---
date: 2021-04-25T02:33
---

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.

![lifting - todo tikz](https://upload.wikimedia.org/wikipedia/en/b/b9/Homotopy_lifting_property.png)

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^[A fiber bundle $E \to B$ is a fibration when $B$ is paracompact.], 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

\begin{tikzcd}
F \arrow[rr, hook] &  & E \arrow[dd, "\pi", two heads] \\
 &  &  \\
 &  & B \arrow[uu, "s", dotted, bend left]
\end{tikzcd}


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 $E \mapsvia{\pi}B$, 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:

\begin{tikzcd}
\pi^{-1}(U) \arrow[dd, "\pi", two heads] \arrow[rr, "\varphi", dashed] &  & U\times F \arrow[lldd, "{(a,b) \mapsto a}"] \\
 &  &  \\
U &  &
\end{tikzcd}

Fiber bundles may admit right-inverses to the projection map $s: B\to E$ satisfying $\pi \circ s = \id_B$, denoted [bundle sections](bundle%20sections). Equivalently, for each $b\in B$, a section is a choice of an element $e$ in the preimage $\pi^{-1}(b) \homotopic F$ (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.^[When $E$ is in fact a product $F\cross B$, this actually is a foliation in the technical sense.]

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](global%20section)?**

A bundle is said to be _trivial_ if $E = F \cross B$, and so another important question is **when is a fiber bundle trivial?**

**Definition**: A fiber bundle in which $F$ is a $k\dash$vector space for some field $k$ is referred to as a _rank $n$ [vector bundle](vector%20bundle)._ When $k=\RR, \CC$, they are denoted real/complex vector bundles respectively. A vector bundle of rank $1$ is often referred to as a _[line bundle](line%20bundle.md)_.


**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

$$ \RR^1 \to E \to S^1$$

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 $E \cong S^1 \cross \RR \cong S^1 \cross I^\circ \in \text{DiffTop}$, 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 $\del M$. Thus the local picture looks like $S^1 \cross I^\circ$, while globally $M\not\cong S^1 \cross I^\circ \in \text{Top}$.^[Due to the fact that, for example, $M$ is nonorientable and orientability distinguishes topological spaces up to homeomorphism.]

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


\begin{tikzcd}
	\RR && M \\
	\\
	&& {S^1}
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to diagram](% https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXFJSIl0sWzIsMCwiTSJdLFsyLDIsIlNeMSJdLFsxLDJdLFswLDFdXQ==)

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 $F \to \wait \to B$.

## Proposition

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

$$I(\RR^1, B) \cong H^1(B; \ZZ_2) \in \text{Ab}$$

in which the RHS is generated by the first [Stiefel-Whitney class](Stiefel-Whitney%20class.md) $w_1(B)$.

_Proof:_

_Lemma:_ The [structure group](structure%20group.md) of a vector bundle is a general linear group. (Or orthogonal group, by Gram-Schmidt)

_Lemma:_ The [classifying space](classifying%20space.md) of $\GL(n, \RR)$ is $\Gr(n, \RR^\infty)$

_Lemma_: $\Gr(n, \RR^\infty) = \RP^\infty \homotopic K(\ZZ_2, 1)$

_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](Grassmannian.md).

\[
I(\RR^1, B) &= [B, \B\Aut_{\Vect}(\RR)]\\
&= [B, \B\GL(1, \RR)]\\
&= [B, \Gr_1(\RR^\infty)] \\
&= [B, \RP^\infty] \\
&= [B, K(\ZZ/2, 1)] \\
&= H^1(B; \ZZ/2)
\]


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 $H^1(S^1, \ZZ_2) = \ZZ_2$ (Note: this computation follows from the fact that $H_1(S^1) = \ZZ$ and an application of both universal coefficient theorems.)

**Note:** 
The [Stiefel-Whitney class](Stiefel-Whitney%20class.md) 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](structure%20group.md)_, 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 _$G\dash$bundle_.

> See next: [2021-04-25_vector_bundles_ug](2021-04-25_vector_bundles_ug.md)