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At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.
(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)
briefly recorded some facts (here) on cohomotopy of 4-manifolds, from Kirby-Melvin-Teichner 12
added pointer to
added also graphics (here) illustrating the $\mathbb{Z}_2$-equivariant version of the previous example.
Am adding the same illustration also to the respective discussion at equivariant Hopf degree theorem
have added pointer to
H. Sati, U. Schreiber:
Equivariant Cohomotopy implies orientifold tadpole cancellation (arXiv:1909.12277)
where those graphics are taken from
added these references on Cohomotopy cocycle spaces:
Vagn Lundsgaard Hansen, The homotopy problem for the components in the space of maps on the $n$-sphere, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 (DOI:10.1093/qmath/25.1.313)
Vagn Lundsgaard Hansen, On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere, Transactions of the American Mathematical Society Vol. 265, No. 1 (May, 1981), pp. 273-281 (jstor:1998494)
Yes! Eventually we need a higher dimensional table. Or a table of tables.
Personally, of course I am eager to go full blown into twisted equivariant differential Cohomotopy of super orbifolds. And Vincent has a bunch of ideas for what to do. But to make sure not to be barking up the wrong tree, we’d first like to finish one or two further consistency checks in the “topological sector” first.
But that’s just me. If you want to go ahead creating more nLab entries on further variants, please do.
Thanks for the pointer. That made me add also the article that it’s based on:
I think the talk is closer to
so have added that. It seems to rely on values of a function bounded away from $0$ being mapped to a sphere.
Thanks! Interesting. I am adding cross-links with persistent homology (in lack of a general mathematical notion of “persistency” of which these two are examples(?))
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