Invariance of the Goresky-Hingston algebra on reduced Hochschild homology

TL;DR

This paper proves that the Goresky-Hingston algebra on reduced Hochschild homology is invariant under quasi-isomorphisms of certain Frobenius algebras, linking algebraic structures to topological invariants of manifolds.

Contribution

It establishes the invariance of the Goresky-Hingston algebra on reduced Hochschild homology under quasi-isomorphisms, connecting algebraic and topological invariants.

Findings

Goresky-Hingston algebra is invariant under quasi-isomorphisms.

The algebra structure on reduced Hochschild homology relates to the homotopy type of manifolds.

The algebra induces an invariant structure on the reduced cohomology of free loop spaces.

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 $\bar{H}^*(LM;\mathbb{Q})$, 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 $\bar{H}^*(LM;\mathbb{Q})$ is an invariant of the homotopy type of $M$.