On Cohen-Jones isomorphism in string topology

TL;DR

This paper refines the proof of Cohen-Jones isomorphism in string topology by providing detailed geometric modifications and establishing a structured version in symmetric spectra.

Contribution

It offers a detailed correction to the Cohen-Jones proof and extends the isomorphism to a structured setting in symmetric spectra.

Findings

Provides a geometric modification of Cohen-Jones proof

Establishes a structured Cohen-Jones isomorphism in symmetric spectra

Enhances understanding of the loop product and Hochschild cohomology connection

Abstract

The loop product is an operation in string topology. Cohen and Jones gave a homotopy theoretic realization of the loop product as a classical ring spectrum $LM^{-TM}$ for a manifold $M$. Using this, they presented a proof of the statement that the loop product is isomorphic to the Gerstenhaber cup product on the Hochschild cohomology $HH^*(C^*(M)\,;C^*(M))$ for simply connected $M$. However, some parts of their proof are technically difficult to justify. The main aim of the present paper is to give detailed modification to a geometric part of their proof. To do so, we set up an "up to higher homotopy" version of McClure-Smith's cosimplicial product. We prove a structured version of Cohen-Jones isomorphism in the category of symmetric spectra.