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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.
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).
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 $X = 1$?
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.
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.
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 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?
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 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.
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?
Maybe synthetic homotopy theory would help; it seems to be good at producing constructive proofs of homotopy-theoretic results.
We have an orphaned page Bott periodicity theorem. Should that just be amalgamated with Bott periodicity?
Mike #11: that thought occurred to me too, but I have next to zero personal (i.e., hands-on) experience with this area.
Well you can find a list of proof methods discussed in this MO question.
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.
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:
$KU \simeq \mathbb{S}[B U(1)][\beta^{-1}] \,.$I added that MO discussion (#14) to Bott periodicity. Re #16, there’s plenty of talk there of the relation to Snaith’s theorem.
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.
cross-linked with equivariant K-theory of projective G-space
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