bundles

cohomology

# Vector bundles

## Idea

A vector bundle is a vector space which “continuously varies” over a topological space $X$.

## Definition

### General

A vector bundle over a space $X$ is a bundle over $X$ which is locally isomorphic to a product with a vector space $V$ as fiber. More precisely, the data is an object $\pi :E\to X$ in $\mathrm{Top}/X$ equipped with a vector space structure internal to $\mathrm{Top}/X$, consisting of maps

$+:E{×}_{X}E\to E\phantom{\rule{2em}{0ex}}\cdot :ℝ×E\to E$+: E \times_X E \to E \qquad \cdot: \mathbb{R} \times E \to E

(where $E{×}_{X}E$ denotes the fiber product or pullback of $\pi$ along itself) satisfying vector space axioms. This vector space object must satisfy the local triviality condition: there exists an open cover

$U=\sum _{\alpha \in A}{U}_{\alpha }⇉X$U = \sum_{\alpha \in A} U_\alpha \rightrightarrows X

and an isomorphism from the pullback $U{×}_{X}E$ to the projection $\pi :U×V\to U$,

$\begin{array}{ccccc}U×V& \stackrel{\varphi }{←}& U{×}_{X}E& \to & E\\ & & ↓& & ↓\pi \\ & & U⇉X\end{array}$\array{ U \times V & \overset{\phi}{\leftarrow} & U \times_X E & \to & E\\ & & \downarrow & & \downarrow \pi \\ & & U \rightrightarrows X }

as vector space objects in $\mathrm{Top}/X$. The projection $U×V\to U$ itself is called a trivial (vector) bundle over $U$.

Equivalently, each fiber ${E}_{x}$ carries a vector space structure, and there exists an open covering $\left\{{U}_{\alpha }{\right\}}_{\alpha \in A}$ of $X$ together with local trivializations: bundle isomorphisms ${\varphi }_{\alpha }$ from a trivial bundle ${U}_{\alpha }×V$ to the pullback of $\pi$ along ${U}_{\alpha }↪X$:

$\begin{array}{ccc}{\pi }^{-1}\left({U}_{\alpha }\right)& \stackrel{{\varphi }_{\alpha }}{\to }& {U}_{\alpha }×V\\ \pi {\mid }_{{U}_{\alpha }}↘& & ↙\mathrm{proj}\\ & {U}_{\alpha }& \end{array}$\array{ \pi^{-1}(U_\alpha) & \overset{\phi_\alpha}{\to} & U_\alpha \times V\\ \pi|_{U_\alpha} \searrow & & \swarrow proj\\ & U_\alpha & }

such that ${\varphi }_{\alpha }$ induces a linear map $V\to {E}_{x}$ between the fibers.

In terms of the local trivialization data, there are transition functions

$\left({U}_{\alpha }\cap {U}_{\beta }\right)×V\stackrel{{\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}}{\to }\left({U}_{\alpha }\cap {U}_{\beta }\right)×V:\left(x,v\right)↦\left(x,{g}_{\alpha \beta }\left(x\right)\left(v\right)\right),$(U_\alpha \cap U_\beta) \times V \overset{\phi_\beta \circ \phi_{\alpha}^{-1}}{\to} (U_\alpha \cap U_\beta) \times V: (x, v) \mapsto (x, g_{\alpha\beta}(x)(v)),

where the ${g}_{\alpha \beta }\left(x\right)$ are linear automorphisms of $V$ and satisfy the Čech 1-cocycle conditions:

${g}_{\beta \gamma }\circ {g}_{\alpha \beta }={g}_{\alpha \gamma }\phantom{\rule{2em}{0ex}}{g}_{\alpha \alpha }=\mathrm{id}$g_{\beta\gamma} \circ g_{\alpha\beta} = g_{\alpha\gamma} \qquad g_{\alpha\alpha} = id

In the converse direction, given such a collection ${g}_{\alpha \beta }:{U}_{\alpha }\cap {U}_{\beta }\to \mathrm{GL}\left(V\right)$ satisfying the 1-cocycle conditions, there is a vector bundle obtained by pasting local trivial bundles together along the ${g}_{\alpha \beta }$, namely the coequalizer of a pair

$i,\mu :\sum _{\alpha ,\beta }\left({U}_{\alpha }\cap {U}_{\beta }\right)×V\stackrel{\to }{\to }\sum _{\alpha }{U}_{\alpha }×V$i, \mu: \sum_{\alpha, \beta} (U_\alpha \cap U_\beta) \times V \overset{\to}{\to} \sum_{\alpha} U_\alpha \times V

in the category of vector space objects in $\mathrm{Top}/X$. Here the restriction of $i$ to the coproduct summands is induced by inclusion:

$\left({U}_{\alpha }\cap {U}_{\beta }\right)×V↪{U}_{\alpha }×V↪\sum _{\alpha }{U}_{\alpha }×V$(U_\alpha \cap U_\beta) \times V \hookrightarrow U_\alpha \times V \hookrightarrow \sum_\alpha U_\alpha \times V

and the restriction of $\mu$ to the coproduct summands is via the action of the transition functions:

$\left({U}_{\alpha }\cap {U}_{\beta }\right)×V\stackrel{\left(⟨\mathrm{incl},{g}_{\alpha \beta }⟩\right)×V}{\to }{U}_{\beta }×\mathrm{GL}\left(V\right)×V\stackrel{\mathrm{action}}{\to }{U}_{\beta }×V↪\sum _{\beta }{U}_{\beta }×V$(U_\alpha \cap U_\beta) \times V \overset{(\langle incl, g_{\alpha\beta} \rangle) \times V}{\to} U_\beta \times GL(V) \times V \overset{action}{\to} U_\beta \times V \hookrightarrow \sum_{\beta} U_\beta \times V

Remarks

• In most applications, the ground field of scalars is assumed to be $ℝ$ or $ℂ$, although sometimes other fields are allowed, as in the study of algebraic vector bundles.

• In most cases (as in K-theory), it is implicitly assumed that the vector space $V$ is finite-dimensional.

• In the context of differential topology or differential geometry, one also assumes that $\pi$ is smooth and that the local bundle isomorphisms ${\varphi }_{\alpha }$ are diffeomorphisms.

### Sheaf-theoretic version

Vector bundles can also be defined via sheaf theory, which permits easy transport to general Grothendieck toposes. Let $\mathrm{Sh}\left(X\right)$ be the category of (set-valued) sheaves on $X$. The sheaf of continuous local sections of the product projection

$X×ℝ\to X$X \times \mathbb{R} \to X

forms a local ring object $R$; when interpreted in the internal logic of $\mathrm{Sh}\left(X\right)$, it is the Dedekind real numbers object. Then, according to a theorem of Richard Swan, in its sheaf-theoretic incarnation a vector bundle is the same thing as a projective $R$-module.

• A theorem of Kaplansky states “every projective module over a local ring is free”. When interpreted in sheaf semantics? (Kripke-Joyal semantics), the existential quantifier implicit in “free” is interpreted locally, so we can consider a vector bundle as a locally free module over the Dedekind reals.

### Virtual vector bundles

In one class of models for K-theorygeneralized (Eilenberg-Steenrod) cohomology theory – cocycles are represented by ${ℤ}_{2}$-graded vector bundles (pairs of vector bundles, essentially) modulo a certain equivalence relation. In that context it is sometimes useful to consider a certain variant of infinite-dimensional ${ℤ}_{2}$-graded vector bundles called vectorial bundles.

Much else to be discussed…

## Literature

• Glenys Luke, Alexander S. Mishchenko, Vector bundles and their applications, Math. and its Appl. 447, Kluwer 1998. viii+254 pp. MR99m:55019

• А. С. Мищенко, Векторные расслоения и их применения (Russian; A. S. Mishchenko, Vector bundles and their applications) Nauka, Moscow, 1984. 208 pp.

• Howard Osborn, Vector bundles. Vol. 1. Foundations and Stiefel-Whitney classes, Pure and Appl. Math. 101, Academic Press 1982. xii+371 pp. MR85e:55001

• Dale Husemoller, Fibre bundles, McGraw-Hill 1966 (300 p.); Springer GTM 1975 (327 p.), 1994 (353 p.).

An exposition with an eye towards gauge theory is in section 16.1 of

• Raoul Bott, Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp.

Discussion with an eye towards K-theory is in

• Max Karoubi, K-theory. An introduction, Grundlehren der Mathematischen Wissenschaften 226, Springer 1978. xviii+308 pp.

• Allen Hatcher, Vector bundles and K-Theory, (partly finished book) web

Revised on December 12, 2012 17:07:20 by Urs Schreiber (71.195.68.239)