On the second homology of the genus 3 hyperelliptic Torelli group
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914263236345856 |
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| author | Spiridonov, Igor |
| author_facet | Spiridonov, Igor |
| contents | Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $Σ_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g)$ consisting of elements that act trivially on $\mathrm{H}_1(Σ_g;\mathbb{Z})$ and commute with $s$. It is generated by Dehn twists about $s$-invariant separating curves, and its cohomological dimension is $g-1$. In this paper we study the top homology group $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. For each pair of disjoint $s$-invariant separating curves there is a naturally associated abelian cycle in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$; we call such cycles \emph{simple}. We show that simple abelian cycles are in bijection with orthogonal (with respect to the intersection form) splittings of $\mathrm{H}_1(Σ_3;\mathbb{Z})$ satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12605 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the second homology of the genus 3 hyperelliptic Torelli group Spiridonov, Igor Geometric Topology Group Theory 20F34 (Primary) 20F36, 57M07, 20J05 (Secondary) Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $Σ_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g)$ consisting of elements that act trivially on $\mathrm{H}_1(Σ_g;\mathbb{Z})$ and commute with $s$. It is generated by Dehn twists about $s$-invariant separating curves, and its cohomological dimension is $g-1$. In this paper we study the top homology group $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. For each pair of disjoint $s$-invariant separating curves there is a naturally associated abelian cycle in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$; we call such cycles \emph{simple}. We show that simple abelian cycles are in bijection with orthogonal (with respect to the intersection form) splittings of $\mathrm{H}_1(Σ_3;\mathbb{Z})$ satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. |
| title | On the second homology of the genus 3 hyperelliptic Torelli group |
| topic | Geometric Topology Group Theory 20F34 (Primary) 20F36, 57M07, 20J05 (Secondary) |
| url | https://arxiv.org/abs/2601.12605 |