nLab
vector bundle

Idea

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

Definition

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

$$+:E{\times }_{X}E\to E\cdot :ℝ\times E\to E$$+: E \times_X E \to E \qquad \cdot: \mathbb{R} \times E \to E

(where $E{\times }_{X}E$ 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

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

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

$$\begin{array}{ccccc}U\times V& \stackrel{\varphi }{\leftarrow }& U{\times }_{X}E& \to & E\\ & & \downarrow & & \downarrow \pi \\ & & U\rightrightarrows X\end{array}$$\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 $U\times V\to U$ 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 $\{U_{\alpha }{\}}_{\alpha \in A}$ of $X$ together with local trivializations: bundle isomorphisms ${\varphi }_{\alpha }$ from a trivial bundle $U_{\alpha }\times V$ to the pullback of $\pi$ along $U_{\alpha }\hookrightarrow X$:

$$\begin{array}{ccc}{\pi }^{-1}(U_{\alpha })& \stackrel{{\varphi }_{\alpha }}{\to }& U_{\alpha }\times V\\ \pi {\mid }_{U_{\alpha }}\searrow & & \swarrow \mathrm{proj}\\ & U_{\alpha }& \end{array}$$\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

$$(U_{\alpha }\cap U_{\beta })\times V\stackrel{{\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}}{\to }(U_{\alpha }\cap U_{\beta })\times V:(x,v)\mapsto (x,g_{\alpha \beta }(x)(v)),$$(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 $g_{\alpha \beta }(x)$ are linear automorphisms of $V$ and satisfy the Čech 1-cocycle conditions:

$$g_{\beta \gamma }\circ g_{\alpha \beta }=g_{\alpha \gamma }{g}_{\alpha \alpha }=\mathrm{id}$$g_{\beta\gamma} \circ g_{\alpha\beta} = g_{\alpha\gamma} \qquad g_{\alpha\alpha} = id

In the converse direction, given such a collection $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \mathrm{GL}(V)$ 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

$$i,\mu :\sum _{\alpha ,\beta }(U_{\alpha }\cap U_{\beta })\times V\overrightarrow{\to }\sum _{\alpha }{U}_{\alpha }\times V$$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:

$$(U_{\alpha }\cap U_{\beta })\times V\hookrightarrow U_{\alpha }\times V\hookrightarrow \sum _{\alpha }{U}_{\alpha }\times V$$(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:

$$(U_{\alpha }\cap U_{\beta })\times V\stackrel{(⟨\mathrm{incl},g_{\alpha \beta }⟩)\times V}{\to }{U}_{\beta }\times \mathrm{GL}(V)\times V\stackrel{\mathrm{action}}{\to }{U}_{\beta }\times V\hookrightarrow \sum _{\beta }{U}_{\beta }\times V$$(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 $\mathrm{Sh}(X)$ be the category of (set-valued) sheaves on $X$. The sheaf of continuous local sections of the product projection

$$X\times ℝ\to X$$X \times \mathbb{R} \to X

forms a local ring object $R$; when interpreted in the internal logic of $\mathrm{Sh}(X)$, it is the Dedekind real number 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-theory – generalized (Eilenberg-Steenrod) cohomology theory – cocycles are represented by ${\mathbb{Z}}_2$-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 ${\mathbb{Z}}_2$-graded vector bundles called vectorial bundles.

Much else to be discussed…