Vector bundle class functor is homotopy-invariant for paracompact

This article describes a result applicable for real vector bundles over a paracompact Hausdorff space. In particular, the result is applicable for real vector bundles over a manifold, CW-space, or metrizable space

Statement

Suppose $A$ and $B$ are paracompact Hausdorff spaces and $f_{0},f_{1}:A\to B$ are continuous maps. Then, the functorially induced maps by the vector bundle class functor, namely:

$Vect^{n}(f_{0}):Vect^{n}(B)\to Vect^{n}(A)$

and:

$Vect^{n}(f_{1}):Vect^{n}(B)\to Vect^{n}(A)$

are equal, i.e. $Vect^{n}(f_{0})=Vect^{n}(f_{1})$.

Corollaries

• Any homotopy equivalence of paracompact Hausdorff spaces induces an isomorphism on the vector bundle classes over them.
• In particular, any vector bundle over a contractible paracompact Hausdorff space is trivial.