cross-linked with *equivariant K-theory of projective G-space*

I suppose the more homotopy-theoretic proof of Snaith’s theorem, the one where synthetic homotopy theory might be useful, is the one by Mike Hopkins.

]]>I added that MO discussion (#14) to Bott periodicity. Re #16, there’s plenty of talk there of the relation to Snaith’s theorem.

]]>The statement that should lend itself to synthetic homotopy theory is *Snaith’s theorem*, which says that K-theory is the result of forming the group ring of $B U(1)$ and then inverting the Bott element:

David, thanks for the alert. I had not been aware of that other entry. (It didn’t really state Bott periodicity, but just the periodicity of the homotopy groups of the stable unitary/orthogonal groups.) I have merged it now. But still the entry *Bott periodicity* remains a stub.

Well you can find a list of proof methods discussed in this MO question.

]]>Mike #11: that thought occurred to me too, but I have next to zero personal (i.e., hands-on) experience with this area.

]]>We have an orphaned page Bott periodicity theorem. Should that just be amalgamated with Bott periodicity?

]]>Maybe synthetic homotopy theory would help; it seems to be good at producing constructive proofs of homotopy-theoretic results.

]]>I am very open to being proven wrong about this, Todd, and would like to be so!

On a very general note, there is rather a lot of analysis and topology going into Atiyah’s proof. If all of that can be done constructively without any problems, I would be astonished!

My worries concerning constructivity were not very deep. I looked at a few things elsewhere, and in those proofs compactness of the unit interval was used. I imagine that something similar would be needed in Atiyah’s proof, and if so, that might be problematical constructively.

In Atiyah’s proof itself, the use of $P^{0}$ and $P^{\infty}$ looks potentially tricky constructively: I imagine it is necessary in Atiyah’s proof that everything in $P(L \oplus 1)$ is in one of these two, and this would be tricky to establish constructively (unless I am missing something).

On a different note, the proof in the case $X=1$ does not seem substantially easier than the general one, so would we gain much from the topos-theoretic account?

]]>I have brought some infrastructure on long exact sequences into *topological K-theory* (here). Then I used this to spell out the proof of Bott periodicity from the product theorem here.

Richard, could you say roughly where you think problems would be encountered? First impressions would be alright…

For example, taking Atiyah’s approach as a template.

]]>I took a quick look at some standard proofs, and my first impression is that a *constructive* proof in the case $X=1$ would be difficult. I would be very happy to be proven wrong; can anybody suggest a classical computation of $K(S^{2})$ that has a chance of going through constructively?

Thanks, Jesse.

Atiyah famously wrote a book on K-theory, here, and I remember getting the impression when I was a graduate student that one could rewrite Atiyah’s proof directly in topos-theoretic language, if one had a mind to do it. Maybe I should look at that again.

]]>I forget who did that one exactly.

According to Johnstone’s *Topos Theory*, this was done by C.J. Mulvey in “A generalization of Swan’s theorem.”, *Math Zeitschrift* **151** (1976), 57-70.

Naively, I just meant the statement/calculation $K(S^2) \cong \mathbb{Z}[H]/(H-1)^2$ in ordinary mathematics, but bearing in mind that the $\mathbb{Z}$ should be thought of as $K(\ast)$. I had this idea a long time ago (1991 or 1992?), and haven’t thought hard or much about it in the meantime. If this isn’t precise enough, maybe you can help.

There was a brief spell of research of a similar flavor, of 70’s vintage or so, where e.g. by internalizing Kaplansky’s theorem that projective modules over local rings are free to a topos of sheaves, you could derive the Serre-Swan theorem. I forget who did that one exactly. I suspect there’s a fair bit of unrealized potential for this type of mathematics, and it seemed to me this result that Urs pointed to in #1 might be an interesting test case.

]]>Hi Todd, this is a very nice idea. I know of basically *no* nice proof of Bott periodicity, and would love a proof of the kind you describe.

Could you possibly try to make precise the statement that would need to be proven in the case $X = 1$?

]]>I’ve always wondered whether this could be given a topos-theoretic proof, where one performs a calculation $K(S^2) \cong K(\ast)[H]/(H-1)^2$ internally in the topos of sheaves over $X$. In other words, give a constructive proof of this for the $Set$ case $X = 1$, which if you do it right would hold for more general Grothendieck toposes. Interpreting $K(\ast)_{Sh(X)}$ and $K(S^2)_{Sh(X)}$ externally, we should have $K(\ast)_{Sh(X)} \cong K(X)_{Set}$ and $K(S^2)_{Sh(X)} \cong K(S^2 \times X)_{Set}$, whereupon one deduces the fundamental theorem (aka complex Bott periodicity).

]]>added details of the statement to *fundamental product theorem in topological K-theory*

(nothing like a proof yet)

One day when I have the leisure, I might follow up on my conjecture that under the translation of K-theory to D-brane physics, the fundamental product theorem in K-theory is the *Myers effect*.