Tags: #homotopy #homotopy/bundles #homotopy/fibrations #homotopy

Fiber Bundles

What is a fiber bundle? Generally speaking, it is similar to a fibration - we require the homotopy lifting property to hold, although it is not necessary that path lifting is unique.

However, it also satisfies more conditions - in particular, the condition of local triviality. This requires that the total space looks like a product locally, although there may some type of global monodromy. Thus with some mild conditions ¹ , fiber bundles will be instances of fibrations (or alternatively, fibrations are a generalization of fiber bundles, whichever you prefer!)

As with fibrations, we can interpret a fiber bundle as “a family of \(F\)s indexed/parameterized by \(B\)s”, and the general shape data of a fiber bundle is similarly given by

where \(B\) is the base space, \(E\) is the total space, \(\pi: E \to B\) is the projection map, and \(F\) is “the” fiber (in this case, unique up to homeomorphism). Fiber bundles are often described in shorthand by the data \(E \xrightarrow{\pi}B\), or occasionally by tuples such as \((E, \pi, B)\).

The local triviality condition is a requirement that the projection \(\pi\) locally factors through the product; that is, for each open set \(U\in B\), there is a homeomorphism \(\varphi\) making this diagram commute:

Fiber bundles may admit right-inverses to the projection map \(s: B\to E\) satisfying \(\pi \circ s = \operatorname{id}_B\), denoted bundle sections. Equivalently, for each \(b\in B\), a section is a choice of an element \(e\) in the preimage \(\pi^{-1}(b) \simeq F\) (i.e. the fiber over \(b\)). Sections are sometimes referred to as cross-sections in older literature, due to the fact that a choice of section yields might be schematically represented as such:

Here, we imagine each fiber as a cross-section or “level set” of the total space, giving rise to a "foliation of \(E\) by the fibers. ²

For a given bundle, it is generally possible to choose sections locally, but there may or may not exist globally defined sections. Thus one key question is when does a fiber bundle admit a global section?

A bundle is said to be trivial if \(E = F \times B\), and so another important question is when is a fiber bundle trivial?

Definition: A fiber bundle in which \(F\) is a \(k{\hbox{-}}\)vector space for some field \(k\) is referred to as a rank \(n\) vector bundle. When \(k={\mathbf{R}}, {\mathbf{C}}\), they are denoted real/complex vector bundles respectively. A vector bundle of rank \(1\) is often referred to as a line bundle.

Example: There are in fact non-trivial fiber bundles. Consider the space \(E\) that can appear as the total space in a line bundle over the circle

\begin{align*} {\mathbf{R}}^1 \to E \to S^1\end{align*}

That is, the total spaces that occur when a one-dimensional real vector space (i.e. a real line) is chosen at each point of \(S^1\). One possibility is the trivial bundle \(E \cong S^1 \times{\mathbf{R}}\cong S^1 \times I^\circ \in \text{DiffTop}\), which is an “open cylinder”:

But another possibility is \(E \cong M^\circ \in\text{DiffTop}\), an open Mobius band.

Here we can take the base space \(B\) to be the circle through the center of the band; then every open neighborhood \(U\) of a point \(b\in B\) contains an arc of the center circle crossed with a vertical line segment that misses \({\partial}M\). Thus the local picture looks like \(S^1 \times I^\circ\), while globally \(M\not\cong S^1 \times I^\circ \in \text{Top}\). ³

So in terms of fiber bundles, we have the following situation

Link to diagram

and thus \(M\) is associated to a nontrivial line bundle over the circle.

Remark: In fact, these are the only two line bundles over \(S^1\). This leads us to a natural question, similar to the group extension question: given a base \(B\) and fiber \(F\), what are the isomorphism classes of fiber bundles over \(B\) with fiber \(F\)? In general, we will find that these classes manifest themselves in homology or homotopy. As an example, we have the following result:

Notation: Let \(I(F, B)\) denote isomorphism classes of fiber bundles of the form \(F \to {-}\to B\).

Proposition

The set of isomorphism classes of smooth line bundles over a space \(B\) satisfies the following isomorphism of abelian groups:

\begin{align*}I({\mathbf{R}}^1, B) \cong H^1(B; {\mathbf{Z}}_2) \in \text{Ab}\end{align*}

in which the RHS is generated by the first Stiefel-Whitney class \(w_1(B)\).

Proof:

Lemma: The structure group of a vector bundle is a general linear group. (Or orthogonal group, by Gram-Schmidt)

Lemma: The classifying space of \(\operatorname{GL}(n, {\mathbf{R}})\) is \({\operatorname{Gr}}(n, {\mathbf{R}}^\infty)\)

Lemma: \({\operatorname{Gr}}(n, {\mathbf{R}}^\infty) = {\mathbf{RP}}^\infty \simeq K({\mathbf{Z}}_2, 1)\)

Lemma: For \(G\) an abelian group and \(X\) a CW complex, \([X, K(G, n)] \cong H^n(X; G)\)

The structure group of a vector bundle can be taken to be either the general linear group or the orthogonal group, and the classifying space of both groups are homotopy-equivalent to an infinite real Grassmannian.

\begin{align*} I({\mathbf{R}}^1, B) &= [B, {\mathbf{B}}\mathop{\mathrm{Aut}}_{{ \mathsf{Vect} }}({\mathbf{R}})]\\ &= [B, {\mathbf{B}}\operatorname{GL}(1, {\mathbf{R}})]\\ &= [B, {\operatorname{Gr}}_1({\mathbf{R}}^\infty)] \\ &= [B, {\mathbf{RP}}^\infty] \\ &= [B, K({\mathbf{Z}}/2, 1)] \\ &= H^1(B; {\mathbf{Z}}/2) \end{align*}

This is the general sort of pattern we will find - isomorphism classes of bundles will be represented by homotopy classes of maps into classifying spaces, and for nice enough classifying spaces, these will represent elements in cohomology.

Corollary: There are two isomorphism classes of line bundles over \(S^1\), generated by the Mobius strip, since \(H^1(S^1, {\mathbf{Z}}_2) = {\mathbf{Z}}_2\) (Note: this computation follows from the fact that \(H_1(S^1) = {\mathbf{Z}}\) and an application of both universal coefficient theorems.)

Note: The Stiefel-Whitney class is only a complete invariant of line bundles over a space. It is generally an incomplete invariant; for higher dimensions or different types of fibers, other invariants (so-called characteristic classes) will be necessary.

Another important piece of data corresponding to a fiber bundle is the structure group, which is a subgroup of \(\text{Sym}(F) \in \text{Set}\) and arises from imposing conditions on the structure of the transition functions between local trivial patches. A fiber bundle with structure group \(G\) is referred to as a \(G{\hbox{-}}\)bundle.

See next: 2021-04-25_vector_bundles_ug