I've heard that there are some problems in developing a good formalism for stable equivariant homotopy theory (either from the spectral mackey functors perspective or from the orthogonal spectra perspective). One of these is the problem that the Burnside ring of finite sets for a profinite group is somehow "not the correct thing" (as the $\pi_0$ of the sphere spectrum).

My questions:

  1. What is, precisely, is wrong/problematic with the Burnside ring for profinite $G$? Is it that we can't construct a stable homotopy theory with it being $\pi_0$ as the sphere spectrum or does this not capture enough aspect of profinite stable equivariant homotopy theory?

  2. What are the other problems with developing such a theory? For example what goes wrong with norms/transfers? The Wirthmuller/Adams isomorphism?

  • 5
    $\begingroup$ There has been some discussion about this at the recent AIM workshop on equivariant homotopy theory. The consensus seems to be that the theory as it is works well if you care only about open subgroups (both via orthogonal spectra and via Mackey functors) but there seems to be some additional subtleties when you want to work with arbitrary closed subgroups. $\endgroup$ – Denis Nardin Jul 25 '16 at 14:30

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