A homotopy theoretic realization of string topology

Abstract

Let $M$ be a closed, oriented manifold of dimension $d$. Let $LM$ be the space of smooth loops in $M$. Chas and Sullivan recently defined a product on the homology $H_*(LM)$ of degree $-d$. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra structure on the $S^1$ equivariant homology $H_*^{S^1}(LM)$. These algebraic structures, as well as others, came under the general heading of the "string topology" of $M$. In this paper we will describe a realization of the Chas - Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov, has a homotopy theoretic realization on the level of Thom spectra. This is Voronov's "cactus operad", which is equivalent to the operad of framed disks in $R^2$. This operad action realizes the Chas - Sullivan BV structure on $H_*(LM)$. We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology.