On the structure of the top homology group of the Johnson kernel
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866916527205253120 |
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| author | Spiridonov, Igor A. |
| author_facet | Spiridonov, Igor A. |
| contents | The Johnson kernel is the subgroup $\mathcal{K}_g$ of the mapping class group ${\rm Mod}(Σ_{g})$ of a genus $g$ oriented closed surface $Σ_{g}$ generated by all Dehn twists about separating curves. In this paper we study the structure of the top homology group ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$. For any collection of $2g-3$ disjoint separating curves on $Σ_{g}$ one can construct the corresponding abelian cycle in the group ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$; such abelian cycles will be called simplest. In this paper we describe the structure of $\mathbb{Z}[{\rm Mod}(Σ_{g})/ \mathcal{K}_g]$-module on the subgroup of ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$ generated by all simplest abelian cycles and find all relations between them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10568 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the structure of the top homology group of the Johnson kernel Spiridonov, Igor A. Geometric Topology Group Theory 20F34 (Primary) 20F36, 57M07, 20J05 (Secondary) The Johnson kernel is the subgroup $\mathcal{K}_g$ of the mapping class group ${\rm Mod}(Σ_{g})$ of a genus $g$ oriented closed surface $Σ_{g}$ generated by all Dehn twists about separating curves. In this paper we study the structure of the top homology group ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$. For any collection of $2g-3$ disjoint separating curves on $Σ_{g}$ one can construct the corresponding abelian cycle in the group ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$; such abelian cycles will be called simplest. In this paper we describe the structure of $\mathbb{Z}[{\rm Mod}(Σ_{g})/ \mathcal{K}_g]$-module on the subgroup of ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$ generated by all simplest abelian cycles and find all relations between them. |
| title | On the structure of the top homology group of the Johnson kernel |
| topic | Geometric Topology Group Theory 20F34 (Primary) 20F36, 57M07, 20J05 (Secondary) |
| url | https://arxiv.org/abs/2111.10568 |