Equivariant homotopy

This is part of an algebraic topology problem list, maintained by Mark Hovey.

I don't know much about equivariant homotopy, but I still have some ideas. They may not be worth much, but here they are.

  1. Prove a nilpotence theorem for the equivariant stable homotopy category. I probably learned this idea from Mike Hopkins, though I don't really remember now. One could start when the group is just Z/2, though the answer may well depend on which universe you take. The idea is that nilpotence should be detected by nonequivariant nilpotence on the fixed point spectra, and this we know how to detect.

  2. Currently we know how to do equivariant stable homotopy theory only when the structure group G is compact Lie. But I bet we can do it when the group G is profinite as well. The main example I am thinking of is the Morava stabilizer group, though one could warm up with the p-adics. Presumably Morava E-theory should be an object of this category, but how?

  3. Figure out how to do equivariant stable homotopy theory without restriction on the group. Here you are going to have to change the current setup a lot, I think. Mike Mandell has some ideas about this. He points out that a lot of the crucial problems in mathematics concern infinite discrete groups, like the Novikov conjecture, and so it may be of value to figure out this problem.

  4. As a simpler model of the equivariant stable homotopy category, construct a derived category of Mackey functors over a Green functor, and analyze its properties. We know a lot about the derived category of a Noetherian ring--thick, localizing, and colocalizing subcategories are completely classified (See Amnon Neeman's paper on the derived category, though in my opinion you should also see Hovey-Palmieri-Strickland for simpler proofs of most of his results). Can we say analogous things about Mackey functors? Gaunce Lewis has thought about this problem a bit.

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Mark Hovey
Department of Mathematics
Wesleyan University