The Serre-Swan theorem states (at least in one form) that the category of real vector bundles over a compact Hausdorff space $M$ is equivalent to the category of finitely generated projective modules over the ring $C(M)$ of continuous functions on $M$. The equivalence is provided by the functor $\Gamma$ sending a bundle to the totality of all its continuous sections.
Are there any classic applications or uses of this theorem? To me right now it seems like a pristine result to be admired from a distance, as I currently know of no actual use for it. I'd love to remedy that!
Best Answer
One application of this is that topological K-theory (i.e., the K-theory of the exact category of vector bundles on the space) is the same thing as the "algebraic" $K_0$ of the ring of continuous functions (i.e., the K-theory of the exact category of finitely generated projective modules over that ring). So topological K-theory is a "special case" of algebraic K-theory (though the tools for proving things like Bott periodicity are very different from those used in proving general results on exact category in algebraic K-theory).
By the way, here is the corresponding result in algebra:
Algebraic vector bundles over an affine scheme $\mathrm{Spec} A$ are the same as finitely generated projective $A$-modules (let's say $A$ is noetherian). So a module is projective if and only if it is locally free, in algebra language.