Free loop space and homology

TL;DR

This paper explores the connection between the (co)homology of free loop spaces and Hochschild homology of singular cochain algebras, providing detailed explanations and technical tools.

Contribution

It clarifies the relationship between free loop space (co)homology and Hochschild homology, offering detailed proofs and technical tools for this connection.

Findings

Established the link between free loop space (co)homology and Hochschild homology.

Provided detailed proofs and technical tools for the relationship.

Served as a companion to Jones's work on cyclic homology.

Abstract

The aim of this paper is to explain the relationship between the (co)homology of the free loop space and the Hochschild homology of its singular cochain algebra. We introduce all the relevant technical tools, namely simplicial and cyclic objects, and we provide the various steps of the proofs, which are scattered around in the literature. This paper can be seen as a companion to the paper "Cyclic homology and equivariant homology" by J.D.S. Jones dealing with the cyclic case.