# Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads

### Ralph M. Kaufmann

University of Connecticut, Storrs

## Abstract

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a Frobenius algebra. We also prove that a there is dg-PROP action of a version of Sullivan chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co-cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply-connected manifold.

In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are: There is a cell model operad for the moduli space of genus g curves with n punctures and a tangent vector at each of these punctures, there exists a CW complex whose chains are isomorphic to a certain type of Sullivan chord diagrams and they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.

## Cite this article

Ralph M. Kaufmann, Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads. J. Noncommut. Geom. 1 (2007), no. 3, pp. 333–384

DOI 10.4171/JNCG/10