fiber bundles

Fiber Bundles

We want to introduce vector bundles and fiber bundles. We firsthave an image representation of a fiber bundle.

Definition of a Fiber Bundle

We start with the definition of a Fiber Bundle

Definition (Definition 1)

A bundle is a quadruple, (E ,B,F , π), where E ,B,F are spacesand π : E → B is a continuous map, called the projection, suchthat for every x ∈ B we have that π−1(x) ∼= F and for every b ∈ Bthere is an open neighborhood U ⊆ B of b such thatπ−1(U) ∼= U × F in a fiber preserving way. E is called the totalspace, B the base spaces, and F the fiber.

Remark: Fibers are closed embedded submanifolds of E.

Tangent Bundle of a Smooth Manifold

Example (Example 1)

Consider first a smooth m-dimensional submanifold, M, of Rn. Toeach point x ∈ m we can associate an m-dimensional linearsubspace of the ambient Rn. Denoted by TxM, this is the tangentplane to M at x . Suppose that near x local coordinates on M aregiven by a map G : Rm → Rn (defined on a neighbourhood U of0 ∈ Rm). Then TxM is the m-plane through x and parallel to theimage of Rm under the linearization of G at x , i.e. under the linearmap defined by the Jacobian [∂Gi

∂xj(x)]. Taking the union over all

x ∈ M we get the tangent bundle

TM =⋃x∈M

TxM .

Tangent Bundle- Continued

We can adapt this construction to the case of an abstract manifoldof dimension m (rather than a submanifold of Rn) if we identifyvectors in TxM with directional derivatives at the point x . Thetangent space TxM then acquires an interpretation as the vectorspace of derivations on the germs of functions at x ∈ M. Ifψ : U → Rm is a local coordinate chart defined on an open setU ⊂ M, and p is a point in U, then a basis for TpM can bedescribed as follows. Let (x1, . . . , xm) be the coordinates on Rm,and let ψ(p) = x . Define local vector fields ei (p) = ψ−1∗ ( ∂

∂xi|x) for

i = 1, . . . ,m. Then {e1(p), . . . , em(p)} is a basis for TpM. Asbefore, we define the tangent bundle TM by taking the union ofthe TpM over all p ∈ M.

Tangent Bundle Structure

Notice the following features of TM:The tangent bundle of a smooth manifold encodes informationabout the C∞ structure of the manifold and is the essentialconstruct for differential geometry.

1. There is a projection map π : TM → M (defined by mappingTpM to p) the fibers of which are m-planes, i.e. are copies ofRm.

2. the map (p; v1, · · · , vm)→∑m

i=1 viei (p) defines anidentification of U × Rm with TM|U =

⋃p∈U TpM. (We say

that TM is locally trivial)

Normal Bundles

Example (Example 2)

Suppose that X ⊂ Y are smooth manifolds. For each x ∈ X , wehave TxX ⊂ TxY as a linear subspace. Define the normal at x tobe the quotient,

Nx = TxY /TxX

If we have a metric/inner product defined, then

Nx = (TxX )⊥

i.e, Nx can be identified with the orthogonal complement to TxM.Define the normal bundle to be

N =⋃x∈X

Nx

The normal bundle encodes information about how X sits inside ofY .

The Hopf Map

Example (Example 3)

Describe S3, the standard 3-sphere as follows,

S3 = {(z1, z2) ∈ C2 ∼= R4 : |z1|2 + |z2|2 = 1}

There is a smooth identification of S2, the standard two sphere,with complex projective space. That is, we may identify

S2 ≈ CP1 = {[z1, z2] : 0 6= (z1, z2) ∈ C2}

where [z1, z2] represents the equivalence class of (z1, z2) under therelation (z1, z2) ∼ (w1,w2) if and only if there is a λ ∈ C such that(z1, z2) = λ(w1,w2). The identification is made usingstereographic projection, say from the north pole N ∈ S2, toidentify pN : S2 − {N} → R2 ≈ C. We can also identifyCP1 − [0, 1] with C via the map

[z1, z2] 7→ z2z1

The map that we want is

[z1, z2] 7→

{p−1N ( z2z1 ) z1 6= 0

N z1 = 0

Define h : S3 → S2 by (z1, z2) 7→ [z1, z2]. It is straight forward toshow that this is a well defined, smooth surjection. The fiber overeach point is

h−1([z1, z2]) = {λ(z1, z2) : for all λ ∈ C with |λ| = 1} ∼= S1

Again, we have a sort of local triviality. Observe thatUi = {[z0, z1] : zi 6= 0},i = 0, 1 is an open set. Then,h−1(Ui ) ∼= Ui × S1 via

(z0, z) 7→ ([z0, z1],zi|zi |

) =

{([1, z1z0 ], z0

|z0|) over U0

([ z0z1 , 1], z1|z1|) over U1

The inverse map on U0 × S1 is given by

([1, z ], λ) 7→ (λ√

1 + |z |2,

λz√1 + |z |2

)

with a similar definition on U1×S1. Observe that we do not have aglobally trivial condition, since this would mean that S3 ∼= S2× S1.Using any number of invariants from algebraic topology (e.g,homology or homotopy groups) yields a contradiction.

Homogeneous Bundles

Example (Example 4)

Let O(n) be the set of n × n orthogonal real matrices. Observethat we can embed O(n − 1) into O(n) by Qn−1 7→ Pn wherePn ∈ O(n) is the matrix with 1 in the upper left corner, zeros inthe rest of the first row and column, and the Qn−1 in theremaining (n − 1)× (n − 1) square, i.e,

Qn−1 7→(

1 ∗∗ Qn−1

)Furthermore, O(n) acts on Sn−1 in the usual way viamulitplication,

A ·

x1x2. . .xn

The action is transitive. Set

e1 =

10. . .0

The isotropy subgroup of e1 is O(n − 1). Themap,O(n)/O(n − 1)→ Sn−1 given by [A] 7→ A · e1 defines adiffeomorphism between Sn−1 and the homogeneous space,O(n)/O(n − 1). Let π : O(n)→ O(n)/O(n − 1) be the naturalmap. For any x ∈ O(n)/O(n − 1), we have thatπ−1(x) = {A ∈ O(n) : A · e1 = x}. Since π−1(e1) = O(n − 1), weget that π−1(x) = O(n − 1). For, if x = Q · e1 for some Q, thenA · e1 = Q · e1 implies that Q−1A ∈ O(n − 1).

Examples of Fiber Bundles

FactA fiber bundle also comes with a group action on the fiber. Thisgroup action represents the different ways the fiber can be viewedas equivalent. For instance, in topology, the group might be thegroup of homeomorphisms of the fiber. The group on a vectorbundle is the group of invertible linear maps, which reflects theequivalent descriptions of a vector space using different vectorspace bases.

While the definition of a bundle is a very general one, we will beapplying the definition of a bundle to several specialized categories.

I (a)Smooth: E ,B,F are smooth manifolds, maps are smooth.I (b)TopM: E ,B,F are manifolds, maps are continous maps.I (c)Holomorphic: E ,B,F are smooth complex manifolds,

maps are holomorphic.

Suppose that (E ,B,F , π) is a bundle. We identify two specialcases by placing restrictions on F and π.

Vector Bundles

What is a vector bundle?


If F is a linear vector space (eg Rn,Cn) and the identificationsπ−1(U) ∼= U × F are linear maps, then we call (E ,B,F , π) avector bundle.

The tangent, normal, and tautological bundles are all vectorbundles.

FactIt is worth noting that any vector bundle has a zero section (whichis in fact a global section).

Further Examples of Vector bundles


If F is a Lie group which has a smooth right action of E such that

I The action is free (i.e, e · g = e if and only if g is the identityelement)

I (b) The action preserves the fibers of E → B.

We then call (E ,B,F , π) a principal F−bundle.

The Hopf bundle is an example of a principal S1 bundle and thehomogeneous bundle O(n)→ O(n)/O(n − 1) is a principalO(n − 1) bundle.

Local Frames

We have already mentioned the trivialization now we want toexplain the notion of a local frame

DefinitionRecall that a frame of an n-dimensional vector bundle E, over anopen subset U ⊂ M, is a family of sections (e1, · · · en) ∈ E (U)that form a basis at each point; thus e1, · · · , en forms a vectorbundle isomorphism between E |U and the trivial bundle.

We can in fact say that a trivialization and a local frame areequivalent.

Direct Sums

Given two vector bundles p1 : E1 7→ B and p2 : E2 7→ B over thesame base manifold B. We can define direct sums of these twovector bundles.

Definition

E1 ⊕ E2 = {(v1, v2) ∈ E1 × E2|p1(v1) = p2(v2)} (1)

There is a projection E1 ⊕ E2 7→ B sending (v1, v2) to the pointp1(v1) = p2(v2)

The fibers of this projection are the direct sum of the fibers of E1

and E2, as we wanted. We can verify this after making twopreliminary observations

Direct sums are defined fiberwise

We need two observations to verify the fiberwise-ness of directsums of vector bundles

1. Given a vector bundle p : E 7→ B and a subspace A ⊂ B thenp : p−1 (A) 7→ A is clearly a vector bundle. We will call thisthe restriction of E over A.

2. Given a vector bundle (v.b.) p1 : E1 7→ B1 and p2 : E2 7→ B2

then p1 × p2 : E1 × E2 7→ B1 × B2 is also a vector bundle,with fibers the product p−11 (b1)× p−12 (b2). For if we havelocal trivialization

hα : p−11 (Uα) 7→ Uα × Rnandhβ : p−12 (Uβ) 7→ Uβ × Rm

for E1 and E2, then hα× hβ is a local trivialization for E1×E2.

Direct Sums- continued

The upshot of the previous observations is that if E1 and E2 havethe same base space B, the restriction of the product E1 × E2 overthe diagonal B = {(b, b) ∈ B × B} is exactly E1 ⊕ E2.

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