The cohomology ring of free loop spaces

Abstract

Let $X$ be a simply connected space and $\Bbbk$ a commutative ring. Goodwillie, Burghelea and Fiedorowicz proved that the Hochschild cohomology of the singular chains on the space of pointed loops $HH^{*}S_*(\Omega X)$ is isomorphic to the free loop space cohomology $H^{*}(X^{S^{1}})$. We prove that this isomorphism is compatible with the usual cup product on $H^{*}(X^{S^{1}})$ and the cup product of Cartan and Eilenberg on $HH^{*}S_*(\Omega X)$. In particular, we make explicit the algebra $H^{*}(X^{S^{1}})$ when $X$ is a suspended space, a complex projective space or a finite CW-complex of dimension $p$ such that $\frac {1}{(p-1)!}\in {\Bbbk}$.