Tropical curves, graph complexes, and top weight cohomology of Mg
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Tropical curves, graph complexes, and top weight cohomology of Mg. / Chan, Melody; Galatius, Søren; Payne, Sam.
In: Journal of the American Mathematical Society, Vol. 34, No. 2, 2021, p. 565-594.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Tropical curves, graph complexes, and top weight cohomology of Mg
AU - Chan, Melody
AU - Galatius, Søren
AU - Payne, Sam
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
U2 - 10.1090/jams/965
DO - 10.1090/jams/965
M3 - Journal article
VL - 34
SP - 565
EP - 594
JO - Journal of the American Mathematical Society
JF - Journal of the American Mathematical Society
SN - 0894-0347
IS - 2
ER -
ID: 257967125