#### Algebraic and Geometric Topology 3 (2003),
paper no. 42, pages 1167-1224.

## On a theorem of Kontsevich

### James Conant, Karen Vogtmann

**Abstract**. In two seminal papers M. Kontsevich
introduced graph homology as a tool to compute the homology of three
infinite dimensional Lie algebras, associated to the three operads
`commutative,' `associative' and `Lie.' We generalize his theorem to
all cyclic operads, in the process giving a more careful treatment of
the construction than in Kontsevich's original papers. We also give a
more explicit treatment of the isomorphisms of graph homologies with
the homology of moduli space and Out(F_r) outlined by Kontsevich. In
[`Infinitesimal operations on chain complexes of graphs',
Mathematische Annalen, 327 (2003) 545-573] we defined a Lie bracket
and cobracket on the commutative graph complex, which was extended in
[James Conant, `Fusion and fission in graph complexes', Pac. J. 209
(2003), 219-230] to the case of all cyclic operads. These operations
form a Lie bi-algebra on a natural subcomplex. We show that in the
associative and Lie cases the subcomplex on which the bi-algebra
structure exists carries all of the homology, and we explain why the
subcomplex in the commutative case does not.
**Keywords**.
Cyclic operads, graph complexes, moduli space, outer space

**AMS subject classification**.
Primary: 18D50.
Secondary: 57M27, 32D15, 17B65.

**DOI:** 10.2140/agt.2003.3.1167

**E-print:** `arXiv:math.QA/0208169`

Submitted: 5 February 2003.
(Revised: 1 December 2003.)
Accepted: 11 December 2003.
Published: 12 December 2003.

Notes on file formats
James Conant, Karen Vogtmann

Department of Mathematics, University of Tennessee

Knoxville, TN 37996, USA

and

Department of Mathematics, Cornell University

Ithaca, NY 14853-4201, USA

Email: jconant@math.utk.edu, vogtmann@math.cornell.edu

AGT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**