On the mapping class group action on the homology of surface covers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909215207981056 |
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| author | Spiridonov, Igor |
| author_facet | Spiridonov, Igor |
| contents | Let $ϕ\in {\rm Mod}(Σ)$ be an arbitrary element of the mapping class group of a closed orientable surface $Σ$ of genus at least $2$. For any characteristic cover $\widetildeΣ \to Σ$ one can consider the linear subspace ${\rm H}_1^{f.o.}(\widetildeΣ, \mathbb{Q})^ϕ\subseteq {\rm H}_1(\widetildeΣ, \mathbb{Q})$ consisting of all homology classes with finite $ϕ$-orbit. We prove that $\dim {\rm H}_1^{f.o.}(\widetildeΣ, \mathbb{Q})^ϕ$ can be arbitrary large for any fixed $ϕ\in {\rm Mod}(Σ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16322 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the mapping class group action on the homology of surface covers Spiridonov, Igor Geometric Topology 57K20 (Primary) 57M10, 57M12 (Secondary) Let $ϕ\in {\rm Mod}(Σ)$ be an arbitrary element of the mapping class group of a closed orientable surface $Σ$ of genus at least $2$. For any characteristic cover $\widetildeΣ \to Σ$ one can consider the linear subspace ${\rm H}_1^{f.o.}(\widetildeΣ, \mathbb{Q})^ϕ\subseteq {\rm H}_1(\widetildeΣ, \mathbb{Q})$ consisting of all homology classes with finite $ϕ$-orbit. We prove that $\dim {\rm H}_1^{f.o.}(\widetildeΣ, \mathbb{Q})^ϕ$ can be arbitrary large for any fixed $ϕ\in {\rm Mod}(Σ)$. |
| title | On the mapping class group action on the homology of surface covers |
| topic | Geometric Topology 57K20 (Primary) 57M10, 57M12 (Secondary) |
| url | https://arxiv.org/abs/2403.16322 |