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splitting this off from Cohomotopy to make room for discussion of the Sullivan models from
Morning wondering, if there’s rational cohomotopy, then there should be p-adic cohomotopy. Google provides only a couple of hits.
True. Haven’t thought about that.
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.
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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