Homotopy invariance of vector bundles

Theorem 9.1.4 (Homotopy invariance of vector bundles, Hatcher (2003), Theorem 1.6 and Corollary 1.8). Let \(p\colon E \to X\) be a complex vector bundle and let \(f_0,f_1\colon Y \to X\) be homotopic continuous maps. If \(Y\) is paracompact Hausdorff, then the pullback bundles \(f_0^*E\) and \(f_1^*E\) are isomorphic.

In particular, every homotopy equivalence \(f\colon X \to Y\) between paracompact Hausdorff spaces induces a bijection \[ f^*\colon \pi _0\Vect (Y) \xrightarrow {\cong } \pi _0\Vect (X). \]