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

TL;DR

This paper develops a hairy graph homology theory linked to cyclic operads, revealing new unstable homology cycles for Out(F_n) and Aut(F_n) and connecting to classical modular forms.

Contribution

It introduces a generalized hairy graph homology theory for cyclic operads and demonstrates its application in discovering new homology classes for automorphism groups.

Findings

Large new summands of the abelianization of hO related to modular forms.

Construction of new cycles for the unstable homology of Out(F_n).

First homology computation of the hairy graph complex for the associative operad.

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 which generalizes the trace map defined by S. 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 SL(2,Z). Using cusp forms we construct new cycles for the unstable homology of Out(F_n), and using Eisenstein series we find new cycles for Aut(F_n). 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.