Mapping class group action on the cohomology of the $\mathrm{SL}_n$ character variety

TL;DR

This paper investigates how the mapping class group acts on the cohomology of the twisted SL_n-character variety of a surface, utilizing endoscopic decomposition and finite cover analysis.

Contribution

It introduces a new approach to analyze the mapping class group action on character variety cohomology using endoscopic decomposition and finite cover techniques.

Findings

Reduced the problem to finite cover cohomology analysis

Applied endoscopic decomposition to the mapping class group action

Provided new insights into the structure of the cohomology

Abstract

We describe the mapping class group action on the cohomology of the twisted $\mathrm{SL}_n$-character variety of a surface $\Sigma_g$ of genus $g$. Our main tool is a relative version of the endoscopic decomposition of Maulik-Shen; this allows us to reduce the problem to the mapping class group action on the cohomology of a canonical finite cover of $\Sigma_g$, which was studied by Looijenga.