Tropical curves, graph complexes, and top weight cohomology of $\mathcal {M}_g$
Tropical curves, graph complexes, and top weight cohomology of $\mathcal {M}_g$
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$mathcal {M}_g$ 的热带曲线、复合图和顶重上同调
DOI：
10.1090/jams/965
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发表时间：
2021
影响因子：
3.9
通讯作者：
Payne, Sam
中科院分区：
文献类型：
--
作者：
Chan, Melody;Galatius, Søren;Payne, Sam
研究了一类带体积的空间参数化稳定热带曲线的拓扑结构，证明了它的简化有理同调不仅与Kontsevich图复调的顶权上同调，而且与同调的属部上同调是正则化的。利用Willwacher关于图复数与grothendieck - teichm<e:1> ller李代数的一个定理，我们推导出，和的非零，事实上它的维数至少在。这推翻了Church、Farb和Putman最近的一个猜想，也推翻了Kontsevich一个更古老、更普遍的猜想。我们还独立证明了Willwacher的另一个定理，即复图的同调在负次域中消失。参考文献
We study the topology of a spaceparametrizing stable tropical curves of genuswith volume, showing that its reduced rational homology is canonically identified with both the top weight cohomology ofand also with the genuspart 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 thatis nonzero for,, and, and in fact its dimension grows at least exponentially in. 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. References