Koszul duality and equivariant cohomology

Abstract

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 $H^{\ast }(BG)$. 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 $C^{\ast }(BG)\to H^{\ast }(BG)$. 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.