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nLab > Latest Changes: fundamental product theorem in topological K-theory

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2017
    • (edited May 28th 2017)

    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.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2017

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

  3. 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=1X = 1?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2017

    Naively, I just meant the statement/calculation K(S 2)≅ℤ[H]/(H−1) 2K(S^2) \cong \mathbb{Z}[H]/(H-1)^2 in ordinary mathematics, but bearing in mind that the ℤ\mathbb{Z} should be thought of as K(*)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.

    • CommentRowNumber5.
    • CommentAuthorjesse
    • CommentTimeMay 28th 2017
    • (edited May 28th 2017)

    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.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2017

    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.

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

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2017

    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.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2017

    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.

    • CommentRowNumber10.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 28th 2017
    • (edited May 28th 2017)

    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 0P^{0} and P ∞P^{\infty} looks potentially tricky constructively: I imagine it is necessary in Atiyah’s proof that everything in P(L⊕1)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=1X=1 does not seem substantially easier than the general one, so would we gain much from the topos-theoretic account?

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 28th 2017

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

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 28th 2017

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

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 29th 2017

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

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 29th 2017

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

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2017

    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.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2017
    • (edited May 29th 2017)

    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 BU(1)B U(1) and then inverting the Bott element:

    KU≃𝕊[BU(1)][β −1]. KU \simeq \mathbb{S}[B U(1)][\beta^{-1}] \,.
    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 29th 2017

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

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2017

    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.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2020

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