The cohomology ring of free loop spaces

Abstract

Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowicz proved that the Hochschild cohomology of the singular chains on the space of pointed loops HH * S * (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 * (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 CWcomplex of dimension p such that 1 (p-1)! k.