Koszul Duality and Equivariant Cohomology

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

2000 Mathematics Subject Classification: Primary 16S37, 55N91; Secondary 16E45, 55N10

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