Homology of configuration spaces of surfaces modulo an odd prime
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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_ | 1866914819899129856 |
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| author | Bianchi, Andrea Stavrou, Andreas |
| author_facet | Bianchi, Andrea Stavrou, Andreas |
| contents | For a compact orientable surface $Σ_{g,1}$ of genus $g$ with one boundary component and for an odd prime number $p$, we study the homology of the unordered configuration spaces $C_\bullet(Σ_{g,1}):=\coprod_{n\ge0}C_n(Σ_{g,1})$ with coefficients in $\mathbb{F}_p$. We describe $H_*(C_\bullet(Σ_{g,1});\mathbb{F}_p)$ as a bigraded module over the Pontryagin ring $H_*(C_\bullet(D);\mathbb{F}_p)$, where $D$ is a disc, and compute in particular the bigraded dimension over $\mathbb{F}_p$. We also consider the action of the mapping class group $Γ_{g,1}$, and prove that the mod-$p$ Johnson kernel $\mathcal{K}_{g,1}(p)\subseteqΓ_{g,1}$ is the kernel of the action on $H_*(C_\bullet(Σ_{g,1};\mathbb{F}_p))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08664 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homology of configuration spaces of surfaces modulo an odd prime Bianchi, Andrea Stavrou, Andreas Algebraic Topology 16E30, 20F36, 55R35, 55R80, 55U25 For a compact orientable surface $Σ_{g,1}$ of genus $g$ with one boundary component and for an odd prime number $p$, we study the homology of the unordered configuration spaces $C_\bullet(Σ_{g,1}):=\coprod_{n\ge0}C_n(Σ_{g,1})$ with coefficients in $\mathbb{F}_p$. We describe $H_*(C_\bullet(Σ_{g,1});\mathbb{F}_p)$ as a bigraded module over the Pontryagin ring $H_*(C_\bullet(D);\mathbb{F}_p)$, where $D$ is a disc, and compute in particular the bigraded dimension over $\mathbb{F}_p$. We also consider the action of the mapping class group $Γ_{g,1}$, and prove that the mod-$p$ Johnson kernel $\mathcal{K}_{g,1}(p)\subseteqΓ_{g,1}$ is the kernel of the action on $H_*(C_\bullet(Σ_{g,1};\mathbb{F}_p))$. |
| title | Homology of configuration spaces of surfaces modulo an odd prime |
| topic | Algebraic Topology 16E30, 20F36, 55R35, 55R80, 55U25 |
| url | https://arxiv.org/abs/2307.08664 |