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Intersection cohomology of representation spaces of surface groups

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arxiv math/0101256 v1 pith:HZSYZ5MF submitted 2001-01-31 math.AG math.SG

Intersection cohomology of representation spaces of surface groups

classification math.AG math.SG
keywords cohomologyintersectionspacesigmaequivariantmodulirepresentationsurface
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We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(\pi_1(\Sigma),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$ equivariant cohomology of $Hom(\pi_1(\Sigma),G)$ where $\Sigma$ is a closed Riemann surface. This enables us to compute the intersection cohomology as a graded vector space with intersection pairing, in terms of the equivariant cohomology ring. The case where $G=SU(2)$ -- the moduli space of rank 2 holomorphic vector bundles of even degree -- is discussed in detail.

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