# Journal of Noncommutative Geometry

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**Volume 1, Issue 3, 2007, pp. 333–384**

**DOI: 10.4171/JNCG/10**

Published online: 2007-09-30

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

Ralph M. Kaufmann^{[1]}(1) University of Connecticut, Storrs

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.

*Keywords: *Moduli spaces, operads, Hochschild cohomology, foliations, cell models, string topology, Sullivan chord diagrams, PROPs, conformal field theory

Kaufmann Ralph: Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads. *J. Noncommut. Geom.* 1 (2007), 333-384. doi: 10.4171/JNCG/10