On Cohen-Jones isomorphism in string topology

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 $L{M}^{-TM}$ 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 $H{H}^{\ast }(C^{\ast }(M);C^{\ast }(M))$ 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.

79 pages, errors corrected, figures and explanations added

References in corpus (4)

• String topology and the based loop space
• String operations on rational Gorenstein spaces
• Derived string topology and the Eilenberg-Moore spectral sequence
• The Hochschild cohomology ring of the singular cochain algebra of a space