7. Unstable Cohomology
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Finding examples of secondary stability
Problem 7.1.
[Jeremy Miller] Can one find any example of secondary stable homology relating to some arithmetic statistics question? For example, can one understand the secondary stable homology in ordered configuration spaces?
One interesting example to look at could be Edward Frenkel’s computation of the cohomology of the commutator subgroup of the braid group. It could also be interesting to try to compute stable cohomology associated to counting $A$-extensions and colored configuration spaces.
Matthew Delangis wrote an exact generating function for abelian $A$ extensions (in as of yet unpublished work) using homology of configuration spaces. One can extract the main term from this and it could be interesting to try to extract secondary terms from this as well. -
Predicting secondary stability
Problem 7.2.
[Jeremy Miller] Can one use the secondary term for counting cubic extensions to make some guess for the secondary stable homology of Hurwitz spaces parameterizing (simply branched) trigonal curves.
In the case of Hurwitz spaces for $S_3$ with transpositions, it would be interesting to understand secondary stability. As a preliminary question, it would be interesting to understand the optimal stable range, which is often a prerequisite for computing secondary terms.
In the $S_3$ case, one can see the class in the cohomology of the Nichols algebra. For counting cubic extensions of $\mathbb F_q(t)$ there is a proposed class which should exist in $H_2(\operatorname{Hur}_6)$ which might lead to secondary stability, as in Gregory Michel’s thesis.
Kevin Chang does have a paper which explicitly exhibits secondary terms for cubic and quartic algebras coming from cohomology.
Homological stability is often paired with a homotopy theoretic description of the stable homology. It seems very difficult to generalize this to compute the secondary stable homology? Perhaps number theorists can tell topologists how to compute the secondary stable homology. -
Unstable cohomology
Problem 7.3.
[Will Sawin] Can one prove that the unstable cohomology in various questions is exponentially growing? Can we prove an exponential lower bound, perhaps using an Euler characteristic argument? Can one quantify how miraculous the cancellation of Frobenius eigenvalues has to be when taking the trace of Frobenius if one wants standard number theory questions to be true? -
Unstable cohomology
Problem 7.4.
[Jeremy Miller] Let $c$ be a union of conjugacy classes in a group $G$. What can we say about the Frobenius action on the cohomology of $\operatorname{Hur}^c_n$? What can we say about top degree cohomology? Can one compute $H_{n-1}(\operatorname{Hur}^c_n)$? Can we use the fact that the Braid group is a duality group? What should we even conjecture about the Frobenius action to be consistent with arithmetic applications? What would give the expected arithmetic results? Some of these questions may also be applicable when $c$ is, more generally, a rack.
Aaron Landesman and Ishan Levy have proven an exponential lower bound via private communication (but this has not been written anywhere). Separately, there is an exponential upper bound with a larger exponent. For the specific classes that are easiest to construct, one may be able to understand the Frobenius action. This is a free Lie algebra.
Jordan Ellenberg comments it would be interesting to compute the trace of Frobenius on this. Separately, it would also be interesting to understand what representations of the mapping class group show up in this cohomology?
Craig Westerland points out the entirety of the top homology is the primitives in the shuffle algebra. If one lets $G$ range over all groups, this limits toward a faithful representation of a certain group ring, even just for the Hurwitz spaces on $3$ points. This has something to do with Belyi’s theorem.
Jeremy Miller was interested in understanding the top degree cohomology associated to a Burau representation. Dan Petersen thinks that there will be a lot of cohomology in the top degree associated to Burau representations.
Cite this as: AimPL: Moments in families of L-functions over function fields, available at http://aimpl.org/momentsoverff.