Asian Journal of Mathematics

Volume 14 (2010)

Number 3

Cohomology of $SL(2,C)$ Character Varieties of Surface Groups and the Action of the Torelli Group

Pages: 359 – 384

DOI: http://dx.doi.org/10.4310/AJM.2010.v14.n3.a5

Authors

Georgios D. Daskalopoulos

Richard A. Wentworth

Graeme Wilkin

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.

Keywords

Character varieties; Higgs bundles; Torelli group

2010 Mathematics Subject Classification

Primary 57M50. Secondary 53C24, 58E20.

Full Text (PDF format)