REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: String topology is concerned with intersection type constructions on the free loop space of a manifold. It was born with Chas and Sullivan’s striking observation that the intersection product on the homology of a compact manifold lifts to the homology of the free loop space of the manifold even if the latter is an infinite dimensional manifold and we do not have Poincaré duality anymore. This was the starting point for the construction of a rich family of operations that satisfy compatibilities which are familiar from other fields of mathematics. The algebraic structures arising in string topology have counterparts in the Hochschild chain complex of a Frobenius algebra. I will describe the main constructions on both the geometric and the algebraic stories and sketch how they relate through iterated integrals. I will also explain how these constructions relate to the known operations which date back to Hochschild and Gerstenhaber. In particular, I will discuss work in progress in which we show that an algebraic analogue of a product originally discovered in a geometric context by Sullivan and Goresky-Hingston defines via cap products a natural extension of the homotopy Gerstenhaber algebra structure on the Hochschild cochains of a differential graded associative algebra to the Hochschild chain complex.