Problems in homotopy theory
Uncategorized problems

This is mostly a holding zone for problems that aren’t categorized yet.

  • (J. Morava) A question on operads and Steinberg modules.

  • (J. Morava) Find an honest, rigorous way to ask: is the Dwyer-Wilkerson 2-compact group DI(4) (an H-space analogous to G 3, cf Ann Math 139 (1994)) defined over F 1 (or maybe F 1 2, cf Connes and Consani, arXiv:0809.2926)?

  • Problems from J. P. May

  • (A. Salch) In algebraic K-theory, in order to know that BGL(R) + is an H-space, you have to use the multiplication maps

    GL m(R)×GL n(R)GL m+n(R).GL_m(R) \times GL_n(R) \to GL_{m+n}(R).

    If you try to show that BG is an H-space for a general group G, you get stuck if G is nonabelian. When can you get an H-space structure on BG + when G isn’t GL(R) or U or any other group with a nice multiplicative filtration like that? More generally, even when BG and/or BG + isn’t an H-space, it may become an H-space after p-completion, or some other more exotic localization/completion. For example, for most finite simple groups G, BG and BG + can’t be loop spaces (I think this follows from the classification of p-compact groups), but either may still be an H-space that doesn’t deloop. Figuring out that kind of thing means examining old work of Harper and Zabrodsky on H-spaces that don’t deloop; it should involve digging through Adem-Milgram and figuring out which of those cohomology rings of finite simple groups admit Hopf algebra structure, since if BG or BG + is an H-space, either way H *(G) gets a coproduct!

    You could also just ask whether BG is an H-space after E(n)-localization or K(n)-localization. Similar flavor: you could ask about doing the p-compact group classification, but instead of finite loop spaces in p-complete spaces, see if there are any exotic finite loop spaces in E(n)-local or K(n)-local spaces.

    • (J. Stasheff) Would like to add to the question that for topological G, there is a series of obstructions, starting with homotopy commutativity.

      Sugawara 1960: If (X,m) is a connected topological monoid, BX admits a multiplication iff m is strongly homotopy multiplicative.

  • Milnor has a problem on studying the homology of Lie groups made discrete: if you take finite coefficients, the homology remains the same. Related to this is the Friedlander-Milnor conjecture.

  • Lipshitz and Sarkar constructed a spectrum-level refinement of Khovanov homology, as have Hu-Kriz-Kriz, and since then there’s been work on studying things like Steenrod operations. It would be really interesting to get some people who are versed in such matters to take a serious look at the homotopy types that arise from these constructions and see if we can find something actually useful to say. See N Kitchloo, The Landweber-Novikov algebra and Soergel bimodules.

  • Is there a set of cohomological Bousfield classes?

  • The dichotomy conjecture of Hovey and Palmieri: every spectrum has either a finite local or a finite acyclic.

Problems from Paul Goerss. (all posted in the wiki now, but thought I would leave this here for now)