Vector bundles and projective modules

Abstract

Serre [9, §50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring.For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring of continuous real-valued functions on X.A number of examples of projective modules have been given using this correspondence.However, no rigorous treatment of the correspondence seems to have been given.I will give such a treatment here and then give some of the examples which may be constructed in this way.1. Preliminaries.Let K denote either the real numbers, complex numbers or quaternions.A X-vector bundle £ over a topological space X consists of a space F(£) (the total space), a continuous map p : E(Ç) -+ X (the projection) which is onto, and, on each fiber Fx(¡z) = p-1(x), the structure of a finite dimensional vector space over K.These objects are required to satisfy the following condition: for each xeX, there is a neighborhood U of x, an integer n, and a homeomorphism <p:p-1(U)-> U x K" such that on each fiber <b is a X-homomorphism.The fibers u x Kn of U x K" are X-vector spaces in the obvious way.Note that I do not require n to be a constant.The dimension of the fiber Fx may vary with x.However, this dimension is clearly locally constant and so will be constant if X is connected.A subbundle of t] is, by definition, a subset Ex <= E (Ç) such that Ex n Fx is a X-subspace of Fx for each x and such that Ex with the projection p | Ex and the X-structure on its fibers induced by that of E forms a X-vector bundle over X.A map of X-vector bundles /:{-»i; is defined to be a continuous map f:E(Ç)-+E(n) such that pf' = p and such that /| Fx(0 : Fx(0 -► Fx(n) is a Xhomomorphism.It is clear that the X-vector bundles over X and their maps form an additive category.A section s of {; over a subset A cz X is a continuous map s : A -+ £(£) such that ps(x) = x.It follows immediately from the definition of a vector bundle that for any xeX, there is a neighborhood U of x and sections sx, ...,s" of £