Cohomology of SL(2,C) character varieties of surface groups and the action of the Torelli group

TL;DR

This paper investigates how the Torelli group acts on the cohomology of SL(2,C) character varieties of surface groups, revealing the structure of trivial and non-trivial parts and computing Betti numbers.

Contribution

It explicitly describes the Torelli group's action on equivariant cohomology and identifies the Prym-Torelli group as the kernel, also computing Betti numbers of the moduli space.

Findings

The trivial part of the action contains the equivariant cohomology of the even component.

The non-trivial part involves even alternating products of Prym representations.

Betti numbers of the moduli space are explicitly computed.

Abstract

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists of the even alternating products of degree two Prym representations, so that the kernel of the action is precisely the Prym-Torelli group. We compute the Betti numbers of the ordinary cohomology of the moduli space of flat SL(2,C) connections. Using results of Cappell-Lee-Miller we show that the Prym-Torelli group, which acts trivially on equivariant cohomology, acts non-trivially on ordinary cohomology.