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
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