A cocyclic construction of $S^1$-equivariant homology and application to string topology

TL;DR

This paper introduces a cocyclic chain complex framework to connect cyclic homology with $S^1$-equivariant homology, providing new insights into string topology and algebraic structures on loop spaces.

Contribution

It develops a cocyclic construction that relates cyclic homology to $S^1$-equivariant homology and refines the gravity algebra structure at the chain level in string topology.

Findings

Established a theorem linking cyclic homology to $S^1$-equivariant homology.

Provided a chain level refinement of the gravity algebra structure.

Connected chain-level string topology with operadic Deligne's conjecture.

Abstract

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to $S^1$-equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) $S^1$-equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an $S^1$-equivariant version of operadic Deligne's conjecture.