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

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.

25 pages