On torsion in the homology of the Torelli group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911712068763648 |
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| author | Vladimirov, Andrei |
| author_facet | Vladimirov, Andrei |
| contents | Let $S_g$ be a closed, oriented surface of genus $g$, and let $\operatorname{Mod}(S_g)$ denote its mapping class group. The Torelli group $\mathcal{I}_g$ is the subgroup of $\operatorname{Mod}(S_g)$ consisting of mapping classes that act trivially on $H_1(S_g)$. For any collection of pairwise disjoint, separating simple closed curves on $S_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k(\mathcal{I}_g)$, which is called an abelian cycle. We prove that the subgroup of $H_k(\mathcal{I}_g)$ generated by such abelian cycles is a $\mathbb{Z}/2\mathbb{Z}$-vector space for all $k$, and that it is finite-dimensional for $k = 2$ and $g \geq 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_25728 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On torsion in the homology of the Torelli group Vladimirov, Andrei Geometric Topology Group Theory 57M07 (Primary) 20J05, 20J06 (Secondary) Let $S_g$ be a closed, oriented surface of genus $g$, and let $\operatorname{Mod}(S_g)$ denote its mapping class group. The Torelli group $\mathcal{I}_g$ is the subgroup of $\operatorname{Mod}(S_g)$ consisting of mapping classes that act trivially on $H_1(S_g)$. For any collection of pairwise disjoint, separating simple closed curves on $S_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k(\mathcal{I}_g)$, which is called an abelian cycle. We prove that the subgroup of $H_k(\mathcal{I}_g)$ generated by such abelian cycles is a $\mathbb{Z}/2\mathbb{Z}$-vector space for all $k$, and that it is finite-dimensional for $k = 2$ and $g \geq 4$. |
| title | On torsion in the homology of the Torelli group |
| topic | Geometric Topology Group Theory 57M07 (Primary) 20J05, 20J06 (Secondary) |
| url | https://arxiv.org/abs/2510.25728 |