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

TL;DR

This paper links the topology of tropical moduli spaces, the cohomology of M_g, and graph complexes, revealing nontrivial cohomology in certain genera and disproving existing conjectures.

Contribution

It establishes a canonical isomorphism between tropical curve homology, top weight cohomology of M_g, and graph complex homology, and applies this to disprove conjectures.

Findings

H^{4g-6}(M_g;Q) is nonzero for g=3, 5, and ≥7

Disproved conjectures of Church-Farb-Putman and Kontsevich

Proved homology of the graph complex vanishes in negative degrees

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.