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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!

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Yeah, you're right; edited! –  Zach Conn Jun 29 '11 at 1:12
Serre-Swan always struck me as more philosophically important than anything else. –  Qiaochu Yuan Jun 29 '11 at 3:12

3 Answers 3

up vote 10 down vote accepted

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.

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You could perhaps do worse than consulting $\S 6.4$ of my commutative algebra notes: "Applications of Swan's Theorem." (You could definitely do better: see below.)

The first application I give is to show that the ring of real-valued continuous functions on $[0,1]$ is a connected ring in which each finitely generated projective module is free but for which there is a nonfree infinitely generated projective module. As I admit myself in the notes, it is possible to prove this purely algebraically and I allude to another proof taken from one of Lam's books, but the topological approach is a nice one.

The second application is a big one: it exhibits stably free non-free modules over the ring $\mathbb{R}[x_0,\ldots,x_n]/(x_0^2+\ldots + x_n^2 - 1)$ of polynomial functions on the $n$-sphere when $n \neq 0,1,3,7$. This is done by reducing to the known behavior of tangent bundles to the $n$-sphere in the usual differential topological setting.

By the way, I got this second application directly from Swan's paper. There are other applications given there as well...

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Re first application: You may remember this thread. Re second application: This blurb of Keith Conrad's is extremely nice. –  t.b. Jun 29 '11 at 8:16

In the theory of spectral triples, Connes's reconstruction theorem tells you that for a commutative spectral triple $(A,H,D)$ of metric dimension $p$, $A \cong C^\infty(X)$ for a (specific!) compact oriented $p$-manifold $X$. You can then use Serre--Swan to conclude that $H \cong L^2(X,E)$ for a Hermitian vector bundle $E \to X$ and that $D$ can be viewed as an essentially self-adjoint first-order differential operator on $E$. Very crudely speaking, you can view this as establishing a bridge between the $K$-homology of the manifold $X$ and the $K$-homology of the $C^\ast$-algebra $C(X)$ compatible with the identification $K^i(C(X)) \cong K_{-i}(X)$ of $K$-theories.

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