Classification of orientable torus bundles over closed orientable surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908732092317696 |
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| author | Kasuya, Naohiko Noda, Issei |
| author_facet | Kasuya, Naohiko Noda, Issei |
| contents | Let $g$ be a non-negative integer, $Σ_g$ a closed orientable surface of genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the group homomorphisms $π_1(Σ_g)\to G$ up to the action of $\mathcal{M}_g$ on $π_1(Σ_g)$ in the following cases; (1) $G=PSL(2;\mathbb{Z})$, (2) $G=SL(2;\mathbb{Z})$. As an application of the case (2), we completely classify orientable $T^2$-bundles over closed orientable surfaces up to bundle isomorphisms. In particular, we show that any orientable $T^2$-bundle over $Σ_g$ with $g\geq 1$ is isomorphic to the fiber connected sum of $g$ pieces of $T^2$-bundles over $T^2$. Moreover, the classification result in the case (1) can be generalized into the case where $G$ is the free product of finite number of finite cyclic groups. We also apply it to an extension problem of maps from a closed surface to the connected sum of lens spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_14138 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of orientable torus bundles over closed orientable surfaces Kasuya, Naohiko Noda, Issei Geometric Topology Algebraic Topology Group Theory 55R15, 57R22, 20F65, 57K43 Let $g$ be a non-negative integer, $Σ_g$ a closed orientable surface of genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the group homomorphisms $π_1(Σ_g)\to G$ up to the action of $\mathcal{M}_g$ on $π_1(Σ_g)$ in the following cases; (1) $G=PSL(2;\mathbb{Z})$, (2) $G=SL(2;\mathbb{Z})$. As an application of the case (2), we completely classify orientable $T^2$-bundles over closed orientable surfaces up to bundle isomorphisms. In particular, we show that any orientable $T^2$-bundle over $Σ_g$ with $g\geq 1$ is isomorphic to the fiber connected sum of $g$ pieces of $T^2$-bundles over $T^2$. Moreover, the classification result in the case (1) can be generalized into the case where $G$ is the free product of finite number of finite cyclic groups. We also apply it to an extension problem of maps from a closed surface to the connected sum of lens spaces. |
| title | Classification of orientable torus bundles over closed orientable surfaces |
| topic | Geometric Topology Algebraic Topology Group Theory 55R15, 57R22, 20F65, 57K43 |
| url | https://arxiv.org/abs/2406.14138 |