simplectic origami

8/11/2019 Simplectic Origami

1/43

Symplectic Origami

Citation Cannas da Silva, A., V. Guillemin, and A. R. Pires. SymplecticOrigami. International Mathematics Research Notices(December 2, 2010).

As Published http://dx.doi.org/10.1093/imrn/rnq241

Publisher Oxford University Press

Version Author's final manuscript

Accessed Fri Aug 01 12:20:10 EDT 2014

Citable Link http://hdl.handle.net/1721.1/80289

Terms of Use Creative Commons Attribution-Noncommercial-Share Alike 3.0

Detailed Terms http://creativecommons.org/licenses/by-nc-sa/3.0/

The MIT Faculty has made this article openly available. Please sharehow this access benefits you. Your story matters.
http://dx.doi.org/10.1093/imrn/rnq241http://hdl.handle.net/1721.1/80289http://creativecommons.org/licenses/by-nc-sa/3.0/http://libraries.mit.edu/forms/dspace-oa-articles.htmlhttp://libraries.mit.edu/forms/dspace-oa-articles.htmlhttp://creativecommons.org/licenses/by-nc-sa/3.0/http://hdl.handle.net/1721.1/80289http://dx.doi.org/10.1093/imrn/rnq241


2/43

arXiv:0909

.4065v2

[math.SG

]21Feb2011

SYMPLECTIC ORIGAMI

A. CANNAS DA SILVA, V. GUILLEMIN, AND A. R. PIRES

Abstract. An origami manifold is a manifold equipped with aclosed 2-form which is symplectic except on a hypersurface whereit is like the pullback of a symplectic form by a folding map and its

kernel fibrates with oriented circle fibers over a compact base. Wecan move back and forth between origami and symplectic mani-folds using cutting (unfolding) and radial blow-up (folding), mod-ulo compatibility conditions. We prove an origami convexity theo-rem for hamiltonian torus actions, classify toric origami manifoldsby polyhedral objects resembling paper origami and discuss exam-ples. We also prove a cobordism result and compute the cohomol-ogy of a special class of origami manifolds.

1. Introduction

This is the third in a series of papers onfoldedsymplectic manifolds.The first of these papers [CGW] contains a description of the basic lo-cal and semi-global features of these manifolds and of the folding andunfoldingoperations; in the second [C] it is shown that a manifold isfolded symplectic if and only if it is stable complex and, in particular,that every oriented 4-manifold is folded symplectic. (Other recent pa-pers on the topology of folded symplectic manifolds are [Ba] and[vB].)

In this third paper we take up the theme of hamiltonian group actionson folded symplectic manifolds. We focus on a special class of foldedsymplectic manifolds which we call origami manifolds. (Jean-ClaudeHausmann pointed out to us that the term origami had once beenproposed for another class of spaces: what are now known as orbifolds.)For the purposes of this introduction, let us say that a folded symplecticmanifold is a triple (M, Z , ) where M is an oriented 2n-dimensional

Date: Revised November 15, 2010. To appear in Int. Math. Res. Not.First published online December 2, 2010. DOI: 10.1093/imrn/rnq241.

1
http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2http://arxiv.org/abs/0909.4065v2


3/43

2 A. CANNAS DA SILVA, V. GUILLEMIN, AND A. R. PIRES

manifold, a closed 2-form and Z i

M a hypersurface. Foldedsymplectic requires that be symplectic on MZ and that the

restriction of to Zbe odd-symplectic, i.e.

(i)n1 = 0 .From this one gets on Z a null foliation by lines and (M, Z , ) isorigami if this foliation is fibrating with compact connected orientedfibers. In this case one can unfold M by taking the closures of theconnected components ofM Zand identifying boundary points onthe same leaf of the null foliation. We will prove that this unfoldingdefines a cobordism between (a compact) M and a disjoint union of(compact) symplectic manifolds Mi:

(1) M i

Mi .

Moreover, ifMis a hamiltonianG-manifold we will prove that the Misare as well. Theorigamiresults of this paper involve reconstructing themoment data ofM (and in the toric case M itself) from the momentdata of the Mis.

Precise definitions of folded symplectic and origami are given inSection 2.1. In 2.2 we describe in detail the unfolding operation (1)and in2.3 how one can refold the terms on the right to reconstruct Mvia a radial blow-up operation. Then in Sections2.4and 2.5 we prove

that folding and unfolding are inverse operations: unfolding followedby folding gives one the manifold one started with and vice-versa.

We turn in Section3 to the main theme of this paper: torus actionson origami manifolds. In 3.1 we define for such actions an origamiversion of the notion of moment polytope, which turns out to be acollection of convex polytopes with compatibility conditions, orfoldinginstructionson facets. We then concentrate in Section3.2 on the toriccase and prove in3.3 an origami version of the Delzant theorem. Moreexplicitly, we show that toric origami manifolds are classified byorigamitemplates: pairs (P, F), whereP is a finite collection of oriented n-dimensional Delzant polytopes and

Fa collection of pairs of facets of

these polytopes satisfying:

(a) for each pair of facets{F1, F2} F the corresponding poly-topes inP have opposite orientations and are identical in aneighborhood of these facets;

(b) if a facet occurs in a pair, then neither itself nor any of itsneighboring facets occur in any other pair;


4/43

SYMPLECTIC ORIGAMI 3

(c) the topological space constructed from the disjoint union of allthe i Pby identifying facet pairs inF is connected.

Without the assumption that Mbe origami, i.e., that the null folia-tion be fibrating, it isnotpossible to classify hamiltonian torus actionson folded symplectic manifolds by a finite set of combinatorial data;why not is illustrated by example 3.11. Nonetheless, Chris Lee hasshown that a (more intricate) classification of these objects by momentdata ispossible at least in dimension four [Lee]. This result of his wefound very helpful in putting our own results into perspective.

Throughout this introduction we have been assuming that our ori-gami manifolds are oriented. However, all the definitions and resultsextend to the case of nonorientable origami manifolds, and that is how

they will be presented in this paper. In particular, the notion of origamitemplate explained above becomes that of Definition3.12, which dropsthe orientations of the polytopes inP and allows for sets of singlefacets inF. Moreover, as we show in Section 3, some of the mostcurious examples of origami manifolds (such as RP2n and the Kleinbottle) are nonorientable.

The final two sections of this paper contain results that hold onlyfor oriented origami manifolds.

In Section4we prove that (1) is a cobordism and, in fact, an equi-variant cobordism in presence of group actions. We show that thiscobordism is asymplectic cobordism, i.e., there exists a closed two formon the cobording manifold whose restriction to M is the folded sym-plectic form on Mand on the symplectic cut pieces is the symplecticform on those manifolds. Moreover, in the presence of a (hamiltonian)compact group action, this cobordism is a (hamiltonian) equivariantcobordism. Using these results and keeping track of stable almost com-plex structures, one can give in the spirit of [GGK] a proof that theequivariant spin-Cquantization ofMis, as a virtual vector space (andin the presence of group actions as a virtual representation), equal tothe spin-Cquantizations of its symplectic cut pieces. However, we willnot do so here. We refer the reader instead to the proof of this resultin Section 8 of [CGW], which is essentially a cobordism proof of thistype.

Section5 is devoted to the origami version of a theorem in the stan-dard theory of hamiltonian actions: In it we compute the cohomologygroups of an oriented toric origami manifold, under the assumptionthat the folding hypersurface be connected.


5/43


Origami manifolds and higher codimension analogues arise naturallywhen converting hamiltonian torus actions on symplectic manifolds into

free actions by generalizations ofradial blow-up along orbit-type strata.We intend to pursue this direction to obtain free hamiltonian torus ac-tions on compact presymplectic manifolds, complementing recent workby Yael Karshon and Eugene Lerman on non-compact symplectic toricmanifolds [KL].

2. Origami Manifolds

2.1. Folded symplectic and origami forms.

Definition 2.1. A folded symplectic formon a 2n-dimensional mani-

foldMis a closed 2-form whose top power n

vanishes transversallyon a submanifoldZ, called the folding hypersurfaceor fold, and whoserestriction to that submanifold has maximal rank. The pair (M, ) isthen called a folded symplectic manifold.

By transversality, the folding hypersurface Zof a folded symplecticmanifold is indeed of codimension one and embedded. An analogue ofDarbouxs theorem for folded symplectic forms [CGW, M] says thatnear any point pZthere is a coordinate chart centered at p wherethe form is

x1dx1

dy1+dx2

dy2+. . . +dxn

dyn .

Let (M, ) b e a 2n-dimensional folded symplectic manifold. Leti : Z

M be the inclusion of the folding hypersurface Z. Awayfrom Z, the form is nondegenerate, so n|MZ= 0. The inducedrestrictionihas a one-dimensional kernel at each point: the line fieldV onZ, called thenull foliation. Note thatV =T ZEiT MwhereE is the rank 2 bundle over Zwhose fiber at each point is the kernelof.

Remark 2.2. When a folded symplectic manifold (M, ) is an orientedmanifold, the complement M Zdecomposes into open subsets M+

wheren

>0 and M where n


6/43


Definition 2.3. Anorigami manifoldis a folded symplectic manifold(M, ) whose null foliation is fibrating with oriented circle fibers, ,

over a compact base,B . (It would be natural to extend this definitionadmittingSeifert fibrationsand orbifold bases.)

ZB

The form is called an origami form and the null foliation, i.e., thevertical bundle of , is called the null fibration.

Remark 2.4. When an origami manifold is oriented we assume thatany chosen orientation of the null fibration or any principal S1-actionmatches the induced orientation of the null foliation V. By definition,

a nonorientable origami manifold still has an orientable null foliation.

Notice that, on an origami manifold, the base B is naturally sym-plectic: As in symplectic reduction, there is a unique symplectic formB onB satisfying

i= B .

Notice also that the foldZis necessarily compact since it is the totalspace of a circle fibration with compact base. We can choose differentprincipalS1-actions onZby choosing nonvanishing (positive) verticalvector fields with periods 2.

Example 2.5. Consider the unit sphereS2n in euclidean space R2n+1 Cn R with coordinatesx1, y1, . . . , xn, yn, h. Let 0 be the restrictiontoS2n ofdx1 dy1+ . . . + dxn dyn= r1dr1 d1+ . . . + rndrn dn.Then 0 is a folded symplectic form. The folding hypersurface is theequator sphere given by the intersection with the plane h = 0. Thenull foliation is the Hopf foliation since

1+...+

n

0=r1dr1 . . . rndrnvanishes on Z, hence a null fibration is S1

S2n1 CPn1. Thus,(S2n, 0) is an orientable origami manifold.

Example 2.6. The standard folded symplectic form 0 on RP2n =

S2n/Z2 is induced by the restriction to S2n of the Z2-invariant form

dx1 dx2 + . . . + dx2n1 dx2n in R2n+1 [CGW]. The folding hypersur-face is RP2n1 {[x1, . . . , x2n, 0]}, a null fibration is the Z2-quotientof the Hopf fibration S1

RP2n1 CPn1, and (RP2n, 0) is anonorientable origami manifold.


7/43


The following definition regards symplectomorphism in the sense ofpresymplectomorphism.

Definition 2.7. Two (oriented) origami manifolds (M, ) and (M, )are symplectomorphic if there is a (orientation-preserving) diffeomor-

phism : MM such that = .This notion of equivalence between origami manifolds stresses the

importance of the null foliation being fibrating, and not a particularchoice of principal circle fibration. We might sometimes identify sym-plectomorphic origami manifolds.

2.2. Cutting. The folding hypersurface Zplays the role of an excep-tional divisor as it can be blown-down to obtain honest symplecticpieces. (Origami manifolds may hence be interpreted as birationallysymplectic manifolds. However, in algebraic geometry the designationbirational symplectic manifoldswas used by Huybrechts [H] in a dif-ferent context, that of birational equivalence for complex manifoldsequipped with a holomorphic nondegenerate two-form.) This process,calledcutting (or blowing-downor unfolding), is essentially symplecticcutting and was described in[CGW,Theorem 7] in the orientable case.

Example 2.8. Cutting the origami manifold (S2n, 0) from Exam-

ple2.5produces CPn and CPn each equipped with the same multipleof the Fubini-Study form with total volume equal to that of an originalhemisphere,n!(2)n.

Example 2.9. Cutting the origami manifold (RP2n, 0) from Exam-ple2.6produces a single copy ofCPn.

Proposition 2.10. [CGW]Let(M2n, ) be an oriented origami man-ifold.

Then the unions M+ B and MB each admits a structure of2n-dimensional symplectic manifold, denoted(M+0 ,

+0) and(M

0 ,

0)

respectively, with+0 and0 restricting to onM+ andMand with anatural embedding of(B, B) as a symplectic submanifold with radially

projectivized normal bundle isomorphic to the null fibrationZ B.

The orientation induced from the original orientation onMmatchesthe symplectic orientation on M+0 and is opposite to the symplecticorientation onM0 .


8/43


The proof relies on origami versions of Mosers trick and of Lermanscutting. Lermans cutting applies to a hamiltonian circle action, de-

fined as in the symplectic case:Definition 2.11. The action of a Lie group G on an origami mani-fold (M, ) is hamiltonian if it admits a moment map, : M g,satisfying the conditions:

collects hamiltonian functions, i.e., d, X = X#,Xg:= Lie(G), where X# is the vector field generated by X;

is equivariant with respect to the given action ofG on Mandthe coadjoint action ofG on the dual vector space g.

(M, , G , ) denotes an origami manifold equipped with a hamiltonian

action of a Lie group G having moment map .

Mosers trick needs to be adapted as in[CGW, Theorem 1]. We startfrom a tubular neighborhood model defined as follows.

Definition 2.12. A Moser model for an oriented origami manifold(M, ) with null fibration Z

B is a diffeomorphism: Z (, ) U

where >0 andUis a tubular neighborhood ofZsuch that(x, 0) =xfor all xZ and

= pi+d

t2p

,

withp: Z (, )Zthe projection onto the first factor, i: ZM the inclusion, t the real coordinate on the interval (, ) and an S1-connection form for a chosen principal S1-action along the nullfibration.

A choice of a principalS1-action along the null fibration, S1

Z B, corresponds to a vector field v on Zgenerating the principal S1-bundle. Following [CGW, Theorem 1], a Moser model can then befound after choices of a connection form , a small enough positivereal number and a vector field w over a tubular neighborhood ofZsuch that, at each xZ, the pair (wx, vx) is an oriented basis of thekernel ofx. The orientation on this kernel is determined by the givenorientation ofT Mand the symplectic orientation ofT M modulo thekernel. Conversely, a Moser model for an oriented origami manifoldgives a connection 1-form by contracting with the vector field12t

t

(and hence gives a vertical vector field v such that v= 1 whichgenerates an S1-action), an from the width of the symmetric realinterval and a vector field w=

t

.


9/43


Lemma 2.13. Any two Moser models i : Z(i, i) Ui, withi= 0, 1, admit isotopic restrictions to Z (, ) for small enough,that is, those restrictions can be smoothly connected by a family ofMoser models.

Proof. Letv0 andv1 be the vector fields generating the S1-actions for

models0and 1, and letw0and w1be the vector fieldswi= (i)

t

.

The vector fields vt = (1t)v0 +tv1 onU0 U1 correspond to aconnecting family of S1-actions all with the same orbits, orientationand periods 2. Connect the vector fields w0 and w1 onU0 U1 bya smooth family of vector fields wt forming oriented bases (wt, vt) ofker over Z. Note that thevt are all positively proportional and theset of all possible vector fields wt is contractible. By compactness of

Z, we can even take the convex combination wt= (1 t)w0+ tw1, for small enough. Pick a smooth family of connections t: For instance,using a metric pick 1-forms t such that t(vt) = 1 and then averageeacht by the S

1-action generated by vt. For the claimed isotopy, usea corresponding family of Moser models t : Z (, ) Ut with sufficiently small so that integral curves of all wt starting at points ofZare defined for t(, ).

With these preliminaries out of the way, we recall and expand theproof from [CGW] for Proposition2.10.

Proof. Choose a principal S1-action along the null fibration, S1

Z B. Let : Z (, ) Ube a Moser model, and letU+ denoteM+ U=(Z (0, )). The diffeomorphism

: Z (0, 2) U+ , (x, s) =(x, s)induces a symplectic form

= pi+d (sp) =:

on Z (0, 2) that extends by the same formula to Z (2, 2).As in standard symplectic cutting [L], form the product (Z(2, 2), )

(C, 0) where 0 = i2dzdz. The product action of S1 on Z(

2, 2)Cby

ei (x,s,z) = (ei x,s,eiz)is hamiltonian and (x,s,z) = s |z|22 is a moment map. Zero is aregular value of and the corresponding level is a codimension-onesubmanifold which decomposes as

1(0) =Z {0} {0}

{(x,s,z)|s >0 , |z|2 = 2s} .


10/43


Since S1 acts freely on 1(0), the quotient 1(0)/S1 is a manifoldand the point-orbit map is a principal S1-bundle. Moreover, we can

view it as1(0)/S1 B U+ .

Indeed,B embeds as a codimension-two submanifold via

j : B 1(0)/S1(x) [x, 0, 0] for xZ

andU+ embeds as an open dense submanifold viaj+ : U+ 1(0)/S1

(x, s) [x,s, 2s].

The symplectic form red on 1(0)/S1 obtained by symplectic re-

duction is such that the above embeddings of (B, B) and of (U+, |U+)are symplectic.

The normal bundle to j(B) in 1(0)/S1 is the quotient over S1-orbits (upstairs and downstairs) of the normal bundle to Z{0}{0}in1(0). This latter bundle is the product bundle Z{0} {0} Cwhere the S1-action is

ei (x, 0, 0, z) = (ei x, 0, 0, eiz) .Performing R+-projectivization and taking theS1-quotient we get thebundleZB with the isomorphism

Z {0} {0} C/S1 [x, 0, 0, rei] eixZ Z {0} {0}/S1 [x, 0, 0] (x)B .

By gluing the rest of M+ alongU+, we produce a 2n-dimensionalsymplectic manifold (M+0 ,

+0) with a symplectomorphismj

+ :M+ M+0 j(B) extending j

+.

For the other side, the map : Z(0, 2) U := M U,(x, s) (x, s) reverses orientation and () = . The baseB embeds as a symplectic submanifold of1(0)/S1 by the previous

formula. The embeddingj : U 1(0)/S1

(x, s) [x,s,

2s]

is an orientation-reversing symplectomorphism. By gluing the rest ofM alongU, we produce (M0 , 0) with a symplectomorphism j :MM0 j(B) extending j.


11/43


Remark 2.14. The cutting construction in the previous proof pro-duces a symplectomorphismbetween tubular neighborhoods 1(0)/S1

of the embeddings ofBin M+0 and M0 , comprisingU

+

U,(x, t)(x, t), and the identity map on B : : 1(0)/S1 1(0)/S1

[x,s,

2s] [x,s, 2s] .Definition 2.15. Symplectic manifolds (M+0 ,

+0) and (M

0 ,

0) ob-

tained by cutting are calledsymplectic cut piecesof the oriented origamimanifold (M, ) and the embedded copies ofB are calledcenters.

The next proposition states that symplectic cut pieces of an origamimanifold are unique up to symplectomorphism.

Proposition 2.16. Different choices of a Moser model for a tubularneighborhood of the fold in an origami manifold yield symplectomorphicsymplectic cut pieces.

Proof. Let0and 1be two Moser models for a tubular neighborhoodU of the fold Z in an origami manifold (M, ). Let (M0 , 0) and(M1 ,

1) be the corresponding symplectic manifolds obtained by the

above cutting. Let t : Z (, ) Ut be an isotopy between(restrictions of) 0 and 1. By suitably rescaling t, we may assumethattis a technical isotopy in the sense of [BJ, p.89], i.e.,t= 0 fort near 0 and t= 1 for t near 1. Let

jt :Ut 1(0)/S1be the corresponding isotopies of symplectic embeddings, where1(0)/S1

is equipped with (red)t, and let (Mt ,

t ) be the corresponding fami-

lies of symplectic manifolds obtained from gluing: For instance, (M+t , +t )

is the quotient of the disjoint union

M+ 1(0)/S1by the equivalence relation which sets each point inU+ equivalent toits image by the symplectomorphism j+t .

Let

Uc :=

Uout

\Uinbe a compact subset of each

Utwhere

Uinand

Uout

are tubular neighborhoods ofZ withUin Uout. LetU+ =U M+where stands for the subscripts c, out or in. Let C be a compactneighborhood ofj+ ([0, 1] U+c ) :=t[0,1]j+t (U+c ) in 1(0)/S1, suchthat B C=.

By Theorem 10.9 in [BJ], there is a smooth family of diffeomorphisms

Ht: 1(0)/S1 1(0)/S1 , t[0, 1] ,


12/43


which hold fixed all points outsideC(in particular, the Ht fix a neigh-borhood ofB ), withH0 the identity map and such that

j+t|U+c =Ht j+0|U+c .The diffeomorphism Ht restricted to D := j

+0(U+out) and the identity

diffeomorphism onM+ \ U+in together define a diffeomorphismt: M

+0 M+t .

All forms in the family t +t on M

+0 are symplectic, have the same

restriction toB and are equal to+0 away from the setDwhich retractstoB. Hence, allt

+t are in the same cohomology class and, moreover,

d

dt

t +t =dt

for some smooth family of 1-forms t supported in the compact setD [MS,p.95].

By solving Mosers equation

wt+t +t= 0

we find a time-dependent vector field wt compactly supported on D.The isotopy t :M

+0 M+0 , tR, corresponding to this vector field

satisfiestid away from D andt

t +t

= +0 for allt .

The map 11 is a symplectomorphism between (M+0 , +0) and(M+1 ,

+1). Similarly for (M

0 ,

0) and (M

1 ,

1).

Cutting may be performed for any nonorientableorigami manifold(M, ) by working with its orientable double cover. The double coverinvolution yields a symplectomorphism from one symplectic cut pieceto the other. Hence, we regard these pieces as a trivial double cover(of one of them) and call their Z2-quotient the symplectic cut spaceof(M, ). In the case whereMZis connected, the symplectic cut spaceis also connected; see Example 2.9.

Definition 2.17. The symplectic cut space of an origami manifold(M, ) is the natural Z2-quotient of symplectic cut pieces of its ori-entable double cover.

Notice that, when the original origami manifold is compact, the sym-plectic cut space is also compact.


13/43


2.3. Radial blow-up. We can reverse the cutting procedure using anorigami (and simpler) analogue of Gompfs gluing construction[G]. Ra-

dial blow-upis a local operation on a symplectic tubular neighborhoodof a codimension-two symplectic submanifold modeled by the followingexample.

Example 2.18. Consider the standard symplectic (R2n, 0) with itsstandard euclidean metric. Let B be the symplectic submanifold de-fined by x1 = y1 = 0 with unit normal bundle N identified with thehypersurfacex21+y

21 = 1. The map :NR R2n defined by

((p, ei), r) = p+ (r cos , r sin , 0, . . . , 0) for pBinduces by pullback an origami form on the cylinder N R S1 R2n1, namely

0= rdr d+dx2 dy2+. . . +dxn dyn .

Let (M, ) be a symplectic manifold with a codimension-two sym-plectic submanifoldB . Leti : BMbe the inclusion map. Considerthe radially projectivized normal bundle over B

N := P+ (iTM/TB) ={x(iT M)/TB,x= 0}/where x x for R+. We choose an S1 action makingN aprincipal circle bundle over B. Let >0.

Definition 2.19. Ablow-up modelfor a tubular neighborhood

UofB

in (M, ) is a map :N (, ) U

which factors as

: N (, ) 0 N S1C U(x, t) [x, t]

whereei(x, t) = (eix,tei) for (x, t) NC and : 0(N (, ))U is a tubular bundle diffeomorphism. By tubular bundle diffeomor-phismwe mean a bundle diffeomorphism covering the identity BBand isotopic to a diffeomorphism given by geodesic flow for some choiceof metric on

U.

In practice, a blow-up model may be obtained by choosing a rie-mannian metric to identifyN with the unit bundle inside the geo-metric normal bundle T B and then by using the exponential map:(x, t) = expp(tx) where p is the projection onto B ofx N.Remark 2.20. From the properties of0, it follows that:


14/43


(i) the restriction of toN (0, ) is an orientation-preservingdiffeomorphism ontoU B;

(ii) (x, t) =(x, t);(iii) the restriction oftoN {0} is the bundle projectionN B;(iv) for the vector fields generating the vertical bundle ofN B

and t

tangent to (, ) we have that D() intersects zerotransversally andD(

t) is never zero.

Lemma 2.21. If :N (, ) U is a blow-up model for theneighborhoodU of B in (M, ), then the pull-back form is anorigami form whose null foliation is the circle fibration :N {0} B.

Proof. By properties (i) and (ii) in Remark 2.20, the form issymplectic away fromN {0}. By property (iii), onN {0} thekernel of has dimension 2 and is fibrating. By property (iv) thetop power of intersects zero transversally.

All blow-up models share the same germ up to diffeomorphism. Moreprecisely, if1 :N (, ) U1 and 2 :N (, ) U2 are twoblow-up models for neighborhoodsU1 andU2 of B in (M, ), thenthere are possibly narrower tubular neighborhoods ofB,Vi Ui anda diffeomorphism :V1 V2 such that 2 = 1. Moreover:

Lemma 2.22. Any two blow-up modelsi :N (i, i) Ui, i =1, 2, are isotopic, that is, can be smoothly connected by a family ofblow-up models.

Proof. By definition, the blow-up models factor as

i= i 0 , i= 1, 2 ,for some tubular neighborhood diffeomorphisms,1 and 2, which areisotopic since the set of riemannian metrics onUis convex and differentgeodesic flows are isotopic.

Let (M, ) be a symplectic manifold with a codimension-two sym-plectic submanifold B .

Definition 2.23. A model involutionof a tubular neighborhoodU ofB is a symplectic involution :U Upreserving B such that on theconnected componentsUi ofUwhere (Ui) =Ui we have |Ui = idUi.


15/43


A model involution induces a bundle involution :N N cov-ering|B by the formula

[v] = [dp(v)] , forvTpM, pB .This is well-defined because (B) = B. We denote by :N Nthe involution [v][dp(v)].Remark 2.24. When B is the disjoint union ofB1 and B2, and cor-respondinglyU=U1 U2, if(B1) = B2 then

1:= |U1 :U1 U2 and |U2 =11 :U2 U1 .In this case, B/B1 andN/ N1 is the radially projectivizednormal bundle to B1.

Proposition 2.25. Let(M, )be a (compact) symplectic manifold, Ba compact codimension-two symplectic submanifold andN its radiallyprojectivized normal bundle. Let :U Ube a model involution of atubular neighborhoodU ofB and :N N the induced bundle map.

Then there is a (compact) origami manifold(M, ) with symplecticpartM\Zsymplectomorphic toM\B, folding hypersurface diffeomor-phic toN/ and null fibration isomorphic toN/ B/.

Proof. Choose :N (, ) Ua blow-up model for the neighbor-hoodUsuch that= . This is always possible: For components

Ui ofU where(Ui) =Ui this condition is trivial; for disjoint neighbor-hood componentsUi andUj such that (Ui) =Uj (as in Remark2.24),this condition amounts to choosing any blow-up model on one of thesecomponents and transporting it to the other by .

Thenis a folded symplectic form onN (, ) with folding hy-persurfaceN {0}and null foliation integrating to the circle fibrationS1

N B. We defineM= M BN (, ) /where we quotient by

(x, t)(x, t) for t >0 and (x, t)((x), t) .The forms on MBand onN (, ) induce onMan origamiformwith folding hypersurfaceN/. Indeedis a symplectomor-phism fort >0, and (, id) onN (, ) is a symplectomorphismaway from t= 0 (since and are) and at points where t= 0 it is alocal diffeomorphism.


16/43


Definition 2.26. An origami manifold (

M,

) as just constructed is

called a radial blow-up of(M, ) through(, B).

The next proposition states that radial blow-ups of (M, ) through(, B) are unique up to symplectomorphism.

Proposition 2.27. Let(M, ) be a symplectic manifold, B a compactcodimension-two symplectic submanifold and :U U a model in-volution of a tubular neighborhoodU of B. Then different choices ofa blow-up model forUyield symplectomorphic radial blow-ups through(, B).

Proof. Let0 and1 be two blow-up models forUsuch that i=i . We restrict them to the same domain with small enough andstill denote

i:N (, ) Ui , i= 0, 1.Lett:N (, ) Ut be a technical isotopy between 0 and1 inthe sense of [BJ, p.89]. Each of the corresponding origami manifolds

(Mt, t) is defined as a quotient of the disjoint union of (MB, ) with(N (, ), t ) by the equivalence relationt given byt and asin the proof of Proposition2.25. Let Cbe a compact neighborhood of([0, 1] N [, ]) :=t[0,1]t (N [, ]) in M\ B, where <

2.

By Theorem 10.9 in [BJ], there is a smooth family of diffeomorphismsHt: MM , t[0, 1],

which hold fixed all points outsideC(in particular, theHtfix B), withH0 the identity map and such that

t|N[,]= Ht 0|N[,] .By property (ii) in Remark2.20, the same holds onN [+, ].

The diffeomorphismHt restricted toD := M\0(N (, )) andthe identity diffeomorphism onN (+, ) together define adiffeomorphism

t:M0Mtwhich fixes the foldN/ .

All forms in the family tt onM0 are origami with the same fold,are equal at points of that fold, and are all equal to0 away from

C 0(N [, ])


17/43


which is a compact neighborhood of the fold in

M0 retracting to the

fold. Hence, by a folded version of Mosers trick (see the proof of

Theorem 1 in [CGW]) there is an isotopy t :M0M0, tR, fixingthe fold such that

t(tt) =0 for allt .

The map 11is a symplectomorphism between (M0, 0) and (M1, 1).

Different model involutions i :Ui Ui of tubular neighborhoodsofB give rise to symplectomorphic origami manifolds as long as theinduced bundle maps i:N Nare isotopic. Examples2.29and2.30illustrate the dependence on the model involution.

Example 2.28. Let Mbe a 2-sphere, B one point on it, and theidentity map on a neighborhood of that point. Then a radial blow-upM is RP2 and a form which folds along a circle.Example 2.29. Let M be a 2-sphere, B the union of two (distinct)points on it, andthe identity map on a neighborhood of those points.

Then a radial blow-upM is a Klein bottle and a form which foldsalong two circles.

Example 2.30. Again, let Mbe a 2-sphere, B the union of two (dis-tinct) points on it, and nowdefined by a symplectomorphism from aDarboux neighborhood of one point to a Darboux neighborhood of theother. Then a radial blow-upM is a Klein bottle and a form whichfolds along a circle.

Remark 2.31. The quotientN (, )/ (, id) provides a collarneighborhood of the fold in (M, ).

When B splits into two disjoint components interchanged by as inRemark2.24, this collar is orientable so the fold is coorientable. Exam-ple2.30illustrates a case where, even though the fold is coorientable,

the radial blow-up (

M,

) is not orientable.

Whenis the identity map, as in Example 2.28,the collar is nonori-entable and the fold is not coorientable. In the latter case, the collaris a bundle of Mobius bands S1 (, )/(id, id) over B.

In general,will be the identity over some connected components ofB and will interchange other components, so some components of thefold will be coorientable and others will not.


18/43


Remark 2.32. For the radial blow-up (

M,

) to be orientable, the

starting manifold (M, ) must be the disjoint union of symplectic man-

ifolds (M1, 1) and (M2, 2) with B = B1B2, Bi Mi, such that(B1) = B2 as in Remark2.24. In this case (M, ) may be equippedwith an orientation such that

M+ M1 B1 and MM2 B2 .The folding hypersurface is diffeomorphic toN1 (orN2) and we have

1 on M1 B12 on M2 B21 onN (, )

We then say that (M, )is the blow-up of(M1, 1)and(M2, 2)through(1, B1) where 1 is the restriction of to a tubular neighborhood ofB1.

Remark 2.33. Radial blow-up may be performed on an origami man-ifold at a symplectic submanifold B (away from the fold). When westart with two folded surfaces and radially blow them up at one point(away from the folding curves), topologically the resulting manifoldis a connected sum at a point, M1#M2, with all the previous foldingcurves plus a new closed curve. Since all RP2n are folded symplecticmanifolds, the standard real blow-up of a folded symplectic manifold

at a point away from its folding hypersurface still admits a folded sym-plectic form, obtained by viewing this operation as a connected sum.

2.4. Cutting a radial blow-up.

Proposition 2.34. Let (M, ) be a radial blow-up of the symplecticmanifolds (M1, 1) and (M2, 2) through (1, B1) where 1 is a sym-plectomorphism of tubular neighborhoods of codimension-two symplec-tic submanifoldsB1 andB2 ofM1 andM2, respectively, takingB1 toB2.

Then cutting(M, ) yields manifolds symplectomorphic to (M1, 1)

and(M2, 2) where the symplectomorphisms carryB to B1 andB2.

Proof. We first exhibit a symplectomorphism 1 between a cut space(M+0 ,

+0) of (M, ) and the original manifold (M1, 1).

LetNbe the radially projectivized normal bundle to B1 inM1 andlet :N (, ) U1 be a blow-up model. The cut spaceM+0 is


19/43


obtained gluing the reduced space

1

(0)/S

1

= (x,s,z)Z [0, 2) C |s = |z|2

2 /S1with the manifold

M1 B1

via the diffeomorphisms

N (0, ) 1(0)/S1 and N (0, ) U1 B1(x, t) [x, t2, t2] (x, t) (x, t)

i.e., the gluing is by the identification [x, t2, t

2] (x, t) for t > 0overU1 B1. The symplectic form+0 on M+0 is equal to the reducedsymplectic form on 1(0)/S1 and equal to1 onM1 B1 (the gluingdiffeomorphism [x, t2, t2](x, t) is a symplectomorphism).

We want to define a map 1 : M1 M+0 which is the identity onM1 B1 and onU1 is the composed diffeomorphism

1 : U1 (N C) /S1 1(0)/S1[x, z] [x, |z|2, z2]

where the first arrow is the inverse of the bundle isomorphism givenby the blow-up model. In order to show that 1 is well-defined weneed to verify that u1 U1 B1 is equivalent to its image 1(u1)1(0)/S1 B. Indeed u1 must correspond to [x, z]

(

N C) /S1

withz= 0. We write zas z= tei with t >0. Since [x, z] = [eix, t],we have u1 = (e

ix, t) and 1(u1) = [eix, |t|2, t2]. These two are

equivalent under(x, t)[x, t2, t2], so 1 is well-defined.Furthermore,M1 andM

+0 are symplectic manifolds equipped with a

diffeomorphism which is a symplectomorphism on the common densesubset M1 B1. We conclude that M1 and M

+0 must be globally

symplectomorphic.

Now we tackle (M2, 2) and (M0 ,

0). The cut spaceM

0 is obtained

gluing the same reduced space 1(0)/S1 with the manifold M2 B2via the diffeomorphisms

N (, 0) 1(0)/S1(x, t) [x, t2, t2]

and

N (, 0) U1 B1 U2 B2(x, t) (x, t) ((x, t))


20/43


i.e., the gluing is by the identification [x, t2, t

2]((x, t)) fort


21/43


with foldN fibering over B. There is a natural two-to-one smoothprojectionMMtakingMsBandMsBeach diffeomorphicallyto M Z where Z is the fold of (M, ), and taking the foldN Nof (M, ) to Z N/ with :N N the bundle map inducedby(the map having no fixed points).

Corollary 2.36. Let(M, )be a radial blow-up of the symplectic man-ifold (Ms, s) through (, B). Then cutting (M, ) yields a manifoldsymplectomorphic to(Ms, s)where the symplectomorphism carries thebase to B.

Proof. Let (Mcut, cut) be a symplectic cut space of (M, ). Let(Ms, s) and (Mcut, cut) be the trivial double covers of (Ms, s) and(Mcut, cut). By Lemma 2.35, the radial blow-up (M, ) of (Ms, s)through (, B) is an orientable double cover of (M, ). As a conse-quence of Definition 2.17, (Mcut, cut) is the symplectic cut space of(M, ). By Proposition2.34, (Ms, s) and (Mcut, cut) are symplecto-morphic relative to the centers. It follows that (Ms, s) and (Mcut, cut)are symplectomorphic relative to the centers.

2.5. Radially blowing-up cut pieces.

Proposition 2.37. Let (M, ) be an oriented origami manifold withnull fibrationZ B.

Let (M1, 1) and (M2, 2) be its symplectic cut pieces, B1 and B2the natural symplectic embedded images ofB in each and1 :U1 U2the symplectomorphism of tubular neighborhoods of B1 and B2 as inRemark2.14.

Let (M, ) be a radial blow-up of (M1, 1) and (M2, 2) through(1, B1).

Then(M, ) and(

M,

) are equivalent origami manifolds.

Proof. Choose a principal S1 action S1 Z B and let :Z (, ) U be a Moser model for a tubular neighborhoodUof Z in M as in the proof of Proposition 2.10. LetN be the radialprojectivized normal bundle to B1 in M1. By Proposition 2.10, thenatural embedding ofB inM1 with image B1 lifts to a bundle isomor-phism fromN B1 to Z B. Under this isomorphism, we pick


22/43


the following blow-up model for the neighborhood 1(0)/S1 ofB1 in(M1, 1):

: Z (, ) 1

(0)/S1

(x, t) [x, t2, t2] .By recalling the construction of the reduced form1on

1(0)/S1 (seeproof of Proposition2.10) we find that 1 = . Hence in this casethe origami manifold (M, ) has

M= M1 B1M2 B2Z (, ) /where we quotient identifying

Z (0, ) 1(0)/S1 U1 B1and

Z (, 0) 1(0)/S1 1(0)/S1 U2 B2and we have

:=

1 on M1 B12 on M2 B21 on Z (, ) .

The symplectomorphisms (from the proof of Proposition 2.10) j+ :M+ M1 B1 and j : M M2 B2 extending (x, t)

x, t2, t

2

make the following diagrams (one for t >0, the other fort


23/43


Then(M, ) and(

M,

) are symplectomorphic origami manifolds.

Proof. We pass to the orientable double covers. By Proposition2.37,the orientable double cover of (M, ) is symplectomorphic to the blow-up of its cut space. By definition, the cut space of the double cover of(M, ) is the double cover of (Mcut, cut). By Lemma2.35, the blow-up

of the latter double cover is the double cover of (M, ). 3. Origami Polytopes

3.1. Origami convexity.

Definition 3.1. IfFi is a face of a polytope i,i = 1, 2, in Rn, we say

that 1 nearF1 agreeswith 2 near F2 whenF1= F2 and there is anopen subsetU ofRn containingF1 such thatU 1 =U 2.

The following is an origami analogue of the Atiyah-Guillemin-Sternbergconvexity theorem.

Theorem 3.2. Let(M, , G , ) be a connected compact origami man-

ifold with null fibration Z B and a hamiltonian action of an m-

dimensional torusG with moment map : M g. Then:(a) The image (M) of the moment map is the union of a finite

number of convex polytopes i, i = 1, . . . , N , each of whichis the image of the moment map restricted to the closure of aconnected component ofM Z.

(b) Over each connected component Z of Z, the null fibration isgiven by a subgroup ofG if and only if(Z) is a facet of eachof the one or two polytopes corresponding to the neighboringcomponent(s) ofMZ, and when those are two polytopes, theyagree near the facet(Z).

We call such images (M) origami polytopes.

Remark 3.3. WhenMis oriented, the facets from part (b) are always

shared bytwo polytopes. In general, a componentZ is coorientable ifand only if(Z) is a facet of two polytopes.

Proof.

(a) Since theG-action preserves, it also preserves each connectedcomponent of the folding hypersurface Zand its null foliation


24/43


V . Choose an oriented trivializing section w ofV . Average wso that it is G-invariant, i.e., replace it with

1|G|

G

gwg1(p) dg .

Next, scale it uniformly over each orbit so that its integralcurves all have period 2, producing a vector fieldv which gen-erates an action ofS1 on Zthat commutes with the G-action.ThisS1-action also preserves the moment map: for anyXgwith corresponding vector field X# on M, we have over Z

Lv, X =vd, X =vX#= (v, X#) = 0 .Using this v, the cutting construction from Section2 has a

hamiltonian version. Let (Mi, i),i= 1, . . . , N , be the resultingcompact connected components of the symplectic cut space.Let Bi be the union of the components of the base B whichnaturally embed in Mi. Each Mi Bi is symplectomorphic toa connected componentWiM Z and Mi is the closure ofMiBi. Each (Mi, i) inherits a hamiltonian action ofG withmoment map i which matches |Wi over Mi Bi and is thewell-defined S1-quotient of|Zover Bi.

By the Atiyah-Guillemin-Sternberg convexity theorem [A,GS], each i(Mi) is a convex polytope i. Since (M) is theunion of the i(Mi), we conclude that

(M) =Ni=1

i .

(b) Assume first thatM is orientable.Let Z be a connected component of Z with null fibration

ZB . LetW1 andW2 be the two neighboring componentsofMZon each side ofZ, (M1, 1, G , 1) and (M2, 2, G , 2)the corresponding cut spaces with moment polytopes 1 and2.

LetU be a G-invariant tubular neighborhood of Z with aG-equivariant diffeomorphism : Z (, ) U such that

= pi+d

t2p

,

where G acts trivially on (, ), p : Z (, ) Z is theprojection onto the first factor, t is the real coordinate on theinterval (, ) and is a G-invariantS1-connection on Z fora chosen principal S1 action, S1 Z B. The existence of


25/43


such follows from an equivariant Moser trick, analogous tothat in the proof of Proposition2.10.

Without loss of generality, Z (0, ) andZ (, 0) corre-spond via to the two sidesU1 :=U W1 andU2 :=U W2,respectively. The involution :U U translating t tinZ (, ) is a G-equivariant (orientation-reversing) diffeo-morphism preservingZ, switchingU1andU2 but preserving .Hence the moment map satisfies = and (U1) =(U2).

When the null fibration is given by a subgroup of G, wecut the G-spaceU at the level Z. The image (Z) is theintersection of(U) with a hyperplane and thus a facet of both1 and 2.

EachUiB is equivariantly symplectomorphic to a neigh-borhoodVi ofB in (Mi, i, G , i) withi(Vi) =(Ui) (Z),i= 1, 2. As a map to its image, the moment map is open [ BK].Since1(V1) =2(V2), we conclude that 1 and 2 agree nearthe facet (Z).

For general null fibration, we cut the GS1-spaceU withmoment map (, t2) at Z, the S1-level t2 = 0. The image ofZ by the G S1-moment map is the intersection of the imageof the fullU with a hyperplane. We conclude that the image(Z) is the first factor projection : gR g of a facetof a polytope

in g R, so it can be of codimension zero or

one; see Example3.5.

If|:1 is one-to-one, then facets of map to facetsof 1 and is contained in a hyperplane surjecting onto g.The normal to that hyperplane corresponds to a circle subgroupofGS1 acting trivially onUand surjecting onto theS1-factor.This allows us to express the S1-action in terms of a subgroupofG.

If| : 1 is not one-to-one, it cannot map the facetFZ of corresponding to Z to a facet of 1: Otherwise,FZwould contain nontrivial verticalvectors (0, x)g R whichwould contradict the fact that theS1 direction is that of the nullfibration on Z. Hence, the normal toFZ in must be trans-verse tog, and the corresponding null fibration circle subgroupis not a subgroup ofG.

When M is not necessarily orientable, we consider its ori-entable double cover and lift the hamiltonian torus action. Thelifted moment map is the composition of the two-to-one projec-tion with the original double map, and the result follows.


26/43


Example 3.4. Consider (S4, 0,T2, ) where (S4, 0) is a sphere as inExample2.5with T2 acting by

(ei1, ei2) (z1, z2, h) C2RR5

= (ei1z1, ei2z2, h)

and moment map defined by

(z1, z2, h) = |z1|2

2x1

,|z2|2

2x2

whose image is the triangle x1

0, x2

0, x1+ x2

12 . The image

(Z) of the folding hypersurface (the equator) is the hypotenuse.

Figure 1. An origami polytope for a 4-sphere

The null foliation is the Hopf fibration given by the diagonal circle

subgroup ofT2

. In this case, Theorem3.2says that the triangle is theunion of two identical triangles, each of which is the moment polytopeof one of the CP2s obtained by cutting; see Example2.8. Likewise, if(S4, 0,T

2, ) was blown-up at a pole, the triangle in Figure 1wouldbe the superposition of the same triangle with a trapezoid.

Example 3.5. Consider (S2 S2, s f, S1, ), where (S2, s) is astandard symplectic sphere, (S2, f) is a folded symplectic sphere withfolding hypersurface given by a parallel, and S1 acts as the diagonal ofthe standard rotation action ofS1S1 on the product manifold. Thenthe moment map image is a line segment and the image of the foldinghypersurface is a nontrivial subsegment. Indeed, the image of is a45o projection of the image of the moment map for the full S1 S1action, i.e., a rectangle in which the folding hypersurface surjects toone of the sides; see Figure2.

By considering the first or second factors ofS1 S1 alone, we getthe two extreme cases in which the image of the folding hypersurfaceis either the full line segment or simply one of the boundary points.


27/43


Figure 2. Origami polytopes for a product of two 2-spheres

The analogous six-dimensional examples (S2 S2 S2, ssf,T

2, ) produce moment images which are rational projections of a

cube, with the folding hypersurface mapped to rhombi; see Figure 3.

Figure 3. An origami polytope for a product of three 2-spheres

3.2. Toric case.

Definition 3.6. A toric origami manifold (M, , G , ) is a compactconnected origami manifold (M, ) equipped with an effective hamil-tonian action of a torusG with dim G= 1

2dim Mand with a choice of

a corresponding moment map .

For a toric origami manifold (M, , G , ), orbits with trivial isotropy theprincipal orbits form a dense open subset ofM [Br, p.179]. Anycoorientable connected component Z ofZ has a G-invariant tubularneighborhood modelled on Z(, ) with a GS1 hamiltonianaction having moment map (, t2). As the orbits are isotropic sub-

manifolds, the principal orbits of the GS1-action must still havedimension dim G. Their stabilizer must be a one-dimensional compactconnected subgroup surjecting onto S1. Hence, over those connectedcomponents of Z, the null fibration is given by a subgroup of G. Asimilar argument holds for noncoorientableconnected components ofZ, using orientable double covers. We have thus proven the followingcorollary of Theorem3.2.


28/43


Corollary 3.7. When (M, , G , ) is a toric origami manifold, themoment map image of each connected component Z of Z is a facet

of each of the one or two polytopes corresponding to the neighboringcomponent(s) ofM Z, and when those are two polytopes, they agreenear the facet(Z).

Delzant spaces, also know as symplectic toric manifolds, are closedsymplectic 2n-dimensional manifolds equipped with an effective hamil-tonian action of ann-dimensional torus and with a corresponding mo-ment map. Delzants theorem [D] says that the image of the mo-ment map (a polytope in Rn) determines the Delzant space (up to anequivariant symplectomorphism intertwining the moment maps). TheDelzant conditionson polytopes are conditions characterizing exactly

those polytopes that occur as moment polytopes for Delzant spaces. Apolytope in Rn isDelzant if:

there are n edges meeting at each vertex; each edge meeting at vertex p is of the form p+ tui, t 0,

where ui Zn; for each vertex, the corresponding u1, . . . , un can be chosen to

be a Z-basis ofZn.

Corollary3.7 says that for a toric origami manifold (M, , G , ) theimage (M) is the superimposition of Delzant polytopes with certaincompatibility conditions. Section3.3will show how all such (compat-ible) superimpositions occur and, in fact, classify toric origami mani-folds.

For a Delzant space,G-equivariant symplectic neighborhoods of con-nected components of the orbit-type strata are simple to infer just bylooking at the polytope.

Lemma 3.8. LetG = Tn be ann-dimensional torus and(M2ni , i, i),i = 1, 2, two symplectic toric manifolds. If the moment polytopesi := i(Mi) agree near facets F1 1(M1) andF2 2(M2), thenthere areG-invariant neighborhoodsUi ofBi =1i (Fi), i= 1, 2, withaG-equivariant symplectomorphism :

U1

U2 extending a symplec-

tomorphismB1B2 and such that2= 1.

Proof. LetUbe an open set containing F1 = F2 such thatU 1 =U 2.

Perform symplectic cutting [L] on M1 andM2 by slicing i along ahyperplane parallel toFi such that:


29/43


the moment polytope

i containing Fi is in the open setU;

the hyperplane is close enough toFi to guarantee that

isatis-

fies the Delzant conditions. (For generic rational hyperplanes,the third of the Delzant conditions fails inasmuch as we onlyget a Qn-basis, thus we need to consider orbifolds.)

Then1 =2. By Delzants theorem, the corresponding cut spacesM1 andM2 areG-equivariantly symplectomorphic, the symplectomor-phism pulling back one moment map to the other. Restricting theprevious symplectomorphism gives us a G-equivariant symplectomor-phism between G-equivariant neighborhoodsUi of Bi in Mi pullingback one moment map to the other.

Example 3.9.The polytopes in Figure4represent four different sym-plectic toric 4-manifolds: twice the topologically nontrivialS2-bundleover S2 (these are Hirzebruch surfaces), an S2 S2 blown-up at onepoint and an S2 S2 blown-up at two points. If any two of thesepolytopes are translated so that their left vertical edges exactly su-perimpose, we get examples to which Lemma3.8applies, the relevantfacets being the vertical facets on the left.

Figure 4. Polytopes agreeing near the left vertical edges

Example 3.10. Let (M1, 1,T2, 1) and (M2, 2,T

2, 2) be the firsttwo symplectic toric manifolds from Example3.9(Hirzebruch surfaces).Let (B, B,T

2, B) be a symplectic S2 with a hamiltonian (nonef-

fective) T2-action and hamiltonian embeddings ji into (Mi, i,T2, i)

as preimages of the vertical facets. By Lemma 3.8, there exists aT2-equivariant symplectomorphism :U1 U2 between invarianttubular neighborhoodsUi of ji(B) extending a symplectomorphism

j1(B)j2(B) such that2= 1. The corresponding radial blow-uphas the origami polytope in Figure5.Different shades of gray distinguish regions where each point repre-

sents one orbit (lighter) or two orbits (darker), which results from thesuperimposition of two Hirzebruch polytopes.

This example may be considerably generalized; see Section 3.3.


30/43


Figure 5. Origami polytope for a radial blow-up of twoHirzebruch surfaces

Example 3.11. Dropping the origami hypothesis gives us much lessrigid moment map images. For instance, take any symplectic toricmanifold (M, , G , ), e.g. S2 S2, and use a regular closed curveinside the moment image to scoop out a G-invariant open subset corre-sponding to the region inside the curve. Let fbe a defining function forthe curve such thatf is positive on the exterior. Consider the manifold

M= (p, x)M R| x2 =f(p) .This is naturally atoricfolded symplectic manifold. However, the nullfoliation on Z is not fibrating: at points where the slope of the curveis irrational, the corresponding leaf is not compact.

For instance, whenM=S2S2 and we take some closed curve, themoment map image is as on the left of Figure6. If instead we discardthe region corresponding to the outside of the curve (by choosing afunction f positive on the interior of the curve), the moment mapimage is as on the right.

Figure 6. Folded (non-origami) moment map images

3.3. Classification of toric origami manifolds. Let (M, , G , ) bea toric origami manifold. By Theorem3.2,the image (M) is the su-perimposition of the Delzant polytopes corresponding to the connectedcomponents of its symplectic cut space. Moreover, maps the foldinghypersurface to certain facets possibly shared by two polytopes whichagree near those facets.

Conversely, we will see that, given a templateof an allowable su-perimposition of Delzant polytopes, we can construct a toric origami


31/43


manifold whose moment image is that superimposition. Moreover, suchtemplates classify toric origami manifolds.

Definition 3.12. Ann-dimensionalorigami templateis a pair (P, F),whereP is a (nonempty) finite collection of n-dimensional Delzantpolytopes and Fis a collection of facets and pairs of facets of polytopesinP satisfying the following properties:

(a) for each pair{F1, F2} F, the corresponding polytopes inPagree near those facets;

(b) if a facetFoccurs in F, either by itself or as a member of a pair,then neitherFnor any of its neighboring facets occur elsewhereinF;

(c) the topological space constructed from the disjoint union

i,

i P, by identifying facet pairs inF is connected.Theorem 3.13. Toric origami manifolds are classified by origami tem-plates up to equivariant symplectomorphism preserving the momentmaps. More specifically, at the level of symplectomorphism classes (onthe left hand side), there is a one-to-one correpondence

{2n-diml toric origami manifolds} {n-diml origami templates}(M2n, , Tn, ) (M).

Proof. To build a toric origami manifold from a template (P, F), takethe Delzant spaces corresponding to the Delzant polytopes in

P and

radially blow up the inverse images of the facets occuring in sets inF:for pairs{F1, F2} F the model involution uses the symplectomor-phism from Lemma 3.8; for single faces F F, the map must bethe identity. The uniqueness part follows from an equivariant versionof Corollary2.38.

Remark 3.14. There is also a one-to-one correspondence between ori-ented origami toric manifolds (up to equivariant symplectomorphism)and oriented origami templates. We say that an origami template isorientedif the polytopes inPcome with an orientation andFconsistssolely of pairs of facets which belong to polytopes with opposite orien-tations; see Introduction. Indeed, for{F1, F2} F, with F11 andF22, the opposite orientations on the polytopes 1 and 2 induceopposite orientations on the corresponding components ofMZwhichpiece together to a global orientation ofM, and vice-versa.

Example 3.15. Unlike ordinary toric manifolds, toric origami mani-folds may come from non-simply connected templates. LetM be the


32/43


manifold S2 S2 blown up at two points, with one S2 factor havingthree times the area of the other: the associated polytope is a rectan-

gle with two corners removed. We can construct an origami template(P, F) wherePconsists of four copies of arranged in a square andFis four pairs of edges coming from the blowups. The result is shown inFigure7. Note that the associated origami manifold is also not simplyconnected.

Figure 7. Non-simply-connected toric origami template

Example 3.16. We can form higher-dimensional analogues of the pre-vious example which fail to be k-connected for k 2. In the casek = 2, for instance, let be the polytope associated to M S2, andconstruct an origami template (P, F) just as before: this gives thethree-dimensional figures on the left and right of Figure 8. We now

superimpose these two solids along the dark shaded facets (the bottomfacets of the top copies of ), giving us a ninth pair of facets and thedesired non-2-connected template.

Figure 8. Non-2-connected toric origami template

Note that even though the moment map image has interesting 2,the template (thought of as the polytopes glued along facets) has trivial


33/43


H2 and 2. This is indeed a general feature of origami templates, seeRemark3.20.

Example 3.17. Although the facets ofFare necessarily paired if theorigami manifold is oriented, the converse fails. As shown in Figure 9,one can form a template of three polytopes, each corresponding to anS2 S2 blown up at two points, and three paired facets. Since eachfold flips orientation, the resulting topological space is nonorientable.

Figure 9. Non-orientable toric origami template withco-orientable folds

Example 3.18. Recall that the polytope associated to CP2 is a triangle(shown on the right of Figure 10). The sphere S4 (shown left) is theorientable toric origami manifold whose template is two copies of thistriangle glued along one edge. Similarly, RP4 (shown center) is thenonorientable manifold whose template is a single copy of the trianglewith a single folded edge. This exhibits S4 as a double cover ofRP4 atthe level of templates.

S4 RP4 CP2

Figure 10. Three origami templates with the sameorigami polytope

Example 3.19. We can classify all two-dimensional toric origami man-ifolds by classifying one-dimensional templates (Figure1114). Theseare disjoint unions of n segments (1-dimensional Delzant polytopes)connected at vertices (the facets of those polytopes) with zero angle:internal vertices and endpoints marked with bullets represent folds.Each segment (resp. marking) gives a component ofM Z(resp.Z),while each unmarked endpoint corresponds to a fixed point. There are


34/43


four families (Instead of drawing segments superimposed, we open upangles slightly to show the number of components. All pictures ignore

segment lengths which account for continuous parameters of symplecticarea in components ofM Z.):

Templates with two unmarked endpoints give manifolds diffeo-morphic toS2: they have two fixed points andn1 componentsofZ (Figure11).

Figure 11. Toric origami 2-spheres

Templates with one marked and one unmarked endpoint givemanifolds diffeomorphic to RP2: they have one fixed point andncomponents ofZ(Figure12).

Figure 12. Toric origami real projective planes

Templates with two marked endpoints give manifolds diffeomor-phic to the Klein bottle: they have no fixed points andn+ 1components ofZ(Figure13).

Figure 13. Toric origami Klein bottles


35/43


Templates with no endpoints give manifolds diffeomorphic toT2: they have no fixed points and an even numbern of compo-

nents ofZ (Figure14).

Figure 14. Toric origami 2-tori

Remark 3.20. As illustrated by Example 3.11, it is not possible toclassify toric folded symplectic manifolds by combinatorial momentdata as is the case for toric origami manifolds. Such a classificationmust be more intricate for the general folded case, and in [Lee] C. Leegives a partial result that sheds some light on the type of classificationthat might be possible:

A toric folded symplectic manifold (M,,T, ) is a compact con-nected folded symplectic manifold (M2n, ) endowed with an effective

hamiltonian action of a half-dimensional torus Tn and a correspond-ing moment map. The orbital moment map is the map on the orbitspaceM/Tinduced by the moment map. Two toric folded symplectic4-manifolds (M,,T, ) and (M, ,T, ) are symplectomorphic (asfolded symplectic manifolds) if H2(M/T,Z) = 0 and there exists adiffeomorphism between orbit spaces preserving orbital moment maps.

When (M,,T, ) is a toric origami manifold, M/T can be real-ized as the topological space obtained by identifying the polytopes ofits origami template along the common facets (see point (c) in Def-inition 3.12). This space has the same homotopy type as the graph

obtained by replacing each polytope by a point and each glued dou-ble facet by an edge between the points corresponding to the polytopesthat the facet belongs to. Therefore,H2(M/T,Z) = 0. The existenceof a diffeomorphism between orbit spaces implies that (M, ,T, ) isan origami manifold as well, and that its origami template is the sameas that of (M,,T, ), which makes the manifolds symplectomorphicby Theorem3.13.


36/43


4. Cobordism

We will now prove the following conjecture of Yael Karshons stat-ing that an oriented origami manifold is symplectically cobordant toits symplectic cut space. By symplectic cobordism we mean, follow-ing [GGK], a cobordism manifold endowed with a closed 2-form whichrestricts to the origami and symplectic forms on its boundary.

Theorem 4.1. Let (M, ) be an oriented origami manifold and let(M0 ,

0) be its symplectic cut pieces.

Then there is a manifold Wequipped with a closed 2-form suchthat the boundary ofW equipped with the restriction of is symplec-tomorphic to

(M, ) (M+0 , +0) (M0 , 0) .Moreover, in the presence of a (hamiltonian) compact group action,this cobordism can be made equivariant (or hamiltonian).

Proof. Choose a S1-action making the null fibration into a principalfibration, S1

Z B . Let L LB be the associated hermitian linebundleZS1 C for the standard multiplication action ofS1 on C. Letr: L Rgiven by r() =

, L() be the hermitian length and let

iZ :Z

L, iZ(x) = [x, 1].

For small enough, let : Z (, ) Ube a Moser model forM (see Definition 2.12). Take a small enough and choose a non-decreasing smooth function g : R+0 R such that g (s) =s for s < 2and g(s) = 1 for s >42.

The set (r, t) R+0 (, )|g(t2)

2

4r2 1

is depicted in Figure15, where R :=

1

2

4.

Then the set

W:=

(, t) L (, )|g(t2)

2

4r()2 1

is the manifold with boundary sketched in Figure16withBrepresentedby a point.


37/43


r

r= 1

r= R

t2 2 /2/2

Figure 15. The shaded area is the set(r, t) R+0 (, )|g(t2) 24r2 1

0

0

1

1

0

0

1

1

0 0

0 0

1 1

1 1

0 0

0 0

1 1

1 1

0

0

1

1

0 0

0 0

1 1

1 1

0 0

0 0

1 1

1 1

C

C C+r= 1

r= R

t2 2 /2/2

t= t=Figure 16. The key portionW of the cobordism man-ifoldW

The boundary ofW is made up of the following three pieces:

C :=

{(, t)

L

(

, )

|r() = 1

}C+ := (, t) L [ 2 , )|g(t2) =r()2 + 24 C :=

(, t) L (,

2]|g(t2) =r()2 + 2

4

The set C is the image ofUunder the diffeomorphism

(2) U 1Z (, ) iZid L (, ).

By the tubular neighborhood theorem, the set C+ is diffeomorphicto a neighborhood ofB in the symplectic cut space M+0 (and similarlyfor C andM

0

); indeed the normal bundle ofB in M+

0

is also L.

We can now extend the cobordism W between C and C+ C

to a global cobordism between M and M+0 M0 (modulo diffeomor-phisms). Let M2 :=

1(Z (2, 2)) be a narrower tubular neigh-borhood ofZ inM. We form the global cobordism Wby gluingW to(M M2) [R, 1] using the restriction toU M2 of the diffeomor-phism (2) and using the identity map on [R, 1]. Note that any subset


38/43


ofU not containing Z, for exampleU M2, can be viewed also as asubset ofM+0 M0 viaj+ j (forj+ andj see proof of Proposition2.10).

We will next exhibit a closed 2-form on Wrestricting to the givenorigami and symplectic forms on the boundary.

Letbe theS1-connection form onZfrom the Moser model. DenotebyB0 the zero-section in L. Since L B0 =ZS1 CZR+, wecan extend to L B0 by pull-back via the projection ZR+ Z.Although is not defined over B0, the productr

2is a smooth 1-formon L: On open sets where the fibration Z

B is trivial, we have = d+ for some 1(B), and r2 = r2d+ r2 is welldefined on the whole L because r2d is no longer singular.

Now leth(r, t) be a smooth function defined on the shaded region inFigure15which is even in t (i.e., h(r, t) = h(r, t)) and in particularequal to t2 on the region|t| > 2 and on the line r = 1, and whoserestriction h to the curve g(t2) = r2 +

2

4 is strictly increasing as afunction oftand equal tor2 for |t|< . An exercise in 2-D interpolationshows such a function exists; note that, by definition ofg, we have thatr2 =t2 24 on the latter curve when|t|< .

We define the following closed 2-form on W:

= B+d(h).

Because the restriction h is unique up to pre-composition with a dif-feomorphism, the restrictions of to C+ and C are unique up tosymplectomorphism. Furthermore, by the Weinstein tubular neighbor-hood theorem [We], |C+ and |C are, up to symplectomorphism, thesymplectic forms+0 and

0 on corresponding open subsets ofM

+0 and

M0 . On the other hand, restricting to C, we have =B+d(t2)

which under the map in (2) is the origami form onU.In particular, on the region |t|> 2in Figure16, the folded symplec-

tic form onC, the symplectic forms onC+ andC, and the cobordingform onWare all identical and equal to

B +dt2. Moreover, theup to symplectomorphism part of these statements involves symplec-tomorphisms which are the identity on the region|t| > 2, and hence can be extended to a cobording 2-form on all ofW by letting it bethe constant (constant on r) cobording 2-form on|t|> .

Suppose now that M is equipped with an action of a compact Liegroup G which preserves . One then gets a G-action on B, Z and


39/43


L, and by averaging we can choose the in the Moser model to beG-invariant. Thus all the data involved in the definition of the cobor-

ding 2-form above areG-invariant and hence itself is G-invariant.Moreover, if isG-hamiltonian, the formB is as well, and since isG-invariant the 2-form d(h) isG-hamiltonian with moment map

vgv =v#(h)wherev# is the vector field on L (, ) associated with the action ofGon this space. Thus Wwith the form is a hamiltonian cobordism,and it extends to a hamiltonian cobordism W.

Remark 4.2. IfMand its cut pieces are pre-quantizable one has anisomorphism of virtual vector spaces (and in the presence of group

actions, virtual representation spaces)

Q(M) =Q(M+0 ) Q(M0 )where Q is the spin-C quantization functor. For a proof of this see[CGW].An alternative proof of this result is based on the quantization com-mutes with cobordism theorem of[GGK].

Remark 4.3. By the cobordism commutes with reduction theoremof [GGK], the symplectic reduction of M at a regular level (for anhamiltonian abelian Lie group action) is cobordant to the symplecticreduction ofM+0 M0 at that level.Remark 4.4. In the nonorientable case, we would obtain an orbifoldcobordism. However, this is not interesting, since any manifold Mbounds an orbifold, (M [1, 1]) /Z2.

5. Cohomology of Toric Origami

In this section we assume connectedness of the folding hypersurfaceZ. This assumption is essential for the argument below: the moregeneral case of a nonconnected folding hypersurface remains open.

Let (M,,T, ) be a 2n-dimensional oriented toric origami manifoldwith null fibration Z

B and connected folding hypersurface Z. Let

S1 T be the circle group generating the null fibration and f :MR a corresponding moment map with f = 0 o n Z and f > 0 onM Z. Note that, on a tubular neighborhood ofZgiven by a Moserdiffeomorphism: Z(, ) U, the origami form is= pi+d(t2p) and hence the moment map is f(x, t) = t

2

2.


40/43


Near the folding hypersurface Z, the functionfis essentially t2

2, and

away from it, frestricts to a moment map on an honest symplectic

manifoldM Z, and is thus Morse-Bott. Furthermore, Z is a nonde-generate critical manifold of codimension one.

Defineg : M Ras

g =

f on M+

0 on Zf on M

We claim that g is Morse-Bott and its critical manifolds are thoseoff, excluding Z. But this follows easily from the fact that, onU, wehave g = t

2, whose derivative never vanishes, while on M Z, dg

vanishes if and only ifdf vanishes:

dg=

df /(2

f) on M+

df /(2f) on M

Morse(-Bott) theory then gives us the cohomology groups HkT(M)in terms of the cohomology groups of the critical manifolds ofg, XM Z:

HkT(M) =

0 ifk odd

XHkrXT (X) ifk even

where rX = Ind(X, g) is the index of the critical manifold X with

respect to the function g, and is Ind(X, f) if X M+

, and 2dInd(X, f) ifXM.Similarly to the symplectic toric manifold case, the cohomology groups

are easily read from the template.

Let (P, F) be the template of a 2n-dimensional oriented toric origamimanifold (M,,T, ) with connected folding hypersurface Z. Sincethe manifold is oriented, the setP is a finite collection of oriented n-dimensional Delzant polytopes and the setFa collection of pairs offacets of these polytopes (see the Introduction). We say that a polytopeispositively orientedif its orientation matches that ofgand negatively

oriented otherwise. Let S1 T be the circle group generating thenull fibration and let HT be a complementary (n 1)-dimensionalsubtorus. Letshbe the induced decomposition of the dual of the Liealgebra ofT, and let refl(ab) =abbe the corresponding reflectionalong s. LetPu be the collection of all positively oriented polytopesinP and of the images by refl of all negatively oriented polytopes inP. The setPu can be thought of in terms of unfolding of the moment


41/43


polytope. LetFu be the set of pairs of facets of polytopes in Pucorresponding to the pairs inF. We will call (Pu, Fu) the unfoldedtemplateof (M,,T, ).Corollary 5.1. Let(Pu, Fu)be the unfolded template of a2n-dimensionaloriented toric origami manifold(M,,T, ) with connected folding hy-persurfaceZ, as defined above. LetXg generate an irrational flow.

Fork even, the degree-kcohomology group ofMhas dimension equalto the number of verticesv of polytopes inPu such that:

(i) atv there are exactly k2 primitive inward-pointing edge vectorswhich point up relative to the projection alongX, and

(ii) v does not belong to any facet inFu.All odd-degree cohomology groups ofM are zero.

Example 5.2.For the toric 4-sphere (S4, 0,T2, ) from Example3.4,

the set Pu contains two triangles, one being a mirror image of the other,as in Figure17, where the dashed hypotenuses form the unique pair offacets in the correspondingFu.

01

01

0

2

X

Figure 17. TheunfoldedsetPu for a toric 4-sphereFor the chosen direction X, the numbers next to the relevant ver-

tices count the edge vectors which point up relative to X. Indeed,dim H4(S4) = dim H0(S4) = 1, all other groups being trivial.

Example 5.3. For the toric 4-manifold (M,,T2, ) from Example3.10,the setPu contains two trapezoids, as in Figure18, where the dashedvertical sides form the unique pair of facets in itsFu.

For the chosen directionX, the numbers next to the relevant vertices

count the edge vectors which point up relative to X. The conclusionis that

Hk(M;Z) =

0 ifk oddZ ifk = 0 or k = 4Z2 ifk = 2

which happen to coincide with the groups for an ordinary Hirzebruchsurface.


42/43


0

1

1

2X

Figure 18. The unfolded setPu for a toric manifoldwith a flag-like moment polytope

Funding. The first author was partially supported by the Fundacaopara a Ciencia e a Tecnologia (FCT/Portugal). The third author waspartially supported by FCT grant SFRH/BD/21657/2005.

Acknowledgements. We would like to express our gratitude to friends,colleagues and referees who have helped us write and rewrite sectionsof this paper and/or have given us valuable suggestions about the con-tents. We would particularly like to thank in this regard Yael Karshon,Allen Knutson, Chris Lee, Sue Tolman, Kartik Venkatram and AlanWeinstein.

References

[A] Atiyah, M. Convexity and commuting Hamiltonians. Bull. London Math.Soc. 14, no. 1 (1982): 115.

[Ba] Baykur, I. Kahler decomposition of 4-manifolds. Algebr. Geom. Topol. 6

(2006): 12391265.[vB] von Bergmann, J. Pseudoholomorphic maps into folded symplectic four-manifolds.Geom. & Topol. 11 (2007): 145.

[BK] Bjorndahl , C., Karshon, Y. Revisiting Tietze-Nakajima local and globalconvexity for maps. Canad. J. Math. 62, no. 5 (2010): 975-993.

[Br] Bredon, G. Introduction to Compact Transformation Groups, Pure and Ap-plied Mathematics 46, Academic Press, 1972.

[BJ] Brocker, T., Janich, K., Introduction to Differential Topology, translated fromthe German by C. B. Thomas and M. J. Thomas, Cambridge University Press,Cambridge, 1982.

[C] Cannas da Silva, A. Fold forms for four-folds. J. Sympl. Geom. 8, no. 2(2010): 189203.

[CGW] Cannas da Silva, A., Guillemin, V., Woodward, C. On the unfolding of

folded symplectic structures.Math. Res. Lett.7, no. 1 (2000): 3553.[D] Delzant, T. Hamiltoniens periodiques et images convexes de lapplication mo-

ment. Bull. Soc. Math. France116, no. 3 (1988): 15339.[G] Gompf, R., A new construction of symplectic manifolds.Ann. of Math.142,

no. 3 (1995): 527595.[GGK] Guillemin, V., Ginzburg, V., Karshon, Y.Moment Maps, Cobordisms, and

Hamiltonian Group Actions, with an Appendix by M. Braverman, AmericanMathematical Society, Providence, 2002.


43/43


[GS] Guillemin, V., Sternberg, S. Convexity properties of the moment mapping.Invent. Math.67 (1982): 491513.

[H] Huybrechts, D. Birational symplectic manifolds and their deformations.J.Differential Geom.45, no. 3 (1997): 488513.

[KL] Karshon, Y., Lerman, E., Non-compact symplectic toric manifolds.arXiv:0907.2891v2.

[Lee] Lee, C., Folded Symplectic Toric Four-Manifolds, Ph.D. thesis, Univ. Illinoisat Urbana-Champaign, 2009.

[L] Lerman, E. Symplectic cuts.Math. Res. Lett.2, no. 3 (1995): 247258.[M] Martinet, J. Sur les singularites des formes differentielles. Ann. Inst. Fourier

(Grenoble) 20, fasc 1 (1970): 95178.[MS] McDuff, D., Salamon, D., Introduction to Symplectic Topology, Oxford Math-

ematical Monographs, Oxford University Press, 1998.[We] Weinstein, A. Symplectic manifolds and their lagrangian submanifoldsAdv.

in Math. 6 (1971): 329-346.

Department of Mathematics, Princeton University, Princeton, NJ

08544-1000, USA and Departamento de Matematica, Instituto Superior

Tecnico, 1049-001 Lisboa, Portugal

E-mail address: [email protected]

Department of Mathematics, Massachussets Institute of Technol-

ogy, 77 Massachussets Avenue, Cambridge, MA 02139-4307, USA


Department of Mathematics, Massachussets Institute of Technol-

ogy, 77 Massachussets Avenue, Cambridge, MA 02139-4307, USA

http://arxiv.org/abs/0907.2891http://arxiv.org/abs/0907.2891

simplectic origami

Documents