The string topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces

Abstract

In 1999 Chas and Sullivan discovered that the homology H_*(LX) of the space of free loops on a closed oriented smooth manifold X has a rich algebraic structure called string topology. They proved that H_*(LX) is naturally a Batalin-Vilkovisky (BV) algebra. There are several conjectures connecting the string topology BV algebra with algebraic structures on the Hochschild cohomology of algebras related to the manifold X, but none of them has been verified for manifolds of dimension n>1. In this work we study string topology in the case when X is aspherical (i.e. its homotopy groups \pi_i(X) vanish for i > 1). In this case the Hochschild cohomology Gerstenhaber algebra HH^*(A) of the group algebra A of the fundamental group of X has a BV structure. Our main result is a theorem establishing a natural isomorphism between the Hochschild cohomology BV algebra HH^*(A) and the string topology BV algebra H_*(LX). In particular, for a closed oriented surface X of hyperbolic type we obtain a complete description of the BV algebra operations on H_*(LX) and HH^*(A) in terms of the Goldman bracket of loops on X. The only manifolds for which the BV algebra structure on H_*(LX) was known before were spheres and complex Stiefel manifolds. Our proof is based on a combination of topological and algebraic constructions allowing us to compute and compare multiplications and BV operators on both H_*(LX) and HH^*(A).