Koszul duality and equivariant cohomology

  • Matthias Franz

    Fachbereich Mathematik Universität Konstanz 78457 Konstanz Germany
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Let GG be a topological group such that its homology H(G)H(G) with coefficients in a principal ideal domain RR is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between GG-spaces and spaces over BGBG to the Koszul duality between modules up to homotopy over H(G)H(G) and H(BG)H^*(BG). This gives in particular a Cartan-type model for the equivariant cohomology of a GG-space with coefficients in RR. As another corollary, we obtain a multiplicative quasi-isomorphism C(BG)H(BG)C^*(BG)\to H^*(BG). A key step in the proof is to show that a differential Hopf algebra is formal in the category of AA_\infty algebras provided that it is free over RR and its homology an exterior algebra.

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Matthias Franz, Koszul duality and equivariant cohomology. Doc. Math. 11 (2006), pp. 243–259

DOI 10.4171/DM/211