Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads

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 that 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 and that there exists a CW complex whose chains are isomorphic to a certain type of Sullivan Chord diagrams and that 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.