# 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 $\mathrm{DI}\left(4\right)$ (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 $\mathrm{BGL}\left(R{\right)}^{+}$ is an $H$-space, you have to use the multiplication maps

${\mathrm{GL}}_{m}\left(R\right)×{\mathrm{GL}}_{n}\left(R\right)\to {\mathrm{GL}}_{m+n}\left(R\right).$GL_m(R) \times GL_n(R) \to GL_{m+n}(R).

If you try to show that $\mathrm{BG}$ 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 ${\mathrm{BG}}^{+}$ when $G$ isn’t $\mathrm{GL}\left(R\right)$ or $U$ or any other group with a nice multiplicative filtration like that? More generally, even when $\mathrm{BG}$ and/or ${\mathrm{BG}}^{+}$ 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 $G$, $\mathrm{BG}$ and ${\mathrm{BG}}^{+}$ 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 $\mathrm{BG}$ or ${\mathrm{BG}}^{+}$ is an $H$-space, either way ${H}^{*}\left(G\right)$ gets a coproduct!

You could also just ask whether $\mathrm{BG}$ is an $H$-space after $E\left(n\right)$-localization or $K\left(n\right)$-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 $E\left(n\right)$-local or $K\left(n\right)$-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 $\left(X,m\right)$ is a connected topological monoid, $\mathrm{BX}$ 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)