A Serre spectral sequence for the moduli space of tropical curves
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929314053750784 |
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| author | Bibby, Christin Chan, Melody Gadish, Nir Yun, Claudia He |
| author_facet | Bibby, Christin Chan, Melody Gadish, Nir Yun, Claudia He |
| contents | We construct, for all $g\geq 2$ and $n\geq 0$, a spectral sequence of rational $S_n$-representations which computes the $S_n$-equivariant reduced rational cohomology of the tropical moduli spaces of curves $Δ_{g,n}$ in terms of compactly supported cohomology groups of configuration spaces of $n$ points on graphs of genus $g$. Using the canonical $S_n$-equivariant isomorphisms $\widetilde{H}^{i-1}(Δ_{g,n};\mathbb{Q}) \cong W_0 H^i_c(\mathcal{M}_{g,n};\mathbb{Q})$, we calculate the weight $0$, compactly supported rational cohomology of the moduli spaces $\mathcal{M}_{g,n}$ in the range $g=3$ and $n\leq 9$, with partial computations available for $n\leq 13$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_01960 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Serre spectral sequence for the moduli space of tropical curves Bibby, Christin Chan, Melody Gadish, Nir Yun, Claudia He Algebraic Geometry Algebraic Topology Combinatorics 14H10, 14Q05, 14T20, 55N30, 55R80, 55T10 We construct, for all $g\geq 2$ and $n\geq 0$, a spectral sequence of rational $S_n$-representations which computes the $S_n$-equivariant reduced rational cohomology of the tropical moduli spaces of curves $Δ_{g,n}$ in terms of compactly supported cohomology groups of configuration spaces of $n$ points on graphs of genus $g$. Using the canonical $S_n$-equivariant isomorphisms $\widetilde{H}^{i-1}(Δ_{g,n};\mathbb{Q}) \cong W_0 H^i_c(\mathcal{M}_{g,n};\mathbb{Q})$, we calculate the weight $0$, compactly supported rational cohomology of the moduli spaces $\mathcal{M}_{g,n}$ in the range $g=3$ and $n\leq 9$, with partial computations available for $n\leq 13$. |
| title | A Serre spectral sequence for the moduli space of tropical curves |
| topic | Algebraic Geometry Algebraic Topology Combinatorics 14H10, 14Q05, 14T20, 55N30, 55R80, 55T10 |
| url | https://arxiv.org/abs/2307.01960 |