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Invariance of the Goresky-Hingston algebra on reduced Hochschild homology

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 $M$.