nLab
vector bundle

Context

Bundles

Linear algebra

Vector bundles

Idea

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

A basic example of vector bundles are tangent bundles of smooth manifolds XX. Here the vector space at each point of XX is the tangent space of that point, the space of all tangent vectors based at that point. The graphics on the right shows one of the tangent space of the 2-sphere.

graphics grabbed from Hatcher

Definition

General

A vector bundle over a space XX is a bundle over XX which is locally isomorphic to a product with a vector space VV as fiber. More precisely, the data is an object π:EX\pi: E \to X in Top/XTop/X equipped with a vector space structure internal to Top/XTop/X, consisting of maps

+:E× XEE:×EE+: E \times_X E \to E \qquad \cdot: \mathbb{R} \times E \to E

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

U= αAU αXU = \sum_{\alpha \in A} U_\alpha \rightrightarrows X

and an isomorphism from the pullback U× XEU \times_X E to the projection π:U×VU\pi: U \times V \to U,

U×V ϕ U× XE E π UX\array{ U \times V & \overset{\phi}{\leftarrow} & U \times_X E & \to & E\\ & & \downarrow & & \downarrow \pi \\ & & U \rightrightarrows X }

as vector space objects in Top/XTop/X. The projection U×VUU \times V \to U itself is called a trivial (vector) bundle over UU.

Equivalently, each fiber E xE_x carries a vector space structure, and there exists an open covering {U α} αA\{U_\alpha\}_{\alpha \in A} of XX together with local trivializations: bundle isomorphisms ϕ α\phi_{\alpha} from a trivial bundle U α×VU_{\alpha} \times V to the pullback of π\pi along U αXU_\alpha \hookrightarrow X:

π 1(U α) ϕ α U α×V π| U α proj U α \array{ \pi^{-1}(U_\alpha) & \overset{\phi_\alpha}{\to} & U_\alpha \times V\\ \pi|_{U_\alpha} \searrow & & \swarrow proj\\ & U_\alpha & }

such that ϕ α\phi_{\alpha} induces a linear map VE xV \to E_x between the fibers.

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

(U αU β)×Vϕ βϕ α 1(U αU β)×V:(x,v)(x,g αβ(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 αβ(x)g_{\alpha\beta}(x) are linear automorphisms of VV and satisfy the Čech 1-cocycle conditions:

g βγg αβ=g αγg αα=idg_{\beta\gamma} \circ g_{\alpha\beta} = g_{\alpha\gamma} \qquad g_{\alpha\alpha} = id

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

i,μ: α,β(U αU β)×V αU α×Vi, \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 Top/XTop/X. Here the restriction of ii to the coproduct summands is induced by inclusion:

(U αU β)×VU α×V αU α×V(U_\alpha \cap U_\beta) \times V \hookrightarrow U_\alpha \times V \hookrightarrow \sum_\alpha U_\alpha \times V

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

(U αU β)×V(incl,g αβ)×VU β×GL(V)×VactionU β×V βU β×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 \mathbb{R} or \mathbb{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 VV is finite-dimensional.

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

Sheaf-theoretic version

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

X×XX \times \mathbb{R} \to X

forms a local ring object RR; when interpreted in the internal logic of Sh(X)Sh(X), 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 RR-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 2\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 2\mathbb{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

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 February 7, 2017 13:20:06 by Urs Schreiber (62.75.239.20)