Problems in homotopy theory Applications

• (J. McClure) Create elliptic cohomology for ${C}^{*}$ algebras. K-theory is one of the basic tools for ${C}^{*}$ algebras, but so far no one seems to have considered elliptic cohomology. There is already quite a bit of work on model categories for ${C}^{*}$ algebras, e.g. Østvaer Homotopy theory of ${C}^{*}$-algebras, MR2723902.

• (J. McClure) Create an intersection form of K-theory (and other generalized cohomology theories). This is a very old problem, e.g. p. 222 of the book Intersection Cohomology edited by Borel. Problems 11 and 12 on page 229 are also interesting.

• (C. Westerland) Problems on moduli spaces.

The cohomology of the moduli space ${M}_{g,n}$ of genus $g$ curves with $n$ marked points is known to stabilize as $g$ tends to infinity (Harer). Madsen-Weiss’s proof of the Mumford conjecture computes this stable cohomology rationally (for the answer over ${𝔽}_{p}$, see Søren Galatius’ thesis). Very little is known about the unstable cohomology of ${M}_{g,n}$, e.g., the cohomology in degrees greater than approximately $\frac{2g}{3}$ (using Søren Boldsen’s improvement of Harer’s argument).

Problem: compute the unstable cohomology of ${M}_{g,n}$.

A possible route to this is through field theories. If we believe that it is possible to understand a group through its representations, then the same should be true of a category. The homology of all of the ${M}_{g,n}$ fit together into a categorical construct whose representations are called homological conformal field theories (HCFT); there is a closely related notion, cohomological field theories (CohFT) which uses the compactified moduli. Gromov-Witten theory is an example of a CohFT, string topology and Hochschild cohomology of ${A}_{\infty }$ algebras are examples of HCFT’s.

Constantin Teleman’s classification of semisimple CohFT’s tells us that they only “see” the stable cohomology of moduli space, and so will be unhelpful in this problem; one must examine non-semisimple algebras. Conversely (or perhaps dually?) Tamanoi has proven that the stable classes in the homology of ${M}_{g,n}$ act trivially in string topology. Perhaps then string topology is exactly the “right” place to find information about the unstable homology of moduli space. Lastly, it’s a result of Wahl, Costello, and others that in some sense all of the homology of ${M}_{g,n}$ should be detected in its action on ${\mathrm{HH}}_{*}\left(A,A\right)$ for some ${A}_{\infty }$ Frobenius algebra $A$. So one could hope to cook up good examples of such $A$ to probe the structure of ${H}_{*}\left({M}_{g,n}\right)$.

• Problems from Ismar Volić on applications of homotopy theory to knot theory.