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

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.