Mapping class groups of exotic tori and actions by ${\rm SL}_d(\mathbb Z)$
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917777326997504 |
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| author | Bustamante, Mauricio Krannich, Manuel Kupers, Alexander Tshishiku, Bena |
| author_facet | Bustamante, Mauricio Krannich, Manuel Kupers, Alexander Tshishiku, Bena |
| contents | We determine for which exotic tori $\mathcal{T}$ of dimension $d\neq4$ the homomorphism from the group of isotopy classes of orientation-preserving diffeomorphisms of $\mathcal{T}$ to ${\rm SL}_d(\mathbb Z)$ given by the action on the first homology group is split surjective. As part of the proof we compute the mapping class group of all exotic tori $\mathcal{T}$ that are obtained from the standard torus by a connected sum with an exotic sphere. Moreover, we show that any nontrivial ${\rm SL}_d(\mathbb Z)$-action on $\mathcal{T}$ agrees on homology with the standard action, up to an automorphism of ${\rm SL}_d(\mathbb Z)$. When combined, these results in particular show that many exotic tori do not admit any nontrivial differentiable action by ${\rm SL}_d(\mathbb Z)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_08065 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mapping class groups of exotic tori and actions by ${\rm SL}_d(\mathbb Z)$ Bustamante, Mauricio Krannich, Manuel Kupers, Alexander Tshishiku, Bena Geometric Topology Algebraic Topology Group Theory 57R10, 57R52, 20C10, 11F06 We determine for which exotic tori $\mathcal{T}$ of dimension $d\neq4$ the homomorphism from the group of isotopy classes of orientation-preserving diffeomorphisms of $\mathcal{T}$ to ${\rm SL}_d(\mathbb Z)$ given by the action on the first homology group is split surjective. As part of the proof we compute the mapping class group of all exotic tori $\mathcal{T}$ that are obtained from the standard torus by a connected sum with an exotic sphere. Moreover, we show that any nontrivial ${\rm SL}_d(\mathbb Z)$-action on $\mathcal{T}$ agrees on homology with the standard action, up to an automorphism of ${\rm SL}_d(\mathbb Z)$. When combined, these results in particular show that many exotic tori do not admit any nontrivial differentiable action by ${\rm SL}_d(\mathbb Z)$. |
| title | Mapping class groups of exotic tori and actions by ${\rm SL}_d(\mathbb Z)$ |
| topic | Geometric Topology Algebraic Topology Group Theory 57R10, 57R52, 20C10, 11F06 |
| url | https://arxiv.org/abs/2305.08065 |