Derived string topology and the Eilenberg-Moore spectral sequence

Abstract

Let $M$ be any simply-connected Gorenstein space over any field. Félix 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 $H_{\ast }(LM)$. 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 $H{H}^{\ast }(A,A^{\vee })$ of a commutative Gorenstein algebra $A$ and show that over $\mathbb{Q}$, $H{H}^{\ast }(A_{PL}(M),A_{PL}(M{)}^{\vee })$ is isomorphic as algebras to $H_{\ast }(LM)$. Thus, when $M$ is a Poincaré duality space, we recover the isomorphism of algebras ${\mathbb{H}}_{\ast }(LM;\mathbb{Q}{)}^{\cong }H{H}^{\ast }(A_{PL}(M),A_{PL}(M))$ of Félix and Thomas.

40 pages, this version is one of two preprints divided from the first version, an appendix on shriek maps is revised