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Tropical curves, graph complexes, and top weight cohomology of $ \mathcal{M}_g$


Authors: Melody Chan, Søren Galatius and Sam Payne
Journal: J. Amer. Math. Soc. 34 (2021), 565-594
MSC (2020): Primary 14H10, 14T20
DOI: https://doi.org/10.1090/jams/965
Published electronically: February 2, 2021
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Abstract: We study the topology of a space $ \Delta _{g}$ 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 $ \mathcal {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-Teichmüller Lie algebra, we deduce that $ H^{4g-6}(\mathcal {M}_g;\mathbb{Q})$ is nonzero for $ g=3$, $ g=5$, and $ g \geq 7$, and in fact its dimension grows at least exponentially in $ g$. 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.


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Additional Information

Melody Chan
Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, RI 02912
Email: melody_chan@brown.edu

Søren Galatius
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen OE, Denmark
Email: galatius@math.ku.dk

Sam Payne
Affiliation: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, PMA 8.100, Austin, TX 78712
Email: sampayne@utexas.edu

DOI: https://doi.org/10.1090/jams/965
Received by editor(s): July 4, 2018
Received by editor(s) in revised form: July 23, 2020
Published electronically: February 2, 2021
Additional Notes: The first author was supported by NSF DMS-1204278, DMS-1701924, CAREER DMS-1844768, a Sloan Fellowship and a Henry Merritt Wriston Fellowship.
The second author was supported by NSF DMS-1405001 and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682922), by the EliteForsk Prize, and by the Danish National Research Foundation (DNRF92 and DNRF151).
The third author was supported by NSF DMS-1702428 and a Simons Fellowship.
Article copyright: © Copyright 2021 American Mathematical Society