K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of the Bott periodicity theorem for topological complex K-theory that I've found in the literature:

- Bott's original proof used Morse theory, which reappeared in Milnor's book
*Morse Theory*in a much less condensed form. - Pressley and Segal managed to produce the homotopy inverse of the usual Bott map as a corollary in their book
*Loop Groups*. - Behrens recently produced a novel proof based on Aguilar and Prieto, which shows that various relevant maps are quasifibrations, therefore inducing the right maps on homotopy and resulting in Bott periodicity.
- Snaith showed that $BU$ is homotopy equivalent to $CP^\infty$ once you adjoin an invertible element. (He and Gepner also recently showed that this works in the motivic setting too, though this other proof relies on the reader having already seen Bott periodicity for motivic complex K-theory.)
- Atiyah, Bott, and Shapiro in their seminal paper titled
*Clifford Modules*produced an algebraic proof of the periodicity theorem.**EDIT:**Whoops x2! They proved the periodicity of the Grothendieck group of Clifford modules, as cdouglas points out, then used topological periodicity to connect back up with $BU$. Wood later gave a more general discussion of this in*Banach algebras and Bott periodicity*. - Atiyah and Bott produced a proof using elementary methods, which boils down to thinking hard about matrix arithmetic and clutching functions. Variations on this have been reproduced in lots of books, e.g., Switzer's
*Algebraic Topology: Homotopy and Homology*. - A proof of the periodicity theorem also appears in Atiyah's book
*K-Theory*, which makes use of some basic facts about Fredholm operators. A differently flavored proof that also rests on Fredholm operators appears in Atiyah's paper*Algebraic topology and operations on Hilbert space*. - Atiyah wrote a paper titled
*Bott Periodicity and the Index of Elliptic Operators*that uses his index theorem; this one is particularly nice, since it additionally specifies a fairly minimal set of conditions for a map to be the inverse of the Bott map. - Seminaire Cartan in the winter of '59-'60 produced a proof of the periodicity theorem using "only standard techniques from homotopy theory," which I haven't looked into too deeply, but I know it's around.

Now, for my question: the proofs of the periodicity theorem that make use of index theory are in some vague sense appealing to the existence of various Thom isomorphisms. It seems reasonable to expect that one could produce a proof of Bott periodicity that explicitly makes use of the facts that:

- The Thom space of the tautological line bundle over $CP^n$ is homeomorphic to $CP^{n+1}$.
- Taking a colimit, the Thom space of the tautological line bundle over $CP^\infty$ (call it $L$) is homeomorphic to $CP^\infty$.
- The Thom space of the difference bundle $(L - 1)$ over $CP^\infty$ is, stably, $\Sigma^{-2} CP^\infty$. This seems to me like a route to producing a representative of the Bott map. Ideally, it would even have good enough properties to produce another proof of the periodicity theorem.

But I can't find anything about this in the literature. Any ideas on how to squeeze a proof out of this -- or, better yet, any ideas about where I can find someone who's already done the squeezing?

Hope ~~this isn't~~ less of this is nonsense!

-- edit --

Given the positive response but lack of answers, I thought I ought to broaden the question a bit to start discussion. What I was originally looking for was a moral proof of the periodicity theorem -- something short that I could show to someone with a little knowledge of stable homotopy as why we should expect the whole thing to be true. The proofs labeled as elementary contained too much matrix algebra to fit into parlor talk, while the proofs with Fredholm operators didn't seem -- uh -- homotopy-y enough. While this business with Thom spaces over $CP^\infty$ seemed like a good place to look, I knew it probably wasn't the only place. In light of Lawson's response, now I'm sure it isn't the only place!

So: does anyone have a good Bott periodicity punchline, aimed at a homotopy theorist?

(Note: I'll probably reserve the accepted answer flag for something addressing the original question.)

18more comments