String topology via the coHochschild complex and local intersections

Abstract

We construct an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct in string topology. The construction takes as its initial input a simplicial complex equipped with a local pairing on its simplicial chains, for instance, a homology manifold with its local intersection pairing. We define the two string topology operations on the coHochschild complex of a suitable coalgebra of chains, making use of local higher homotopies that control the compatibility of the pairing with the diagonal approximation coproduct. In the case of a closed oriented smooth manifold, we prove that our algebraic operations coincide, up to chain homotopy, with their geometric counterparts. The local nature of our constructions allows for arguments based on the method of acyclic models.