Surface configuration kernels

TL;DR

This paper investigates the action of the mapping class group on the cohomology of configuration spaces of punctured surfaces, revealing new abelian subgroups and refuting a prior conjecture in the field.

Contribution

It introduces a comparison between the kernel of the group action on cohomology and Johnson subgroups, identifying high-rank abelian subgroups and disproving a conjecture.

Findings

Identification of high-rank abelian subgroups in the quotient

Refutation of the Bianchi–Miller–Wilson conjecture

Connection between Johnson images and symplectic representation theory

Abstract

Let $\Sigma_{g,*}$ be a once-punctured oriented surface of genus $g$. We study the action of the mapping class group $\Gamma_{g,*}$ on the $n^{th}$ rational cohomology of the configuration space $\text{Conf}_n(\Sigma_{g,*})$ of injections $\{1,\ldots, n\}\hookrightarrow \Sigma_{g,*}$, and compare the kernel $J_{g,*}^{cfg}(n)$ of this action with the $n^{th}$ Johnson subgroup $J_{g,*}(n)$. We find high-rank abelian subgroups in the quotient $J_{g,*}^{cfg}(n)/J_{g,*}(n)$ arising from the higher Johnson images and from symplectic representation theory. In particular we refute a conjecture due to Bianchi--Miller--Wilson.