On Cohen-Jones isomorphism in string topology

2020/03/08 by Moriya, Syunji
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.03704

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.

Related

• A homotopy theoretic realization of string topology
• Derived string topology and the Eilenberg-Moore spectral sequence
• The loop homology algebra of spheres and projective spaces
• Product and coproduct in string topology
• String topology and the based loop space