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arXiv · 2110.05711

Weight two compactly supported cohomology of moduli spaces of curves

Sam Payne·Thomas Willwacher
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Abstract

We study the weight 2 graded piece of the compactly supported rational cohomology of the moduli spaces of curves $M_{g,n}$ and show that this can be computed as the cohomology of a graph complex that is closely related to graph complexes arising in the study of embedding spaces. For $n = 0$, we express this cohomology in terms of the weight zero compactly supported cohomology of $M_{g',n'}$ for $g' \leq g$ and $n' \leq 2$, and thereby produce several new infinite families of nonvanishing unstable cohomology groups on $M_g$, including the first such families in odd degrees. In particular, we show that the dimension of $H^{4g-k}(M_g)$ grows at least exponentially with $g$, for $k \in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$.

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Sam Payne, Thomas Willwacher. 2024-11-19. Weight two compactly supported cohomology of moduli spaces of curves. https://doi.org/10.1215/00127094-2024-0003

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