Mathematics / Fiber Bundles / Local triviality and gluing

Fiber Bundles notes

Chern Classes and Bloch Bundles Fiber Bundles, Connections, and Curvature Principal Bundles and Connections Topological Fiber Bundles Vector Bundles, Frames, and Sections

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Topological Fiber Bundles

The formal definition

We now define fiber bundles formally, before specializing to vector bundles.

Definition 11 (Topological fiber bundle). A topological fiber bundle consists of topological spaces $E$ , $B$ , and $F$ , together with a continuous surjective map

\pi:E\to B,

such that there exists an open cover $\{U_i\}_{i\in I}$ of $B$ and homeomorphisms

\phi_i:\pi^{-1}(U_i)\to U_i\times F

satisfying

\mathrm{pr}_1\circ \phi_i=\pi

on $\pi^{-1}(U_i)$ . Here $\mathrm{pr}_1:U_i\times F\to U_i$ is projection onto the first factor.

The homeomorphism $\phi_i$ is called a local trivialization. The condition

\mathrm{pr}_1\circ \phi_i=\pi

means that the trivialization respects the base point: points lying over $b\in U_i$ are identified with points in $\{b\}\times F$ .

The fiber over $b\in B$ is

E_b=\pi^{-1}(b).

The local trivialization identifies it with $F$ :

E_b\cong F.

What “locally product” means The definition does not say that $E$ is globally $B\times F$ . It says that there exists at least one open cover $\{U_i\}$ such that each restricted piece $\pi^{-1}(U_i)$ is a product $U_i\times F$ .

Does every open cover need to trivialize the bundle?

No. The definition requires the existence of a trivializing open cover. Equivalently, for every $b\in B$ , there must exist some open neighbourhood $U$ of $b$ such that

\pi^{-1}(U)\cong U\times F.

It is not required that every open set $U\subset B$ trivialize the bundle.

However, if $U_i$ is a trivializing open set and $V\subset U_i$ is a smaller open set, then the restriction also trivializes:

\pi^{-1}(V)\cong V\times F.

Thus any refinement of a trivializing cover is again a trivializing cover.

Möbius strip The Möbius strip is locally $U\times\mathbb{R}$ over small arcs $U\subset S^1$ , but it is not globally $S^1\times\mathbb{R}$ . Therefore the open set $U=S^1$ itself does not trivialize the Möbius bundle.

Transition functions

On an overlap $U_i\cap U_j$ , we have two trivializations:

\phi_i:\pi^{-1}(U_i\cap U_j)\to (U_i\cap U_j)\times F,

\phi_j:\pi^{-1}(U_i\cap U_j)\to (U_i\cap U_j)\times F.

The change of trivialization is

\phi_i\circ\phi_j^{-1}:(U_i\cap U_j)\times F\to (U_i\cap U_j)\times F.

Because both trivializations preserve the base point, this map has the form

\phi_i\circ\phi_j^{-1}(b,f)=(b,\tau_{ij}(b)(f)),

where

\tau_{ij}(b):F\to F

is a homeomorphism of the fiber.

If one chooses a structure group $G\subset \mathop{\mathrm{Aut}}(F)$ acting on $F$ , then transition functions are maps

g_{ij}:U_i\cap U_j\to G

such that

\phi_i\circ\phi_j^{-1}(b,f)=(b,g_{ij}(b)\cdot f).

On triple overlaps $U_i\cap U_j\cap U_k$ , consistency gives

g_{ij}g_{jk}=g_{ik}.

Also,

g_{ii}=1, \qquad g_{ji}=g_{ij}^{-1}.

Local data and global object A fiber bundle can be described by local products $U_i\times F$ plus transition functions on overlaps. The nontrivial global information is contained in how the local products are glued.

Constructing a bundle from transition functions

Conversely, suppose we are given:

a base space $B$ with open cover $\{U_i\}$ ,
a fiber $F$ ,
transition functions $g_{ij}:U_i\cap U_j\to G\subset\mathop{\mathrm{Aut}}(F)$ satisfying $g_{ij}g_{jk}=g_{ik}$ .

Then one can construct the total space by gluing:

E=\left(\bigsqcup_i U_i\times F\right)/\sim,

where

(b,f)_j\sim (b,g_{ij}(b)\cdot f)_i

for $b\in U_i\cap U_j$ .

This construction is important because it shows that the total space $E$ is not usually guessed first as a product. It is often built by gluing local products.

Smooth fiber bundles

If $E$ , $B$ , and $F$ are smooth manifolds, a smooth fiber bundle is defined similarly, with all relevant maps smooth and local trivializations diffeomorphisms:

\phi_i:\pi^{-1}(U_i)\to U_i\times F.

The projection $\pi:E\to B$ is then a smooth map, and locally it looks like projection

U_i\times F\to U_i.

Base space assumptions The base $B$ of a topological fiber bundle can be a general topological space. In differential geometry and physics, $B$ is usually a smooth manifold. Compactness is not required. If $B$ is compact, one can often choose finite subcovers; if it is not compact, the definitions still work.

Sections of a fiber bundle

Let $\pi:E\to B$ be any fiber bundle. A section is a map

s:B\to E

such that

\pi\circ s=\mathrm{id}_B.

This means

s(b)\in E_b

for each $b\in B$ .

A section chooses one point in each fiber. When the fiber is a vector space, sections are fields. When the fiber is a group or frame space, sections are gauge choices or frame choices.