > Lh (wx~) ; n
a~((f,b): (M,DM) }(X,~X))
h , ~o.((f,b) × isl : (M,~M) x S I . )(X,~X) x S I)
(= o~(f,b)®o~(S I) by Proposition 5.3 ii))
and the alaebraic, split surjection [:L~(~x~) ~Lh(~)n need not be
zero: there is defined a finitely dominated null-bordism with
~l(X x D 2) = ~l(X) =
(f,b) x I(D2,SI ) : (M,~M) × (D2,S I) ~ (X,~X) x (D2,S I)
of the relative (homotopy) finite surgery problem
(f,b) x isl : (M,~M) × S I, ~ (X,~X) x S 1
with finiteness obstruction
IX x m 2] = IX] ~ K0(~[7])
It follows that cB'q~(f,b)~ Lh(~) is the image of
[X] e Hn-I(~2;K0!~[~])
Hn÷I(~2;Ko(~[~])}
sequence
... ~P (~) ~n+l
Hn+I(2z2;Ko(2Z[~])) under the map
~Lh('~) in t h e g e n e r a l i z e d R o : h e n b e r g e x a c t n
~Hn~I(zz2;K0 (zZ[~])) "~Lh(~)n ~ LP(~)n ) . . . .
Thus {' and e de not in general belong to the same direct sum system.
In fact ~ belongs to the algebraically significant direct sum
decomposition of Lh(~x~) described above, while B' belongs to the n
geometrically defined direct sum decomposition
B Lh(~----------~Lh(~x~)~ ~>L p ~ (~
n ~ n-~
with B as defined in [18 ,§4] and ~' the split surjectlon determined
by -£,B,B'. It is the latter direct sum system which is meant when h h
to "the geometric splitting L.(~xZZ)= L,(~)@LP_I(~)_ of 118]". referring
354
Define the geometrically significant splitting
to be the one given by the algebraic isomorphism
~L' = : LS(~×2Z)n > LS(~)OLn -i (~)
() e
B~ = : Lh(zxZZ) - ~Lh(n)OLPn_ 1 (z) B n
with inverse
where
and
8L -I = (£ B') : LS(~)~Lh ) " n n -i (~ ~LS (~xZg)
B~ -I = (C B') : mh(n)ehn p i(~) .... >mh(nxZ~) n - n
{ B' = -~o*(S I) : L n r h-l(~); >LS (zxZg}
~, = -®Or(S I) : L p ~(n)> .~Lh(~×ZZ) n-± n
[ ~' = £(I-B'B) : LS(~×~) ,~LS(~) n n
£' = ~ (I-B'B) : Lh(~x~)n >~L~(z)
Proposition 6.1 The geometric splitting Lh -- L~(~×=) n(~)ee~_l(~)
I Shaneson [29] is the geometrically significant splitting Pedersen and Ranicki [18]
in algebra.
of
[]
The algebraically significant split injections
h were defined in Ranicki [20] using the forms B:LP(~)~ ~ L.+l(~X2Z)
and formations of Ranicki [19] ; for example
B : LPi(~)>--------+Lhi+l(~X2Z) ;
(Q,~) ~, ~ (M@M,~@-J2 ;A, (l@z) A)@ (H (_) i (N) ;N,N)
sends a projective non-singular (-)l-quadratic form over ~[~] (Q,~)
355
tO a free non-singular (-)i-quadratic formation over 2Z[~×Tz] =2Z[~] [z,z -I]
w{th M = Q[z,z -I] the induced f.g. projective ZZ[~×2Z]-module,
& = {(x,x) ~M(gMIxdM} CM@M the diagonal lagrangian of (M@M,~@-~), and
H(_)i(N) = (N~N*,~ O io1)the (-)i-hyperbolic (alias hamiltonian)form kO
on a f.g. projective ZZ[~×ZZ]-module N such that M@N is a f.g. free
7z[~×~Z]-module. The geometrically significant split in3ections [~, h s
:L, (~)~ )L,+I (~×2Z) ~,:Lp(~) ; h were defined in ~i0 of Ranicki [22] using
~L,+ I (~ ×2Z)
algebraic Poincare complexes. It is easy to translate from complexes
to forms and formations (or the other way round); for example, in
terms of forms and formations
~. : LPi(~)> ' , h L2i+l (~×2Z) ;
(Q,%)k--------+ (M@M,9@-~;A, (I(gz)A)(9(H(_)i(N) ;N,N*) ,
making apparent the difference between B and B' in th~s case.
For any group ~ the exact sequence
O > HO(zz2;Ko(~))-
splits, with the injection
~O(zz2;Ko(2Z) } = 2Z2~
) Llrh(~) > LI(~) ~ 0
;L I rh(~) ; I ,
. {S 1
Now
°*(Sl)r - °~ (SI) = ~E°r (SI) £ LI(2z) ' r
split by the rational semicharacteristic
Ll(~)r )) 2Z 2 ; (C,¢)~ ~ X½(2Z®2z[~]C;~)
By the discussion at the end of Ranicki [22,§i0]
LI(2z) = LI({I))~LO({I}) = 2Z2(92Z ,
with (O,I) = 0*(S I) C LI(zz) the symmetric signature of S ] . Let
o* (S I) C L l(Tz) be the image of o* (S I) C L I(2Z) under the splitting map q r
LI(2z)> ~LI(2z) so that o*(S I) = (I-~)~*(S I) and ~o*(S I) =OC LI({I}}. r ' q r q r
The algebraically significant injections are defined by
n+l
356
so that
- = l) =
By analogy with the map of algebraic K-groups defined in §3
=-®T(-l) : KO(TZ[~])
define maps of algebraic L-groups
co = -®~Or(Sl) : Lh(~) n
~m =-®eo*(S 1) : LP(~) r n
~Wh (~)
~LS+I (~)
}Lhn+l (~) ,
where ¢Or(S I) = (i,i) ~ Ll({l})r = ZZ2(~ZZ2" As ~=(S I) = ~(-i) eKI(ZZ)= ZZ 2
the various maps co t o g e t h e r d e f i n e a m o r p h i s m o f g e n e r a l i z e d R o t h e n b e r g
exact sequences
.... >L h (~) ~ L p (~) n n
~L h (~) ~.. > Hn(~2;Ko(ZZ hi)) " n-i ""
" ~ H n + l ( 2 z 2 ; W h ( ~ ) ) ' LS(~)n .... > . . . .
Proposition 6,2 The algebraically and geometrically signiflcant split
injections of L-groups differ by
- co
{ B' -B = ¢co : L (z)
B ' - B = 7~ : LP(~) ~)
The split surjections differ by
L s n+l (~)}
Lh+l(~)>
£ • LS+I (r x2Z)
L h n+l (~x2Z)
m ¢ ' - e = eJB : LS(nxZZ) ;.~ L h I n n-i (~ >LSn (~
B co ¢' - ¢ = COB : Lh(wxZ~) >>'L p (~ >Lh(~)
n -1 n
The L-theory maps ~ factor as
I w : Lh(~) >Hn(Zz2;Wh(~) = Hn+2(ZZ2;Wh(~)) "L s n n+l (~)
co LP(~ln ~fin(z~2;.Ko(Z~[~])) =Hn+2(~Z2;Y'o(Z~I~I)I-------~Lh+I(~)
The K-theory map co is the sum of the composites
~n(2z2;~o(2Z[~]) ) ~ L h ~n-i ~n+l n_l (~) ~ , (2Z2;Wh (~)) = (ZZ2;Wh(~))
Hn(z~2;Ko(2Z[~])) = Hn+2(ZZ2;Ko(2Z[~])) ~Lh+l(~) ~n+l(2z2;Wh(~) ) .
357
I L~'S(~) Proof: Let (n ~ O) be the relative cobordism group of
I ( f i n i t e , s i m p l e ) n - d i m e n s i o n a l q u a d r a t i c P o i n c a r ~ p a i r s
( f i n i t e l y d o m i n a t e d , f i n i t e )
o v e r ~ [ ~ ] ( f : C , D , ( 6 5 , ~ ) e Q n ( f ) ) , so t h a t t h e r e i s d e f i n e d an e x a c t
sequence
L s L h n
L h L p (~) ~ (~1 n n
~Lh'S(~) n
~L p ' h ~) n
~L s n_l(~)= .~ ...
; L h n_l(~) ~ . ..
and there are defined isomorphisms
Lh'S(~)-------eHn(~2~Wh(~)) ; n
(f:C----~D,(6~,@))~ ~ ((I+T) (6¢,~)o:C(f)n-* }D)
L~'h(~) ~ Hn{~2;Ko(~Z[~]) ) ; {f:C--~D, (6,,*)1} > [D]
Product with the 2-dimensional symmetric Poincare pair 0*(D2,S I) over
defines isomorphisms of relative L-groups
{ -®o*(D2,SI) : Lh'S(~) ' L~;~(~) n
-~o*(D2,S I) ; LP'h(~) ~,LP'~(~) n n+Z '
corresponding to the canonica] 2-periodicity isomorphisms of the Tate
~2-cohomology groups
(~n(~2;Wh(~)) ~n+2(~2;Wh(~))
I Hn(~2;Ko(~[~])) ~ Hn+m(~2~Ko(~[~]))
The boundary of ~*(D2,S I) is EC*(SI) . r
In particular, the algebraic and geometric splitt'ing maps in
L-theory differ in 2-torsion only, since 2~ = O (cf. Proposition 3.3).
The splitting maps in the algebraic and geometric splittings of
Wh(Tx~) given in ~3 and the duality involutions * are such that
~* = *-6 : Wh(~) '~ Wh(~x2Z)
~* = *~ , ~'* = *c' : Wh(~×2Z) ~ Wh(~)
B* =-*B : Wh(~x2Z) ;Zo(ZZ[~])
= Nil(TAil]) ~ Wh(~×2Z) ZJ *Z : -
358
The. involution *:Wh(~x~) ............. ~Wh(zx~) interchanges the two Nil summands,
so that they do not appear in the Tare ~2-cohomology groups and there
are def-ined two splittings
Hn(~2;Wh(~x~)) = ~n(~2;Wh(~))o~n-l(~2;Ko(~[~]) ) ,
the algebraically significant direct sum decomposition
Hn(ZZ2;Wh(~))~ ...'Z > Hn(zz2;Wh(~xZZ)) 4 ,~n-i (ZZ2 ;~o(2Z [~ ] ) )
and the geometrically significant direct sum decomposition
n{2z2;w h 5 ~ ~n-i {2Z2 ;~o(ZZ [~ ] ) ) (~))~ { , _> Hn(zg2;Wh{~×ZZ)) ( <
Proposition 6.3 The Rothenberg exact sequence of a polynomial extension
... ~LS(~x2Z) ~Lh(~xZZ) ~Hn(Tz2;Wh(~x?z)) ~L s l(~XTz) ~" ... n n
has two splittings as a direct sum of the exact sequences
• • m ~.LS (~) ~Lh(~) ' ~ Hn(zz2;Wh(z)) ~L s (7) > n n-l "'" '
,Lhn i<~) ~P ,(~) ,~n-i(=2;~O(=I~])) ,L~ a(~) , . - n - I - '
an algebraically and a geometrically significant one.
[]
The split injection of exact sequences in the appendix of
Munkholm and Ranicki [16] is the geometrically significant injection
... ;, Lhn_l ( ~ ) ~L p n_l (~)
. . . >LS(~×=) , Lh(~×Zg)
>Hn-i (2Z2 ;Ko (ZZ [~ ] ) ) --~Chn_2 ([) ) ...
~ ~n (ZZ2;Wh (~xZZ)) ~L s n_l (z x 7z)---~ ....
As for algebraic K-theory (cf. the discussion 3ust after
Proposition 3~3) it is tempting to identify the geometrically
{e ':LS(~×~) ~LS(~)
significant split surjection n n with the split e' Lh(~×~) ~Lh(~)
n n
surjection of L-groups induced functorially by the split surjection of
rings with involution
[z,z -1] ~_ ajz3~ S_=aj n : ~[~] = ~[nx~] ~>.~[~] ; ) (-i) j J J
and indeed
359
¢'[ (=l) = nl : im(~:L (~); >LS(~x2Z) : n ¢'I(=l) = ql : im(~:Lh(~) ~ ~Lh(~ x2Z)
n n
However, q o ; ( S l ) = (1 ,O) ~ 0 e L l r ( { 1 } ) = ~2~2~ 2
2 Zg-module chain complex is ~ >2Z) and in general
IE'I(=o) "~-nl :
so that
..... ~ L s (~) n
..~ Lh (~) n
since the underlying
= .L h (~ b---~ LS (~ x~ ) ) im(B' -;D°r (SI)" n-i
-L p (~ p---~Lh (~x2Z)) im(B' =-~°r (SI)" n-i
[' ~ q : LS(~×ZZ) ~LS(~) n n
Lh(gX~)n ~,L~(n) e' ~ ~ :
~LS(~)
h > Ln (I~)
For q = s,h,p the type q total surgery obstruction groups
~(X) were defined in Ranicki [21] for any topological space X to
fit into an exact sequence
oq
. . . )Hn(X;~_O) * ........ >L~{~I(X)) ~ ~(X) ' Hn_I<X;IL O)
with -~-~0 an algebraic 1-connective fl-spectrum such that
~.(~0 ) = L,({I})
and o~ an algebraic version of the Quinn assembly map. If X is a
I simple
finite
finitely dominated
n-dimensional geometric Poincar6 complex the
S(x) s(x) ~ ~ n
total surgery obstruction s(X) ~h(x) is defined, and is such that n
s(×) e ~ nP(X)
s(X) = 0 if (and for n >,5 only if) X is - homotopy
XxS 1
f equivalent to a compact n- dimensional topological manifold. For a
(n+l) -
compact n-dimensional topological manifold M with n >5 the exact sequence
°q q °q , Lq(Zl (M)) " " " ---~Hn+l (M;---~O) ~*Lq+l (~I(M)) > ~n+l (M) .... ~ Hn (M:ILO) n
is isomorphic to the type q Sullivan-Wall surgery exact sequence
360
~q ... ~ [MxDI,MxSO;G/TOP,, ] ~ L q ~qTOP n+l(~l(M))--~ (M)
8q ) [M,G/TOP] ........ ) Lq(z (M)) n 1
with 8 q the type q surgery obstruction map and ~qTOP(M) the type q
topological manifold structure set of M.
Proposition 6.4 For any connected space X with ~I(X) = n the commutative
braid of algebraic surgery exact sequences of a polynomial extension
~n+l(~2;Wh(zx~)) 6:(X x S l) Hn_ I(X x S1; _~O )
LS(~×~) ~h(X × S l) n n
• Hn(X x SI;~o ) Lh(~xZS"; Hn(~2;Wh(~x~)) -- n
has a geometrically significant splitting as a direct sum of the braid
in+I(ZKz;W h(I) ) ~S(x)
<I×l /
H n (X ; ~ 0 ) n
Hn_I(X;__~ O)
and the braid
~n(zK2;~O(Z~[~ ]) ) ~n-l(X) Hn_ 2(x;_~_O)
LI-I(~) ~Pn-I (X}
/\. Hn-I(X;ILo)-- LPn-l'(~ Hn-I(z{2;Ko(Z~[~]))
[]
361
It is appropriate to record here (in the terminology of this
paper) a footnote from the preprint version of Cappell and Shaneson [3 ]:
"it is not completely obvious that the maps given in Ranicki [20] give
a splitting
LS(~×~)n = Ls(~)~L~-I ( ~ ) n
respected by the surgery map
e s : [M × SI,G/TOP] = [M× D I,M × sO;G/TOP,*]e[M,G/TOP] ...... ~L sn+l(~×~)
with M a compact n-dimensional topological manifold and ~ = ~I(M)."
Department of Mathematics,
Edinburgh University
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Pseudo free actions I.,
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The obstruction to fibering a manifold ove[ the circle
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[ll]I.Hambleton, A.Ranicki and L.Taylor
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[12]K.Kwun and R.Szczarba
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Proper simple h0motopy theory versus simple homotopy
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The projective class group transfer induced by an
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Projective surgerji_~heory Topology 19, 239- 254 (1980)
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Lecture Notes 1126, 199- 237 (1985)
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Wall's surgery groups for G x
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The obstruction to finding a boundary for an open
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Princeton Ph.D. thesis (1965)
A torsion invariant for bands
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Infinite simple homotopy types
Indag. Math. 32, 479 - 495 (1970)
A total Whitehead torsion obstruction to f ibering over
the circle Comm. Math. Helv. 45, 1 -48 (1970)
[34] C.T.C.Wall
Finiteness condition s .for CW comple..xes
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[35] Surg_ery on compact manifolds Academic Press (1970)
[31]
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[33]
Coherence in Homotopy Group Actions
R. Schw~nzl and R. M. Vogt
I. Introduction
In the effort to construct an action of a group G on a homotopy type
one encounters the problem of having to realize a homotopy action of
G on a space X by a genuine G-action on a space Y of the same homo-
topy type as X.
1.1 Definition: A homotopy action of a group G on a space X is a
homomorphism e: G • ~o(AUtX) , where AutX is the space of self-
homotopy equivalences of X. A realization of ~ is a G-space Y
together with a homotopy equivalence f: X ~ Y which is equi-
variant in the homotopy category TOPh. If Y is a free G-space,
we call (Y,f) a f ree r e a l i z a t i o n .
This problem has been solved by Cooke [C] for discrete groups:
1.2 Theorem: ~: G - ~ ~o(AUtX) admits a realization iff there is a
lift (up to homotopy),
B (AutX)
B (n ° AutX)
where B denotes the classifying space functor.
A rational version has been studied by Oprea [O].
Zabrodsky took up this problem in [Z] with a different attitude. He
investigated the relations induced by AutX on the space of homeo-
morphisms of a realization Y. He indicated an obstructions theory for
realizing a homotopy G-map from a G-space to a homotopy G-space.
1.3 Definition: Let ~: G • ~o(AUtX) and 8: G ~ ~o(AUtY) be
homotopy actions of G on X respecitively Y. A homotopy G-map from
X to Y is a map f: X ~ Y which is G-equivariant in the homotopy
365
category. A r e a l i z a t i o n of a homotopy G-map f: X ~ Y is a
homotopy commutative diagram
f X ) Y
f ' X' ~ Y'
where (X',h X) and (Y',hy) are realizations of ~ and 6 and f' is
a G-equivariant map.
A draw-back of Zabrodsky's theory is that he works in the category of
based topological spaces, so that all group actions have to leave the
base point fixed.
The aim of the present paper is to tackle these problems with the
methods of homotopical coherence theory as developed in [B -V]. We
interprete Cooke's obstructions as obstructions to higher coherence.
Our proofs allow a generalization to topological groups. We deal with
relative versions in the sense of [Z] and with relative versions with
respect to subgroups.
Throughout this paper we work in the category Top of compactly
generated spaces in the sense of [Vl].
Organization of the paper: We introduce the notions of n-coherent
homotopy G-actions and n-coherent homotopy G-maps (in Section 2) using
the W-construction of [B-V] and state some of their fundamental pro-
perties, we formulate the main results (in Section 3) and discuss the
universal property of the W-construction (in Section 4). In order to
keep this paper fairly self-contained all constructions from coherence
theory are executed and proofs of almost all statements are indicated
so that a knowledge of the more complicated theory of [B -V] is not
required. The proofs (in Sections 5) of our main results make use of
homotopy-homomorphisms of monoids and functors of topologized cate-
gories which have close connections with Fuchs'stheory of H -maps and
G -maps [FI], [F2], [F3]. We discuss this relationship in a final
Section 6.
Our interest in this subject was initiated by a problem posed to us
by T. tom Dieck. We want to thank him for suggesting to apply cohe-
rence theory to this type of problems. Finally we want to draw the
366
reader's attention to work of Dwyer and Kan [D -K]. We could equally
well have used their methods to obtain our results in the ~-coherent
case, which after all is the most interesting one.
2. n-coherent homotopy actions and homotopy q,maps
Given a hemotopy action ~: G " ~o AutX on X, we choose a representa-
tive g £ ~(g) and a path
w(g I ,g2 ) : •
g1"g2 gl ° g2
for each pair (gl,g2) 6 G x G. If e 6 G is the neutral element, we make
the spacial choices ~ = idx, and w(e,g) = w(g,e) = trivial path on g.
We call the resulting structure a l-coherent homotopy G-action on X.
Of course, a l-coherent homotopy action is not uniquely determined
by a homotopy action.
Given three elements gl,g2,g 3 in G different from e, a 1-coherent
homotopy G-action on X gives rise to a loop l(gl,g2,g 3) in AutX
gl g2 ° g3 q
w(g I,g2) o g3
w(glg2,g 3 )
gl ° g2 ~ g3
g1° w(g2,g3 )
glg2g3 gl ° g2g3
w(gl,g2g 3 )
Sometimes it is possible to fill in all loops l(gl,g2,g 3) by a disk
d(gl,g2,g3). We add these disks to the data and call the structure
thus obtained a 2-coherent homotopy ac t i on . Playing this game with
more group elements we can define arbitrarily high coherence. We now
formalize this concept.
Let C be an arbitrary small category. Throughout this paper we assume
all small categories to have sets of objects but topologized morphism
spaces such that composition is continuous. We call C well-pointed
if ob Cc mor C is a closed cofibration. Let Cat be the category of
such topologized categories. We construct a functor (see [B-V])
367
W: Cat ~ Cat
as follows: Let C £ Cat; then ob WC = ob C , and
WC(A,B) = __~ Cn+I(A,B) × In/~ n k o
where Cn+I(A,B) is the space of all composable morphisms
fo fl fn A = A O • A I ~ A 2 . ?... . An+ I = B
in C with the obvious subspace topology from (mor C)
The relations are
n+l , and I = [0,1].
(2.1)
(I)
(2)
(3)
(4)
(fn,tn,-..,fl,tl,f O)
= (fn,tn,...,fi o fi-l' ti-l'''''f1'tl'fo) if t i = 0
= (fn,tn,...,fl) if fo = id
= (fn,tn .... ,fi+1,max(ti+1,ti),fi_1,...,fo) if fi = id
= (fn_1,tn_1, .... fl,tl,fo) if fn = id
Composition in WC is given by
(fn,tn,...,fo) o (gk' Uk'''''go) = (fn'tn'''''fo ' I, gk,Uk,...,go).
The n-skeleton subcategory wnc of WC is the subcategory generated by
all morphisms having a representative (fk' tk .... ,fo ) with k ~ n.
2.2 Definition: An n - c o h e r e n t homotopy a c t i o n of a topological group
G on a space X is a homomorphism a: WnG . AutX of topological
monoids.
Explanation: A topological monoid can be considered as a topological
category with one object and vice versa.
Since G is a group, an n-coherent homotopy G-action determines and
is determined by a continuous functor WnG • Top sending the unique
object to X. We often call such a functor (and consequently
~: WnG ~ . AutX) a WnG-strueture on X, or X a WnG-space.
2.3 Notation: A C-space is a continuous functor C. Top. A homomor-
phism of C-spaces is a natural transformation of such functors.
368
As indicated in the introduction we want to investigate maps which are
homomorphisms up to homotopy, possibly with coherence conditions. To
find the appropriate definition, observe that a natural transformation
y: F o . FI: C ~ Top
of functors Fo,F I determines and is determined by a continuous functor
C × L! '~ . Top, where i I is the category O • I. This leads to
2.4 Definition: Let ~: WnG --- Top and 8: WnG ...... Top be n-coherent
homotopy actions of a topological group G on spaces X and Y. An
n-coherent homotopy G-map from (X,~) to (Y,8) is a continuous
functor y: Wn(G × il) ~ Top with yIWn(G × O) = ~ and yIwn(G × I)=8.
The map y ((e × (0 - I))): X ..... Y is called the underlying map
of ¥.
We recall from [B-V; chapt.4] that homotopy classes (through functors)
of ~-coherent homotopy G-maps form a category, where G may be any well-
pointed topological group. The same holds (by the same arguments) for
discrete groups and n-coherent homotopy G-maps. Moreover we shall
use [B-V; (4.20) , (4.21) ]:
2.5 Propositign: Let H be a subgroup of G such that HoG is a closed
cofibration. Let e': WH -- Top and 8,y: WG • Top be ~-coherent
homotopy actions of H on X and of G on Y and Z. Suppose further
we are given an u-coherent homotopy H-map p': (X,~') ~ (Y,81WH)
and an ~-coherent homotopy G-map i : (Y,~) ~ (Z,y) whose under-
lying maps are homotopy equivalences. Then:
(1) ~' extends to a WG-structure ~ and p' to an ~-coherent homo-
topy G-map @: (X,~) ....... (Y,B)
(2) Any homotopy inverse <': W(H × Ll) .... Top of II (WH × i 7)
extends to a homotopy inverse ~: W(G x il) ~ Top of I.
2.6 Remark: Of course, Definition 2.4 still makes sense if G is
replaced by an arbitrary topological category C, and Proposition
2.5 holds with G replaced by an arbitrary well-pointed category C
and H replaced by a subcategory D of C such that mor Dc mor C is
a closed cofibration.
369
3. Main results
Throughout this section let G be a discrete group unless stated other-
wise.
We first interprete the obstructions to a lift of Be in (1.2) as ob-
structions to higher coherence
3.1 Theorem: A homotopy action ~: G ~ Zo(AUtX) of G on X is induced
by an n-coherent homotopy action iff there is a lift up to homo-
topy
B Bn+IG ~ B(AutX)
Ba BG ...... ~ B (T ° AutX)
where Bn+IG is the (n+1)-skeleton of BG.
An --coherent homotopy action can always be realized (see (3.2)) so
that (3.1) and (3.2) imply Cooke's result (1.2):
3.2 Theorem: Given an ~-coherent homotopy action 8: WG ~ AutX,
there is a free G-space Y8 and an ~-coherent homotopy G-map
iB: X -- Y8 with the following properties
(I) i s embeds X as a strong deformation retract
(2) any ~-coherent homotopy G-map p : (X,~) - (Z,y) into a
genuine G-space Z factors uniquely as through i B and a
genuine G-equivariant map Y8 , Z.
If one starts with a G-space X and drags it through the machines of
(3.1) and (3.2) Cooke already showed that one ends up with X made free.
We prove a corresponding result in our set-up by giving a complete
classification of all free realizations of a given homotopy action:
Let ~: G ~ ~o(AUtX) be a homotopy action of G on X. We call two
realizations (Y,f) and (Z,g) of e equivalent iff there is a G-homotopy
equivalence h: Y .......... Z, i.e. a homotopy equivalence in the category
of G-spaces and equivariant maps, such that h° f ~g.
370
3.3 Theorem: There is a bijective correspondence between the equiva-
lence classes of free realizations (Y,f) of a homotopy G-action
on X and the homotopy classes of lifts
~ ~ B (AutX)
BG ~ > B(~ ° AutX) B~
As a generalisation we now consider the case that a homotopy G-action
extends a given genuine H-action of a subgroup H of G.
3.4. Theorem: Let H c G be a subgroup of G. Let e: G ~ no(AUtX) be
a homotopy action of G on X such that elH is induced by a genuine
H-structure B: H .... AutX. Then ~ is induced by an n-coherent
homotopy action y: WnG ~ AutX extending B iff there is a filler
up to homotopy
BB B n+1H c BH
I B a Bn+ I G c BG
• B (AutX)
B(~ O AutX)
Of course, (3.2) has its analogue in the relative case.
3.5 Theorem: Let y: WG ...... AutX be an ~-coherent homotopy action of
G on X extending a strict H-action 8. Then there is a free G-space
Y and an H-equivariant map f: Y ~ X which is an ordinary homo-
topy equivalence and whose H-structure extends to an ~-coherent
homotopy G-map.
Remark: We have defined realizations as maps X ~ Y into a G-space.
Since f in (3.5) is H-equivariant and Y is free we cannot expect to
obtain such a map from X to Y unless X is H-free. In this case, we
indeed may choose f as H-map from X to Y by (2.5) and (4.5) below.
3.6 Corollary: Let H be a p-Sylow subgroup of a finite group G, and
let X be a p-local space of the homotopy type of a CW-complex
with an H-action compatible with a homotopy G-action ~ on X. If
AutlXcAut X denotes the component of id x and if H~(BH; {~_2AutlX])
coincides with its G-invariant part [Br; p.84] then there is a free
371
G-space Y and a H-equivariant "realization" f:Y--X of ~ (i.e. f is
H-equivariant and a homotopy equivalence).
We now turn to relative versions in the sense of [Z]. We need some
preparations to state our results:
A topological space X with a right Mo-aCtion and a left M1-action of
topological monoids M o and M I gives rise to a category C(MI,X,M o) with
two objects O,1 and morphism spaces mor(i,i) = M., mot(O,1) = X and 1
mor(1,0) = ~. Composition is defined by monoid multiplication and the
actions. Conversely, any such category C makes C(O,I) into a right
C(O,O) - and left C(1,1)-space. A
For a topological group G let G denote the space G with its left and ^
right G-action from multiplication. Since C(G,G,G) = G x 51, an n-coher-
ent homotopy G-map is a functor
~: wnC(G,~,G) ....... Top.
Since G is a group such functors ~ with ~(0) = X and ~(I) = Y are in
I-I correspondence with functors
A A e: wnC(G,G,G) ........ C(AutY, F(X,Y) , AutX)
where F(X,Y) is the space of maps from X to Y.
In particular, a homotopy G-map (X,e o) - - (Y,e I) of spaces with
homotopy G-actions is a functor
A A e: C(G,G,G) - - C(~o(AUtY) , ~o F(X,Y) , ~o(AUtX))
(AutX) and ~I: G . ~ (AutY). extending So: G ~o o
A This functor defines a map ~: G . ~ F(X,Y) of the left G x G°P-space A O G to the left ~o(AUtY) x ~o(AUtx°P)-space ~oF(X,Y) which is equivariant
op The pair (e I x eoOP,e) and with respect to the homomorphism el x n ° .
the obvious projections induce maps of 2-sided bar constructions
[M; section 7]
Be op (3.7) BG °p o > B (~oAUtX°P)
Po qo
Be B(~,G x G°P,~) ~ B(~,~o(AutY ) x ~o(AutX°P) ,~oF(X,Y))
Bet 1 BG > g(~ AutY)
o
372
(Recall BG = B(~,G,~)).
3.8 Theorem: Let G be discrete. A homotopy G-map
A A ~: C(G,G,G) ~ C(~o(AUtY) ,~oF(X,Y) ,~o(AUtX))
from a homotopy G-space (X,~ o) to a homotopy G-space (Y,el) is
induced by an n-coherent homotopy G-map
A ~: Wnc (G,G,G) > C(AutY, F(X,Y) , AutX)
B (AutX °p)
~ O
h n ~ B ( * , A u t Y × AutX ° p , F ( X , Y ) )
~ lql g n + l
~ B(AutY)
iff there is a lift (up to homotopy)
f n+1 Bn+IGOP
Po
Bn(*,G × G°P,~)
Bn+IG .......
of (3.7) on the indicated skeletons.
Moreover, if fn+1 and gn+1 are obtained from WG-stuctures on X and
Y according to (3.1), y can be chosen to be compatible with these
structures.
The analogue of (3.2) is
3.9 Theorem: Given an ~-coherent homotopy G-map
A y: WC(G,G,G) • C(AutY, F(X,Y), AutX)
there exists a homotopy commutative diagram
X' f,
X
i x
y
y,
where f is the underlying map of ¥ (i.e. f = y((e,o - 1))),
373
i x and iy are the underlying maps of ~-coherent homotopy G-maps
which embed X and Y as strong deformation retracts into free
G-spaces X' and Y', and f' is a strict G-map. Moreover, if X and
Y are G-spaces there are homotopy inverses of i x and iy which are
G-equivariant.
This answers the realization problem for homotopy G-maps.
3.10 Extensions of Our results:
(1) The proofs will show that in the most interesting case of infinite
coherence our results hold for any well-pointed topological group G
of the homotopy type of a CW-complex. If G is not well-pointed we have
to substitute WG by WG', where G' is the monoid obtained from G by
attaching a whisker.
3.11 Theorem: If n = ~ all our results hold for a (well-pointed)
topological group G of the homotopy type of a CW-complex. In the
cases (3.4) and (3.5) well-pointed subgroups H of G of the homo-
topy type of a CW-complex are admitted if H c G are closed co-
fibrations.
(2) In the case of finite coherence, an analysis of our proofs gives
results similar to (3.1), (3.4), and (3.8) for finite-dimensional
CW-groups but with dimension shifts. The details are left to the
reader.
(3) It is not difficult to state and prove classificationresults of
the type of (3.3) in the relative cases.
(4) In (3.5) one often wants the stronger result that we have an H-
equivariant realization in the strong sense, i.e. a realization in the
category of H-spaces. We prove this in the case that X is H-free. If
this does not hold one has to take care of the fixed point structure
of X which makes the analysis more complicated. We shall deal with
this problem in a subsequent paper [S-V].
4. Basic properties 0~ the W-construction
The correspondence (fn,tn,...,fo) .... fn °fn-1°'''°fo defines a natural
transformation e: W .... Id. Pulling back a G-structure via ~ to a WG-
structure we can make the notion of an n-coherent homotopy G-map into
374
a genuine G-space (see (3.2)) formally precise. The same holds for
n-coherent homotopy G-actions extending genuine H-actions in (3.4).
4.1 Proposition: (I) e: W C . C is a homotopy equivalence (~n
morphism spaces).
(2) If C is well-pointed, c n = £1wnc: wnc . C is n-connected.
Proof: e has a natural, non-functorial section ~: C ~ WC sending f
to (f), and
ht(fn,tn,...,fl,tl,fo) = (fn,t . t n .... ,fl,t • tl,fo)
is a fibrewise deformation of ~ ~ e to the identity. This proves (I).
wr+Ic is obtained from wrc by attaching (r+1)-cubes Cr+2(A,B) x I r+1
along DCr+2(A,B) x I r+1 u Cr+2(A,B ) x ~I r+1 and products of those cubes
as upper faces of some higher dimensional cubes. Here DCr+2(A,B) is the
space of all strings (fr+2,...,fo) containing an identity. Since
DCr+2(A,B ) c Cr+2(A,B ) is a closed cofibration, the homotopy excision
theorem implies that wrc ~ wr+Ic is r-connected. Hence the inclusion
wnc ~ WC is n-connected. So (2) follows from (I).
Proposition 4.1 can be interpreted as follows: The relations in C hold
in WC up to a contractible choice of homotopies. An inspection of the
relations (2.1) shows that WC is obtained from the free category W°C
on the graph defined by C by putting back the relations up to com-
patible homotopies. The next result will show that WC is universal
with respect to the properties in (4.1).
Let V c wC be a subcategory, and Vn+I(A,B) c Cn+I(A,B ) x I n the subspace
of all elements respresenting morphisms in V. We call V an admissible
subeategory of WC provided each morphism in V that decomposes in WC
also decomposes in V, and
Vn+I(A,B) U Cn+I(A,B) x DI n U DCn+I(A,B ) x I n c Cn+I(A,B) x I n
is a closed cofibration for all n,A,B. Note that the empty subcategory
is admissible if C is well-pointed.
4.2 Proposition: Consider the diagram of categories and functors
V
wc ~ ~ A t c i
C F ~ B
375
Assume (i) V is admissible
(ii) L is a homotopy equivalence
(iii) K~ is a homotopy through functors from F ° (elV] to
L ° Hi.
Then there exist extensions H: WC ~ A and Kt: WC ..... B of H'
' such that Kt: F o e ~Lo H. Moreover, any two such extensions and K t
are homotopic relV.
The extensions are constructed by induction over the n-skeletons wnc.
For details see [B-V, p.84 ff]. This proves the universality of c.
4.3 Proposition: Consider the diagram of categories and functors
C > B
L t61
where wnD c WC is the subcategory generated by V and WnC, and
E n = ~IwnD.
Assume (O) C has discrete morphism spaces
(i) V is an admissible subcategory of WC
(ii) L is n-connected
(iii) K t' is a homotopy through functors from F° (E nIV) to
L oH '
Then there are extensions H: wnD ..... A and Kt: WnD .... B of H'
and K t' such that Kt: F ° en ~ L° H. Moreover, the restrictions of
any two extensions to wn-ID are homotopic rel V .
Note that the morphism spaces of wnD are CW-complexes so that (4.3) is
an immediate consequence of classical homotopy theory.
Another important result is the homotopy extension property of the
W-construction. We use the terminology of (4.3).
4.4 Proposition: Let V be an admissible subcategory of WC and let n
be a natural number or ~. Suppose we are given a functor
Fo: wnD -- E and a homotopy thrDugh functors Ht: V -- - E such
376
that H ° = FolV. Then there exists an extension F t of F O and H t.
This follows directly from the definition of an admissible subcategory.
We now turn to the problem of "realizing" a WC-space by a C-space: Let
Y: C ~ Top be a C-space. From (2.6) we deduce that any collection
of homotopy equivalences fA: XA . Y(A), A6 ob C, can be extended
to a homotopy C-map X • Y. In particular, the correspondence
A - X A extends to a WC-structure X. For the proof of (3.2) we need
the converse of this fact.
4.5 Proposition: There is a functor M from the category of WC-spaces
and homomorphisms to the category of C-spaces and homomorphisms
together with an ~-coherent homotopy C-map ix: X , MX with the
following properties
(i) ix(A): X(A) ~ MX(A) embeds X(A) as a strong deformation
retract into MX(A)~A £ ob C
(ii) Any ~-coherent homotopy C-map a: X - Y from a
WC-space X to a C-space Y factors uniquely as ~ = h ° ix, where
h: MX : Y is a homomorphism of C-spaces.
Proof: Define
MX(B) = i W(C x LT)((A,O), (B,I)) x X(A)/~ A
with the relation
(a ~b oc,x) ~ (s(a)ob, X(c) (x))
if a 6W(C x I) and c £W(C x O). The ~-coherent homotopy C-map i x is
given by the adjunctions of the projections
W(C x iT) ((A,O), (B,I)) x X(A) - ~ MX(B).
Its underlying map is
X(A) ........... ~ MX(A) x ~ ((id A , O ~ I ; x) .
The C-structure on MX is the obvious left action of C on MX, and the
universal property of i X follows from the construction.
It remains to show that X is a strong deformation retract of MX. For
this we filter MX(A) by skeletons F n. For convenience we use the
symbol
377
(fn'tn '''''fi+1'ti+1'fi'ti '''''fo ;x)
for the representative
((fn,idl),tn,...,(fi+1,idl) ,ti+1,(fi,O- I) ,ti,...,(fo,ido) ;x)
of an element of MX. Let K c MX(A) denote the space of all those
elements which have a representative of the form (fk,tk,...,fo;X).
In a first step we deform MX(A) into K. Since Fn_ I c FniS a closed
cofibration, it suffices to constr~t deformations of F n U K into
Fn_l u K. Observe that (fn,tn,...,fi,ti,...,fo;X) represents an
element in Fn_ 1 u K iff i = n, or some fj = id, j % i, or
(tn,...,tl) 60 x I n-1 u I x ~I n-1. Since the latter space is a deforma-
u K to u K tion retract of I n the required deformation of F n Fn_ I
exists. The deformation h t of K into X(A) is defined by
ht(~k,t k ..... fo;X) = (idA,t,fk,t k ..... fo;X) •
5. Proofs
Part of the proof of (3.1) in the case of n= ~ consists of constructing
a homomorphism WG --- AutX from a map BG ~ B(AutX), i.e. we have to
pass from the classifying space of a monoid back to the monoid itself.
One way of doing this is to compare the fibers of the "universal G-
fibration" PG: EG " - BG, where EG = B(~,G,G) is a free contractible
right G-space, and the path space fibration n: P(BG;~,BG) ~ BG. Here
P(X;A,B) denotes the space of Moore paths in X, starting in A and
ending in B. Its elements are pairs (m,r) 6 F~R+,X) x~+ such that
~(o) £ A, ~(r) 6 B, and ~(t) = ~(r) for t ~r.
The inclusion G c EG of the simplicial O-skeleton is an equivariant
map of right G-spaces. Using the G-structure, we define a monoid
structure on P(EG;e,G) by setting (p,s) + (v,r) = (m,r + s) with
(5.1) ~(t) O<t<r F
~(t) = i p(t-r) • v(r) r <t <r+s
The endpoint projection ~: P(EG;e,G) ~ G is a homomorphism. P(EG;e,G)
is the homotopy fiber of G c EG. Since EG is contractible, z is a
homotopy equivalence. Hence, from (4.2) we obtain
378
5.2 Proposition: If G is a well-pointed topological monoid, there is
a homotopy commutative diagram of homomorphisms
3 G WG > P (EG;e,G)
G
Moreover, ~G is natural up to homotopy with respect to homomor-
phisms G -- H.
Convention: Homotopies of homomorphisms or functors are always homo-
topies through homomorphisms or functors.
The last statement of (5.2) is a consequence of the uniqueness part of
(4.2) applied to the following diagram of homomorphisms
Wf WG ~ WH
I ~ p(f) e ,H)~H 1 ~G --~ P (EG;e,G) ....... > P(EH; --~ e H
f G ............. > H
Since nH ~ P(f) ~ ~G~H° ~H° Wf, both homomorphisms P(f) o ~G and
~H ° Wf lift f~ E G (up to homotopy) and hence are homotopic.
(5.2) together with the following well-known fact establishes the
comparison of fibers mentioned above.
5.3 Proposition: If G is a grouplike well-pointed topological monoid,
the homomorphism P(pG) : P(EG;,e,G) . ~BG: = P(BG;~, *) is a homo-
topy equivalence (as a map).
Remark: we call a monoid G grouplike if its multiplication admits a
homotopy inverse. If G is of the homotopy type of a CW-complex this
is equivalent to the usual definition that ~o G be a group [tD-K-P;
(12.7)]
Hence, for well-pointed grouplike monoids G we have a homomorphism
379
(5.4) JG: WG - > ~BG
which is Oa homotopy equivalence (as a map) and natural in G up to
homotopy.
Applying (4.2) twice we obtain homomorphisms 1 G und k G which are
homotopy equivalences (as maps)
1 G (5.5) WG ..... ~ WWG
~[eG ~ eWG
E G G < WG
WJ G k G > W~BG -~ WG
JG > ~BG
The uniqueness part of (4.2) implies that
(5.6) ~WG ° IG ~id k Go WJGo 1 G~id
Moreover, k G and 1 G are natural up to homotopy in G. For 1 G
clear from (4.2). For k G it follows from the diagram
W~Bf WnBG > WnBH
~I~ WG Wf ~ WH ~
~Bf ~BG > ~BH
this is
All these results hold for well-behaved monoids. But if X is too big,
AutX could be nasty. In this case we substitute it by the CW-monoid
R(AutX) where R is the topological realization of the simplicial
complex functor. The back adjunction R(AutX) ...... AutX is a homo-
morphism and a weak equivalence. Since in all our statements (in-
cluding (3.11)) BG is of the homotopy type of a CW-complex, each
map BG- ~ B(AutX) factors uniquely up to homotopy through BR(AutX).
Moreover, each homomorphism WG ~ AutX factors uniquely up to homo-
topy through R(AutX). This follows from the fact that (4.2) also holds
if L is a weak equivalence and mor C is of the homotopy type of a
CW-complex.
So from now on we assume that AutX is a CW-monoid.
380
5.7 Proofs. of (3.1) and (3.4) :
(3.1) follows from (3.4). We prove (3.4). Suppose ~: G ~ ~o(AUtX)
is induced by an n-coherent homotopy action y: WnG ........ AutX, compatible
with 8 ° elWH. Let wn~ c WG be the subcategory generated by WnG and WH.
Since WG is obtained from wnp by attaching cubes of dimensions greater
than n, the functor e defines a commutative square
WH c wn?
H c G
with e H an equivalence and e' n-connected (see 4.1). We obtain a map
of pairs
(Be',BeH) : (BWn~,BWH) - (BG,BH)
with Be H a homotopy equivalence and Be' (n+1)-connected. Hence the
inclusion (Bn+IG u BH,BH) c (BG,BH) factors up to homotopy
(BWnD,BWH)
Bn+IG U BH,BH) / (Bc',BeH)
(BG,BH)
where n is any chosen homotopy inverse of Be H. The composite
B(y u 6 o e) ~ Pn+1 is a required filler.
Conversely, suppose we are given a filler f. Since Bn+IHcBn+IG is a
cofibration we may assume that f and B~ together define a map
Bn+IG u BH .... B(AutX) , which we also denote by f. Consider the dia-
gram WJHO IH WH > W~BH
wn~ ..... W~(Bn+IGu BH) ~ W~B(AutX) ~ W(AutX) ~ AutX
l e n G
W~BG
E G ii WG
381
where i: Bn+IG o BH c BG is the inclusion. Since it is (n+l)-connected,
W~i and hence the composite eGO kG0 WSi is n-connected. By (4.3), F
exists. (5.6) and the naturality of k G provide homotopies
eAutXO kAutXO W~BB~ WJHo 1 H ~ EAutX o W~o kHO WJHO 1 H ~ 8o eH.
By (4.4), we can extend this homotopy to a homotopy of homomorphisms
from eAutX ° kAutX o Wgfo F to a functor y: WnD ~ AutX with
yIWH = S ~ e H.
5.8 Proof of (3.2) and (3.5) : (3.2) is a special case of (4.5). For
(3.5) we apply (4.5) to obtain a free H-space MHX and a free G-space
MGX together with an ~-coherent homotopy H-map iH: X ~ MHX and an
~-coherent homotopy G-map iG: X ~ MGX. Since W(H × £7) c W(G x [7) we
have a cofibration
j: MHX > MGX
which is H-equivariant. Since j o iH = iG as maps of spaces, j is a
homotopy equivalence and hence an H-equivariant homotopy equivalence,
because both spaces are H-free. By (4.5.2), the retraction r: MHX ~ X
can be chosen to be H-equivariant. If j-1 denotes an H-equivariant
homotopy inverse of j, the composite
,--I ro 3 : MGX > MHX > X
is the required H-equivariant map. By (2.5.2) its H-structure can be
extended to an ~-coherent homotopy G-map, because it is homotopy in-
verse to i G.
5.9 Proof of (3.3): Let l(y) : BG • ÷ B(AutX) be the lift obtained
from y: WG ~ AutX, and let a(f) : WG .... AutX be the functor induced
by f: BG , B(AutX). By construction
l(y) = By0 1G
a(f) = SAutX ° kAutX ° W~f ° WJGO i G
where 1 G is a chosen homotopy inverse of Be G . The diagram
382
W~I G WeB7
W~BG ....... m W~BWG
W~BG WWG Wy
kG ~WG
id y WG 7 WG
WnB(AutX)
l k A u t X
W (AutX)
eAutX
AutX
implies that a(l(y)) = y.
By definition, 1 o a: [BG,B(AutX)] - [BG,B(AutX) ] is the composite
BW~ w B~ [BG,B(AutX)] --~ [BW~BG,BW~B(AutX) ]------~- [BWG,B(AutX) ]~---[BG,B(AutX) ]
where [ , ] denotes homotopy classes,
and w[h] = [B(eAutX Q kAutX) o h ~ B(WJGO 1G) ].
Consider
Be n UBG BW~BG ~ B~BG > BG
B~ u B BW~B (AutX) > B~B (AutX) , (AutX)> B (AutX)
Clearly (I) commutes. By [M; (14.3) ] there is a homotopy equivalence
Ux: B~X • X, X a connected space of the homotopy type of a CW-complex,
which is natural up to homotopy. Hence BW~ is bijective. This proves
that 1 is inverse to a.
We have shown
5.10 Proposition: a: [BG,B(AutX)]
inverse i.
. [WG,AutX] is bijective with
Any WG-structure y: WG ~ AutX inducing ~: G . ~o(AUtX) determines
a free realization iy:X . : MyX by (4.5). Conversely, a free realization
f: X -- Y of ~ by (2.5) gives rise to a WG-structure p(f) on X, which
induces e. We shall show below that i and p induce maps
383
P
i: [WG,AutX] e Real(a) : p
where Real(e) is the set of equivalence classes of free realizations
of e and [WG,AutX]~ the set of h©motopy classes of WG-structures
inducing e.
If ~ = p(i7) , we are given ~-coherent homotopy G-maps
(X,y) M X (X,~) Y
having the same homotopy equivalence as underlying map. By (2.5), we
obtain a composite ~-coherent homotopy G-map (X,y) -- . (X,~) whose
underlying map may be chosen to be the identity (composites are de-
fined up to homotopy). Conversely, given a free realization f: X ....... Y,
we by (2.5) and (4.5.2) have a commutative diagram of w-coherent homo-
topy G-maps
(X,p (f))
/ \ h
M X Y P(f)
with h strictly G-equivariant. Since Mp(f)X and Y are free, h is a
G-equivariant homotopy equivalence. Hence i o p = id. So (3.3) is
proved once we have shown.
5.11Lemma: Two WG-structures e and 6 on X are homotopic iff there is
an ~-coherent homotopy G-map (X,e) ~ (X,~) with id x as under-
lying map.
Proof: Suppose ~ ~ 8. In (4.4) let C = G x i I and V be the subcategory
of WC generated by W(G x O) , W(G x I) and the morphism ((e,O-1)). We
extend the identity homotopy G-map (X,~) (X,e) and the homotopy on
V given by the constant homotopy on W(G x O) and ((e,O~l)), and by
~6 on W(G x I), to obtain an ~-coherent homotopy G-map (X,~) (X,8)
over id x-
Conversely, suppose id X has the structure of an ~-coherent homotopy
G-map y: W(G x i;) ...... Top from (X,~) to (X,6). For the rest of the
proof we have to recall the basic idea of the proof of (2.5). Let 16
be the category
384
0 I
J
A homotopy inverse of y is constructed by first extending the under-
lying map to a functor z: WIs ~ Top and then extending ~ andy to
9:W(G × IS) -- Top. The inclusion of L1 into Is as j defines the homo-
topy inverse. In our case we may choose u to be the constant functor.
Let C' be the full subcategory of W(G × IS) consisting of the object
O, and let C be obtained from C' by adding the relation
fn'tn "fo fn_l o ... o f ( '''" ) = fn ° n
if each of the fk is of the form (e,i) or (e,j). By our choice of ~,
the functor ~ induces a functor I: C • Top. The functors
Fo,FI: WG • C
Fo(gn,t n ..... go ) = ((gn,ido) ,tn,...,(go,ido))
F1(gn,tn,...,g O) = ((e,j) ,I, (gn,idl) ,t n ..... (go,idl) ,1,(e,i))
both make
WG C
G
commute and hence are homotopic (4.2). By construction, v o Fo =
and ~o FI = 8.
5.12 Proof of (3.6): By (3.4) and (3.5) we have to find a filler for
BB BH > B (AutX)
n t Bc~
BG ) B (~oAUtX)
The obstructions for its existence lie in Hn(BG,BH; {~n_2(AUtlX)]) ,
n~ 3, where AutlX is the component of the identity in AutX. By
[C;Cor.2.2], ~n_2(AutiX) is p-local. Hence the transfer ensures the
vanishing of Hn(BG,BH; {Zn_2(AutIX)}) for n ~ 3.
385
The idea of the proof of (3.8) is the same as the one of (3.]): We
have to pass from BG back to G (or rather WG) and from B(,,Gx H°P,x)
to X, where X is a left G-right H-space. For BG this has been done in
the proof of (3.1), for B(*,G x HOP,x) we compare the fiber sequence
X ~ B(*,G x H°P,x) ) BG x BH Op
with the fiber sequence
Fib • B(*,G × H°P,x) ) BG x BH Op
where Fib is the homotopy fiber. We represent Fib by a space having
a natural action of the Moore-loops ~BG x ~BH °p, and, similar to the
proof of (3.1), we construct a functor
WC(G,X,H) ~ C(~BG,Fib, (~BH°P) °p)
extending JG and (JHoP) Op
We proceed as far as possible in analogy to (3.1): Let G and H be well-
pointed monoids. Let EX = B(*,G x H°P,x), EGX = B(*,G,X) , and
EHx = B(*,H°P,x). The injections G -- G x Hop and HoP -- G x HoP make
EGX and EHx subspaces of EX with intersection X, the simplicial O-ske-
leton of all three spaces. We have pairings
EG x EHx ~ B(*,G x H°P,G x X) > B(*,G x HOP,x) = EX
EH °p x EGX ~ B(~,G x H°P,H °p x X) ) B(*,G x H°P,x) = EX
which commute on EGX N EHx = X and extend the pairing on the O-ske-
letons given by the G x H°P-action on X.
Let Sq c F~R+ x ~+,EX) x ~+x 5+ be the subspace of all "Moore-squares"
(w,r,s) in EX such that
w(r,u) 6 EGX for t ~ r and all u
w(t,u) = w(t,s) £ EHx for s Au and all t .
Consequently, w(r,s) 6 X. We define a left P(EG;e,G) x P(EH°P;e,H °p) -
action on Sq by
((m,l),(~,k)) * (w,r,s) = (v,r +k,s+l)
where
386
v(t,u) =
l w(t,u)
v(t- r) " w(r,u)
~(u- s) • w(t,s)
~(u- s) • ~(t- r) • w(r,s)
0~tSr , OSuSs
r ~t~r+k, OSuSs
0stSr , s~uSs+l
r~t~r+k, s~uSs+l
where •
s+l
denotes the pairings
in EHx in X
w • w(-,s)
in EHx
in X
~n EGX
w" ~ "w(r,s)
" w(r,-)
in EGX
r r+k
The endpoint projection ~: Sq ~ X, (w,r,s) ~ w(r,s) together with
the endpoint projections P(EG;e,G) ~ G and P(EH°P;e,H °p) ~ H °p
define a functor
~: C(P(EG;e,G) ,Sq, (P(EH°P;e,H°P) °p) . C (G,X,H)
which is a homotopy equivalence (on morphism spaces). Hence we obtain
the analogue of (5.2).
5.13 Lemma: If G and H are well-pointed topological spaces and X is a
left G-right H-space, there is a diagram of categories and
functors
WG u WH > P(EG;e,G) u P(EH°P;e,H°P) °p
1 .... °o 1 ~G ~ t3HO p)
JX op WC(G,X,H) ....... ~ C(P(EG;e,G) ,Sq, (P(EH°P;e,H °p) )
C(G,X,H)
where ~ is the functor of (5.2).
387
Let PG: EX .... BG and pH: EX . BH Op. As model for the h-fiber of
H) (pG, p : EX .... BG x BH op we take the space
Fib(PG,pH ) = {(m,9,z) 6 P(BG;BG,*)xP(BH°P;BH°P,~)xEX;m(O) =pG(z),~(O)=PH(z)]
There is an obvious left action of ~BG x ~BH °p on Fib(PG,pH) . The map
Sq .. Fib(PG,p H) sending (w,r,s) to the triple
(pG o w(O,-) ;p Ho w(-,O),w(O,O)) together with the maps of (5.3) define
a functor Sq(pG,PH) : C(P(EG;e,G) ,Sq, (P(H°P;e,H°P) °p) , C(~BG,Fib(PG,PH),(~BH°P) °p)
5.14 Lemma: If G and H are group-like, Sq(pG,pH) is a homotopy
equivalence.
This follows immediately from (5.3) and [P; Thm.], where Fib(PG,p H)
is proved to be homotopy equivalent to X.
Hence if G and H are grouplike and well-pointed we obtain functors
Jx: WC(G,X,H) ~ C(~BG,Fib(PG,pH) ,(DBH°P) °p)
KX: WC(~BG,Fib(PG,pH) , (~BH°P) °p) m WC(G,X,H)
LX: WC (G,X,H) ~ WWC (G,X,H)
like in (5.4) and (5.5). They are homotopy equivalences, natural up to
homotopy in (G,X,H) , and satisfy (5.6).
5.15 Proof of (3.8): Suppose A A e: C(G,G,G) ............. C(~o(AUtY),ZoF(X,y) ,~o(AUtX)) is induced by an n-
coherent homotopy G-map A
¢: wnC(G,G,G) , C(AutY,F(X,Y),AutX) . Then ¢ defines a map
(compare (3.7))
BY o
(5.16) BWnG °p ) B(AutX °p)
^ By B(* ,WnG x WnG°P,wnG) ~ B ( * , A u t Y x A u t X ° P , F ( X , y ) )
BWnG ,, By 1 ~ B (AutY)
which sits over (3.7). By [P; Thm.] the rows in the following diagram
are h-fibration sequences
388
A WnG
len
A G
B (* ,WnG x WnG Op,Wn~) - -
B(e~) n
> B(*,G x G°P,~) ....
B(WnG x WnG °p)
I BE n
B(G x G °p) Hence B(E~) n is n-connected because E n is n-connected and Be n is (n+1)-
connected. So there exist maps
kn: Bn(*,G x G°P,~) ~ B(*,WnG x WnG°P,Wn~)
rn+1: Bn+IG > BWnG
~) . . . . • such that B( n kn Jn and B~ n rn+ I ±n+1' where In+I:Bn+IGcBG and
Jn: Bn(*'G x G°P,~) c B(*,G x G°P,~). The diagram
op rn+ I
Bn+IGOP ~ BwnGOP
n nop n A Bn(*,G x G°P,~) > B(*,WnG x W G ,W G)
lPl rn+1 IPl
Bn+IG ~ BWnG
commutes up to homotopy because
Be n ) :[Bn(*,G~ G°P,~) BWnG] > [Bn( * G x G°P,~) ,BG] t t
Is bijective. Together with (5.16) it provides the required lift.
: Bn+IG°P . . B(AutX°P) , Conversely, suppose we are given lifts fn+l^
gn+l:Bn+IG -- B(AutY) , and hn:Bn(*,G x Gop,G) ---~ B(*,AutYxAutX°P,F(X,Y))
as in (3.8). The inclusions of skeletons and the triple (gn+1,hn,fn+1)
give rise to maps of h-fibration sequences
Fib (PG'pG) .....
f' w Fib(j) !
Pibn (PG ,pG) I
I Fib (h) I
'~ x Fib (py,p)
G°P ~" B(*,G x , ) ~BG x BG °p
~I I . op Jn in+1 × in+1
Bn( *, G × G°P,~) > Bn+IG x Bn+IG Op
i h n lgn+ I x fn+1
B(*,AutY x AutX °p,F(X,Y))--->B(AutY) x B(AutX °p)
389
We take the model described above as h-fiber. This diagram in turn
defines functors
C(2Bn+IG,Fibn(pG,pG),(2Bn+IG°P)°P )
c (~m ,r ib (pc,p c) , (~m °p) op)
J o op
C (2B(AutY),Fib(py,pX), (2B(AutX P) )
Since in+ I is (n+1)-connected and Jn is n-connected, Fib(j) is
n-connected. Hence the functor S is n-connected. The rest of the proof
now is exactly the same as in (5.7).
(517) The proof of (3.9) is just another application of (4.5).
6. Final remarks
The methods of § 5 are related to the theories of H -maps of topo-
logical monoids and G -maps of G-spaces in the sense of Fuchs [F2].
An analysis of the definitions (1.3) and (1.4) of [F2] shows that an
H -map from a monoid G to a monoid H can be interpreted as a homo-
morphism FG • H and a G -map from a G-space X to an H-space Y as
a "functor" FC(G,X, {e}) ~ C(H,Y,{e}) , where FC is the semicategory
(i.e. category without identities) obtained from a category (or semi-
category) C in the same way as WC but with relations (2.1.2),-,(2.1.4)
dropped. For our purposes we need the stronger structure WC.
If C is a well-pointed category, with our methods it is easy to show
that a "functor" FC . D into a category D is homotopic to a "functor"
which factors through the projection "functor" FC . WC. Hence our
results in § 5 give quick proofs of many results of IF1], [F2], [F3]
and make explicit constructions unnecessary. In particular, the
preparations for the proof of (3.8) in § 5, applied to the case H= {e}
where Moore squares may be replaced by the more familiar Moore paths,
can be used to correct a flaw in [F3; Section 5].
[B-V]
[c]
[tD-K-P]
[D-K]
[FI]
[F2]
[F3]
[M]
[0]
[p]
Is-v]
[Vl]
[V2]
[z]
390
References
J.M. Boardman and R.M.Vogt, Homotopy invariant algebraic
structures on topological spaces, Springer Lecture Notes
in Math. 347 (1973)
G. Cooke, Replacing homotopy actions by topological actions,
Trans. Amer. Math. Soc. 237 (1978), 391-406
T. tom Dieck, K.H. Kamps, and D. Puppe, Homotopietheorie,
Springer Lecture Notes in Math. 157, (1970)
W. Dwyer and D. Kan, Equivariant homotopy classification,
J. Pure and Applied Algebra 35 (1985), 269-285
M. Fuchs, Verallgemeinerte Homotopie-Homomorphismen und
klassifizierende R~ume, Math. Ann. 161 (1965), 197-230
~ , Homotopy equivalences in equivariant topology,
Proc. Amer. Math. Soc. 58 (1976), 347-352
~ , Equivariant maps up to homotopy and Borel spaces,
Publ. Math. Universitat Aut6noma de Barcelona 28 (1984),
79-102
J.P. May, Classifying spaces and fibrations, Memoirs A.M.S.
155 (1975)
J.F. Oprea, Lifting homotopy actions in rational homotopy
theory, J. Pure and Applied Algebra 32 (1984), 177-190
V. Puppe, A remark on homotopy fibrations, Manuscripta
Math. 12 (1974), 113-120
R. Schw~nzl and R.M. Vogt, Relative realizations of homo-
topy actions, in preparation
R.M. Vogt, Convenient categories of topological spaces for
homotopy theory, Arch. der Math. 22 (1971), 545-555
~ , Homotopy limits and colimits, Math. Z. 134 (1973),
11-52
A. Zabrodsky, On George Cooke's theory of homotopy and
topological actions, Canadian Math. Soc. Conf. Proc.,
Vol. 2, Part 2 (1982), 313-317
EXISTENCE OF COMPACT FLAT
RIEMANNIAN MANIFOLDS WITH THE
FIRST BETTI NUMBER EQUAL TO ZERO
AndrzeJ Szczepa6ski
Gda~sk, Poland
0. Let M n be a compact flat Riemannian manifold of dimension n .
From Bieberbach's Theorems (see [3,8]) we know that its fundamental
group nl(M) = F has the following properties:
1) F is a torsion free, discrete and cocompact subgroup of E(n) ,
the group of isometries of R n
In particular, F acts freely and properly discontinuously as
a group of Euclidean motions
2) There exists a short exact sequence
0 - Z n ~ r - G - 1 (*)
where Z n is a maximal abelian subgroup in r and G is finite.
The sequence (*) defines by conjugation a faithful representation
p : G - GL(n,Z) and is classified by an element ~ E H~(G,Z n) .
Lemma 0.1 [2]. Let Z n be a G-module. The extension of G by Z n
corresponding to ~ E H2(G,Z n) is torsion free if and only if
res~ ~ 0 , where H runs over representatives of conjugacy classes
of subgroups of prime order •
We have the following construction due to E. Calabi
Theorem 0.2 [1,8]. If M is an n-dimensional flat manifold with
b1(M ) = q > 0 then there exist an (n-q)-dimensional flat manifold N
and a finite abelian group F of affine automorphisms of N of rank
q so that
M = N × Tq/F ,
where T q is a flat q-torus on which F acts by isometries •
This construction suggests a programme for an inductive classifi-
cation of flat manifolds with positive first Betti number. Those with
b I = 0 must necessarily be handled separately.
Remark 0. 3 . It can be proved ~5] that b1(M) = 0 if and only if
dimQ[Qn] G = 0 , where G acts on Z n by con0ugation in the short
exact sequence
392
0 - Z n -- ~I(M) - G - I
and
Qn = Z n ®Z Q "
Definition 0. 4 [5]. Let H be finite group. We say H is primitive
if H is the holonomy group of a flat manifold M with bl(M ) = 0 .
Recently H. Hiller and C.H. Sah L5] have determined the primitive
group s.
.Theorem o.>. A finite group H is primitive if and only if no cyclic
Sylow p-subgroup of H has a normal complement •
In this note we shall consider properties of the short exact se-
quence (*) for G = Z n (cyclic), G = D n (dihedral), G = Q(2 n) (gene-
ralized quaternion 2-group).
1. Let g(G) denote the smallest degree of a faithful integral
representation of G . It is easy to see that such a "minimal" integral
representation has no fixed points. Therefore we can ask the following
question :
Question 1.1. Suppose 0 ~ Z n ~ F - G ~ I is a short exact
sequence, such that the integral representation induced by conjugation
G - GL(n,Z) is "minimal" and faithful.
Can r be a fundamental group of a flat manifold?
Conjecture 1.2. Suppose 0 - Z n ~ r - G ~ 1 is a short
exact sequence and the integral representation induced by conjugation
is irreducible and faithful. Then F is not a fundamental group of a
flat manifold.
For generalized quaternion 2-groups
question coincide.
Now we formulate our main result.
Theorem 1.~. If G = Z n , G = D n ,
the question 1.1 is negative.
Proof.
a)
b)
Q(2 n)
G = Q(2 n)
the conjecture and our
then the answer to
I. Let G = Z n be a cyclic group. The number g(Z n) is equal~4~:
g(Zpk) = pk pk-1 for any k , where p-prime number
if m and n are relatively prime then g(Zm, n) = g(Zm)+g(Z n) ,
unless m = 2 and n is odd, in which case g(Z2n) = g(Z n)
From this and from the fact that the representation of Z n of de-
393
gree g(Z n) has no fixed points we have that H2(Zn,Z g(zn)) - 0 . Now
the theorem follows from lemma 0.1.
2. Let G = D n = (x,ylx n = 1,yxy -1 = x-l,y 2 = 1) . We shall sketch
the proof that
g(D n) = g(Z n) (**) k
It is well known E7] that g(Dp) = g(Zp) = p-1 For n = p
(k > 1) the result (**) follows from the inclusion Dpk o Dpk-1
and theorem about the dimension of the induced representation. Finally
for an arbitrary n the equality (**) follows from the first part of
the proof /for cyclic groups/ and the definition of the Dihedral group.
Now we may consider a homomorphism:
reszDn n H2(Dn,Z g(Dn) ) H2(Zn,Z g(Dn) ) : ~ = 0
of abelian groups where the second one is equal to zero by (**).
The theorem follows from lemma 0.1.
3. Let G = Q(2n), a generalized quaternion 2-group. It is well
known that g(Q(2n)) = 2 n . From the preprint of E6] it can be proved
that minimal dimension of a flat manifolds with b I = 0 and Q(2 n)
as holonomy group is equal to 2n+3 . It completes the proof of the
theorem |
REFERENCES:
~1] CALABI, E.: Closed locally euclidean four dimensional manifolds, Bull. Amer. Math. Soc. 63, 135 (1957)
L2] CHARLAP, L.S.: Compact flat Riemannian manifolds I. Ann. Math. 81, 15-30 (1965)
C3] FARKAS, D.R.: Crystallographic groups and their mathematics. Rocky mountain J. Math. li. 4.511-551 (1981)
~4] HILLER, H. : Minimal dimension of flat manifolds with abelian holonomy - preprint
~5] HILLER, H., SAH, C.H.: Holonomy of flat manifolds with b I = O , to appear in the Quaterly J. Math.
~6] HILLER, H., MARCINIAK, Z., SAH, C.H., SZCZEPANSKI, A.: Holonomy of flat manifolds with b I = O,II - preprint
~7] PU, L.: Integral representations of non-abelian groups of order pq , Mich. Math. J. 12, 231-246 (1965)
~8] WOLF, J.A.: Spaces of constant curvature, Boston, Perish 1974
WHICH GROUPS HAVE STRANGE TORSION?
Steven H. Weintraub Department of Mathematics Louisiana State University
Baton Rouge, Louisiana 70803-4918 U.S.A.
The purpose of this note is to ask what we think is a natural question, and
to provide some examples which suggest that it should have an interesting answer.
I. STRANGE TORSION
DEFINITION i. A group G has strange p-torsion if
a) H*(G;Z) has p-torsion, but
b) G does not have an element of order p.
It has strange torsion if it has strange p-torsion for some p. (We take coeffi-
cients in Z as a trivial ZG-module.)
There are admittedly some reasonably natural groups which have strange
torsion:
EXAMPLE -4. Let B k be Artln's braid group on k strands (in ~) and let
B be the direct limit B~ = ---+lim B k. Then B~ is torslon-free but for every
prime p, Hi(B ;Z) has p-torsion for arbitrarily large i. This is a result of
F. Cohen [CLM, III. Appendix].
EXAMPLE -3. If G is a one-relator group, then Hi(G;Z) may have strange
torsion for i = 2 (but not for i ¢ 2). This follows from Lyndon's computation
[Ly].
EXAMPLE -2. (A special case of example -3.) G = the fundamental group of a
non-orlentable surface of genus g ~ i, or of the mapping torus of f: S 1 --+ S 1 n
by z--+ z , n # 0,1,2.
EXAMPLE -1. Many Bieberbach groups, e.g. the following group considered by
A. Szczepanski: 1 --+ Z 3 --~ G--+ Z/2 + Z/2 --+ 1 where the two generators a
and b of Z 2 + Z 2 act on Z 3 by a(x,y,z) = (x,-y,-z), b(x,y,z) = (-x,y,-z).
On the other hand, here are some examples of groups which do not have
strange torsion:
395
EXAMPLE 0. All finite groups. (The existence of the transfer implies that
the cohomology of a finite group is annihilated by multiplication by the order of
the group.)
EXAMPLE I. Any subgroup of SL2(Z) or PSL2(Z). This follows from the
following well-known theorem (There are some polnt-set theoretical conditions
here, which we suppress.):
THEOREM I. Let a group G act on a contractible space X with the iso-
tropy group G x of x finite for every x e X. Then if p is prime to IGxl
for all x, H*(G;Zp) is isomorphic to H*(X/G;Zp).
Proof: Let EG be a contractible space on which G acts freely. Then G
acts freely on X × EG by the diagonal action, so H*(G;Zp) = H*((X x EG)/G;Zp).
Let f: X--+ Y = X/G and 7: X x^ EG = (X × EG)/G--+ Y be the projections.
* -I * H~(pt;Zp) Then H (z (y);Zp) = H (BGx;Z p) = (by example 0) for all y, where
y = f(x), so by the Vietoris-Begle mapping theorem, H~(G;Zp) = H~(X/G;Zp).
COROLLARY 2. If H*(X/G;Z) has no p-torsion, G has no strange p-torsion.
(In particular, this holds for G acting on ~ in an orientation-preserving
way.)
The following examples require a lot more work:
EXAMPLE 2. G = SP4(Z) and G = F(2), the principal congruence subgroup of
level 2, as well as PSP4(Z) and PF(2). This is proven in [LW].
EXAMPLE 3. G = SL3(Z). This follows from Soule's computation [So].
The interesting thing about example n, n > 0, is that the non-existence
of strange torsion is proven geometrically, by studying a G-action on a con-
tractible space X satisfying the hypothesis of Theorem I. Thus we ask the
question:
QUESTION I. Which groups have strange torsion?
2. VERY STRANGE TORSION
S. Jackowskl has suggested that it might be better to ask about very strange
torsion.
DEFINITION 2. A group G has very strange p-torsion if
a) HI(G:Z) has p-torslon for i arbitrarily large, but
b) G does not have an element of order p.
It has very strange torsion if it has very strange p-torslon for some p.
396
In this connection we have the following well-known result.
THEOREM 2. Let G be a group with vcd(G) < ~. Then G has no very
strange torsion.
Proof. Recall the following from Is]: A group G' has finite cohomologi-
cal dimension n = cd(G') < ~ if for every module M, Hi(G';M) = 0 for i > n.
A group G has virtually finite cohomological dimension, vcd(G) < ~, if G
has a subgroup G' of finite index with cd(G') < ~. In this case we set
vcd(G) = n = cd(G'), and vcd(G) is well defined (i.e. independent of the
choice of G').
If n = vcd(G) < ~, we have Farrell cohomology ~i(G:~) defined, with the
property that ~i(G:Z) = Hi(G:Z) for i > n. Furthermore, by [B, p. 280, ex. 2]
~i(G:Z) has p-torsion only for primes for which G has an element of order p,
so G has no strange torsion above dimension n.
There are many important classes of groups G for which vcd(G) < ~. A host
of examples are given in [B, Sec. VIII.9]. In particular, all arithmetic groups
G satisfy vcd(G) < ~.
Example n has finite vcd for n ~ -2. Example -3 has strange torsion but
not very strange torsion, while example -4 has very strange torsion. Thus we
conclude with the question:
~UESTION 2. Which groups have very strange torsion?
References
[B] Brown, K° Cohomology of Groups. Springer, Berlin, 1982.
[CLM] Cohen, F. R., Lada, T. J., and May, J. P. The hom019gY of iterated loop spaces, Lecture notes in math. no. 533, Springer, Berlin, 1976.
[LW] Lee, R., and Weintraub, S. H. Cohomology of SP4(Z) and related groups and spaces, Topology 24(1985), 391-410.
[Ly] Lyndon, R. C. Cohomology theory of groups with a single defining relation, Ann. Math. 52(1950), 650-665.
[Q] Quillen, D. The spectrum of an equivariant cohomology ring, Ann. of Math. 94(1971), 549-602.
[S] Serre, J. -P. Cohomologie des groupes discrets, in Prospects!n Mathematics, Ann. of Math. Studies vol. 70, Princeton Univ. Press, Princeton NJ, 1971, 77-169.
[So] Soul~, C. Cohomology of SL3(Z) , Topology 17(1978), 1-22.