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vector bundle

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Vector bundles

Idea

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

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 March 29, 2014 04:22:42 by Urs Schreiber (185.37.147.12)