SMS scnews item created by Boris Lishak at Fri 1 Feb 2019 1613
Type: Seminar
Modified: Tue 12 Feb 2019 1356
Distribution: World
Calendar1: 12 Feb 2019 1100-1200
CalLoc1: Carslaw 159
CalTitle1: Galatius -- Tropical curves, graph homology, and top weight cohomology of $$M_g$$
Auth: borisl@dora.maths.usyd.edu.au

# Tropical curves, graph homology, and top weight cohomology of $$M_g$$

### Soren Galatius (Copenhagen)

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.