Weight two compactly supported cohomology of moduli spaces of curves
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929658232045568 |
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| author | Payne, Sam Willwacher, Thomas |
| author_facet | Payne, Sam Willwacher, Thomas |
| contents | 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 $k \in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_05711 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Weight two compactly supported cohomology of moduli spaces of curves Payne, Sam Willwacher, Thomas Algebraic Geometry Algebraic Topology Geometric Topology 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 $k \in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$. |
| title | Weight two compactly supported cohomology of moduli spaces of curves |
| topic | Algebraic Geometry Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2110.05711 |