Tropical curves, graph complexes, and top weight cohomology of M_g

Abstract

We study the topology of a space parametrizing stable tropical curves of genus g with volume 1, showing that its reduced rational homology is canonically identified with both the top weight cohomology of M_g and also with the genus g part of the homology of Kontsevich's graph complex. Using a theorem of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least 7. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of another theorem of Willwacher, that homology of the graph complex vanishes in negative degrees.

31 pages. v2: streamlined exposition. Final version, to appear in J. Amer. Math. Soc

References in corpus (6)

• Tropical curves, graph complexes, and top weight cohomology of M_g
• Topology of moduli spaces of tropical curves with marked points
• Equivariant geometry and the cohomology of the moduli space of curves
• P. Etingof's conjecture about Drinfeld associators
• Oriented hairy graphs and moduli spaces of curves
• Factorization for stacks and boundary complexes

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