Vector bundles

Definition 9.1.1. Let \(X\) be a topological space. A finite-rank complex vector bundle over \(X\) is a continuous map \(p\colon E\to X\), together with a complex vector space structure on every fiber \(E_x:=p^{-1}(x)\), satisfying the following local triviality condition: every point \(x\in X\) admits an open neighborhood \(U\), an integer \(k\geq 0\), and a homeomorphism \[ \phi \colon U\times \C ^k\xrightarrow {\cong }p^{-1}(U) \] over \(U\) whose restriction \(\{y\}\times \C ^k\to E_y\) is complex-linear for every \(y\in U\). We will usually simply call this a complex vector bundle.

The rank function \[ \rk _E\colon X\longrightarrow \N , \qquad x\longmapsto \dim _{\C }(E_x), \] is locally constant. We say that \(E\) has rank \(k\) if this function is constant with value \(k\). More generally, each subspace \(X_k:=\rk _E^{-1}(k)\) is clopen in \(X\), and the restriction of \(E\) to \(X_k\) has rank \(k\).

Given another complex vector bundle \(q\colon E'\to X\), a morphism of vector bundles \(E\to E'\) is a continuous map \(f\colon E\to E'\) over \(X\) whose restriction \(E_x\to E'_x\) is complex-linear for every \(x\in X\). We write \(\pi _0\Vect (X)\) for the set of isomorphism classes of finite-rank complex vector bundles over \(X\), and \(\pi _0\Vect _k(X)\) for the subset represented by bundles of rank \(k\).