III.96 Vector Bundles

Let X be a TOPOLOGICAL SPACE [III.90]. A vector bundle over X is, roughly speaking, a way of associating a vector space with each point x of X in such a way that these spaces “vary continuously” as you vary x. As an example, consider a smooth surface X in ³. Associated with each point x is the tangent plane at x, which varies continuously with x and can be identified in a natural way with a two-dimensional vector space. A more precise definition is as follows: a vector bundle of rank n over X is a topological space E, together with a continuous map p : E → X, such that the inverse image p^-1 (x) of each point x (that is, the set of points in E that map to x) is an n-dimensional vector space. Moreover, for every sufficiently small region U of X, the inverse image of U is homeomorphic to ⁿ × U (this property is called local triviality). The most obvious vector bundle of rank n over X is the space ⁿ × X with the map p(υ,x) = x; this is called the trivial bundle. However, the interesting bundles are the nontrivial ones, such as the tangent bundle of the 2-sphere. One can learn a great deal about a topological space by understanding its vector bundles. For this reason, vector bundles are central to algebraic topology. See ALGEBRAIC TOPOLOGY [IV.6 §5] for more details.