On a theorem of Kontsevich

Abstract

In [12] and [13], 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 [4] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [3] to the case of all cyclic operads. These operations form a Lie bialgebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bialgebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.