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

Georgios Daskalopoulos; Richard Wentworth; Graeme Peter Desmond Wilkin

doi:10.4310/AJM.2010.v14.n3.a5

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

Georgios Daskalopoulos, Richard Wentworth, Graeme Peter Desmond Wilkin

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	359-384
Number of pages	26
Journal	Asian Journal of Mathematics
Volume	14
Issue number	3
DOIs	https://doi.org/10.4310/AJM.2010.v14.n3.a5
Publication status	Published - 1 Jan 2010

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Cite this

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

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.",

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AB - 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.

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