Vector bundle

A vector bundle is a geometric structure that attaches a vector space to each point of a topological space in a continuous way, generalizing the tangent bundle of a manifold. Formally, it consists of a total space E, a base space M, a projection π: E → M, and a vector space structure on each fiber π⁻¹(p), such that locally the bundle looks like a product U × V.

Vector bundles are the objects on which characteristic classes like the Chern class and Pontryagin class are defined. They are the natural setting for the Atiyah-Singer index theorem and for the formulation of gauge fields in physics. Every force in the Standard Model is described by a connection on a principal bundle, whose associated vector bundles carry the matter fields.

The classification of vector bundles over a space is one of the central problems of K-theory and algebraic topology. A vector bundle is trivial — globally a product — if and only if all of its characteristic classes vanish, but the converse is not always true: there exist nontrivial bundles with vanishing Chern classes.