nLab
vector bundle

Context

Bundles

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Vector bundles

Idea

Given some context of geometry, then a vector bundle is a collection of vector spaces that varies in a geometric way over a given base space XX: over each element xXx \in X there is a vector space V xV_x, called the fiber over xx, and as xx varies in XX, the fibers vary along in a geometric way. One also says that vector bundles are fiber bundles whose fiber carries vector space-structure. Hence the theory of vector bundle is parameterized linear algebra.

For example

and so on.

One requires that “locally”, on small enough patches of the base space XX, the variation of the fibers is constant up to isomorphism (one says the vector bundle is “locally trivial”), but the key point of vector bundles is that there may be non-trivial structure in how the collection of vector spaces “globally glues together”.

For example if X=S 1X = S^1 is the circle regarded as a topological space in the standard way, and if we consider real vector spaces, then there are up to isomorphism two different \mathbb{R}-vector bundles over S 1S^1 whose fibers look like the 1-dimensional real vector space \mathbb{R} itself, namely

  1. the cylinder

  2. the Möbius strip:

(In these pictures each vertical interval is to be thought of as a stand-in for a copy of the real line \mathbb{R}.)

Clearly for the cylinder nothing special happens to the fibers as one moves around the cricle (one says this is a trivial vector bundle) while the Möbius strip is “locally trivial” but globally has a twist: as one moves once around the circle the original fiber comes back identified with its reflection at the origin.

graphics grabbed from Hatcher

An important class of exmaples of vector bundles are tangent bundles of differentiable 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.

Dually, given an embedding of differentiable manifolds into a Euclidean space, then the normal vectors to the tangent bundle span a vector bundle called the normal bundle of the embedding.

All the usual operations on finite dimensional vector spaces in linear algebra generalize to vector bundles by applying them fiber-wise. For instance there is direct sum of vector bundles and the tensor product of vector bundles over the same base space.

To the extent that the base space XX is encoded in its algebra of functions (tautologically in algebraic geometry or via Gelfand duality in topology), the Serre-Swan theorem asserts that vector bundles over XX are equivalently encoded in the projective modules over these algebras constitutes by their sections.

Vector bundles have various applications and uses:

  1. their Grothendieck group under direct sum of vector bundles yields topological K-theory, an interesting generalized (Eilenberg-Steenrod) cohomology theory;

  2. a reduction of the structure group of vector bundles encodes actual geometry on the base space; when applied to tangent bundles such G-structures on vector bundles encode for instance orthogonal structure, Riemannian geometry, complex geometry, symplectic geometry, conformal geometry etc. (in general: Cartan geometry); when applied to normal bundles these G-structures give rise, via Thom's theorem, to Thom spectra and cobordism theory;

  3. equipping differentiable vector bundles with connection on a vector bundle is the basis for Chern-Weil theory and for the application of vector bundles in physics, where they model gauge fields and instanton sectors; see also at fiber bundles in physics.

Definition

Standard

See at topological vector bundle

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 R-module.

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

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 August 10, 2017 13:16:58 by Urs Schreiber (46.183.103.17)