Invariance of the Goresky-Hingston algebra on reduced Hochschild homology

Abstract

We prove that two quasi-isomorphic simply connected differential graded associative Frobenius algebras have isomorphic Goresky-Hingston algebras on their reduced Hochschild homology. Our proof is based on relating the Goresky-Hingston algebra on reduced Hochschild homology to the singular Hochschild cohomology algebra. For any simply connected oriented closed manifold $M$ of dimension $k$, the Goresky-Hingston algebra on reduced Hochschild homology induces an algebra structure of degree $k-1$ on ${\bar{H}}^{\ast }(LM;\mathbb{Q})$, the reduced rational cohomology of the free loop space of $M$. As a consequence of our algebraic result, we deduce that the isomorphism class of the induced algebra structure on ${\bar{H}}^{\ast }(LM;\mathbb{Q})$ is an invariant of the homotopy type of $M$.

arXiv commentFinal version to appear in the Proceedings of the London Mathematical Society. The presentation and exposition were improved following the suggestions of an anonymous referee. In particular, we have added an outline of the proof of our main result in the introduction