Vector bundle class functor is homotopy-invariant for paracompact

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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.