Free loop space and the cyclic bar construction

TL;DR

This paper constructs a homotopy coherent map linking the cyclic bar construction of cochains to the cochains on free loop spaces, utilizing $E_ abla$-structure to address a gap in prior work.

Contribution

It introduces a new homotopy coherent map connecting cyclic bar constructions to free loop space cochains, filling a gap in Jones's earlier research.

Findings

Established a homotopy coherent map using $E_ abla$-structure.

Bridged a gap in cyclic homology and free loop space theory.

Enhanced understanding of the relationship between cyclic bar constructions and free loop spaces.

Abstract

Using the $E_\infty-$structure on singular cochains, we construct a homotopy coherent map from the cyclic bar construction of the differential graded algebra of cochains on a space to a model for the cochains on its free loop space. This fills a gap in the paper "Cyclic homology and equivariant homology" by John D.S. Jones.