Abstract:We study the topology of a space parametrizing stable tropical curves of genus with volume , showing that its reduced rational homology is canonically identified with both the top weight cohomology of and also with the genus 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 is 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.

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中文翻译：

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curves_ {ℊ}的热带曲线，图形复杂度和权重同调

摘要：我们研究了一个空间的参数化拓扑结构，该空间参数化了属的稳定热带曲线，具有体积，表明它的减少的有理同源性可以通过Kontsevich图复合体的同源性的最高权重同构性和属类部分来规范地识别。使用将这个图复合体与Grothendieck-TeichmüllerLie代数相关联的Willwacher定理，我们推导了，和的非零值，实际上，它的维数在 。这反驳了最近对Church，Farb和Putman的猜想以及对Kontsevich的较旧的更普遍的猜想。我们还给出了Willwacher另一个定理的独立证明，即图复杂度的同源性以负数消失。

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