On Cohen-Jones isomorphism in string topology

Abstract

The loop product is an operation in string topology. Cohen and Jones [15,16] gave a homotopy theoretic realization of the loop product as a classical ring spectrum LM −T M 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 is 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.