Derived string topology and the Eilenberg-Moore spectral sequence

Abstract

Let $M$ be any simply-connected Gorenstein space over any field. F\'elix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space $H_*(LM)$. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively. We also define a generalized cup product on the Hochschild cohomology $HH^*(A,A^\vee)$ of a commutative Gorenstein algebra $A$ and show that over $\mathbb{Q}$, $HH^*(A_{PL}(M),A_{PL}(M)^\vee)$ is isomorphic as algebras to $H_*(LM)$. Thus, when $M$ is a Poincar\'e duality space, we recover the isomorphism of algebras $\mathbb{H}_*(LM;\mathbb{Q})^\cong HH^*(A_{PL}(M),A_{PL}(M))$ of F\'elix and Thomas.