On the mapping class group action on the homology of surface covers

Igor Spiridonov

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arXiv · 2403.16322

On the mapping class group action on the homology of surface covers

Igor Spiridonov

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Abstract

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 $ϕ\in {\rm Mod}(Σ)$.

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BibTeX RIS

Igor Spiridonov. 2024-06-02. On the mapping class group action on the homology of surface covers. https://arxiv.org/abs/2403.16322

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