Vector bundles and projective modules

Vaserstein, Leonid N.

doi:10.1090/S0002-9947-1986-0825734-3

Vector bundles and projective modules
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by Leonid N. Vaserstein PDF

Trans. Amer. Math. Soc. 294 (1986), 749-755 Request permission

Abstract:

Serre and Swan showed that the category of vector bundles over a compact space $X$ is equivalent to the category of finitely generated projective modules over the ring of continuous functions on $X$. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space $X$. Also the well-known homotopy classification of the vector bundles over compact $X$ up to isomorphism is extended to arbitrary $X$. It is shown that the ${K_0}$-functor and the Witt group of the ring of continuous functions on $X$ coincide, and they are homotopy-type invariants of $X$.

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