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
Department of Mathematics, Cornell University
Ithaca, NY 14853-4201, USA

AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to