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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 6th 2019

    splitting this off from Cohomotopy to make room for discussion of the Sullivan models from

    • Jesper Møller, Martin Raussen, Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (jstor:2000242)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 5th 2021

    Morning wondering, if there’s rational cohomotopy, then there should be p-adic cohomotopy. Google provides only a couple of hits.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2021

    True. Haven’t thought about that.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 9th 2021
    • (edited Mar 9th 2021)

    Another morning thought. Presumably at some point your path out from Hypothesis H meets the kind of ideas you once sketched in an MO question on p-adic string theory. Presumably whatever’s p-adic in tmf is inherited from what’s p-adic in stable cohomotopy.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2021

    Yes. The usual lore (here) that “tmf approximates the sphere spectrum” means (Example 4.13 here) that on 10+1-dimensional spacetimes the Boardman homomorphism from 4-Cohomotopy to 4-tmf is an equivalence (tweet).

    So, on M-theory spacetimes, C-field flux quantization in Cohomotopy is indistinguishable from flux quantization in tmf. Hence all discussion of the latter applies to the former.

    (Notice that in both cases the shift to degree 4 is fixed by the fact that this is the only degree with a non-torsion homotopy group to accomodate the degree 4-charge of the C-field).

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