# Koszul duality and equivariant cohomology

### Matthias Franz

Fachbereich Mathematik Universität Konstanz 78457 Konstanz Germany

## 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 $H_{∗}(BG)$. 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 $C_{∗}(BG)→H_{∗}(BG)$. A key step in the proof is to show that a differential Hopf algebra is formal in the category of $A_{∞}$ algebras provided that it is free over $R$ and its homology an exterior algebra.

## Cite this article

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

DOI 10.4171/DM/211