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

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

2011-07-25

arxiv.org/abs/1107.4839v2

Mathematics

Algebraic Topology

In version 2, errors were corrected in section 5.2 and Lemma 8.3. The proof of Theorem 6.7 was corrected and section 6.2 was e

Scientific paper

For every cyclic operad O there is an associated Lie algebra hO of positive degree symplectic derivations. For the associative and Lie operads, Kontsevich used his graph homology theory to prove that the Lie algebra homology of hO computes the cohomology of mapping class groups of punctured surfaces and outer automorphism groups of free groups, 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 new pieces 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. This paper is the first of two. In the second paper we use representation theory to determine the image of the trace map precisely and show how the theory we develop is related to Loday's dihedral homology and to Getzler and Kapranov's theory of modular operads.

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