Hairy graphs and the unstable homology of Mod(g , s ), Out(F n ) and Aut(F n )

Abstract

We study a family of Lie algebras hO which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper, we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map that generalizes the trace map defined by Morita. For the Lie operad, we use the trace map to find large new summands of the abelianization of hO which are related to classical modular forms for SL2(ℤ). Using cusp forms, we construct new cycles for the unstable homology of Out(Fn), and using Eisenstein series, we find new cycles for Aut(Fn). For the associative operad, we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of hO in the associative case.