String topology via the coHochschild complex and local intersections

TL;DR

This paper develops an algebraic framework for string topology operations using the coHochschild complex and local intersection pairings, establishing their equivalence to geometric operations on manifolds.

Contribution

It introduces a novel algebraic model for string topology operations based on local pairings and higher homotopies, extending the applicability to simplicial complexes.

Findings

Algebraic operations match geometric string topology operations up to chain homotopy.

Framework applies to simplicial complexes with local pairings, including homology manifolds.

Uses acyclic models method for local and compatible constructions.

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.