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

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JournalAsian Journal of Mathematics
DatePublished - 1 Jan 2010
Issue number3
Volume14
Number of pages26
Pages (from-to)359-384
Original languageEnglish

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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