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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2305.13109 |
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| _version_ | 1866909088830455808 |
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| author | Boggi, Marco Putman, Andrew Salter, Nick |
| author_facet | Boggi, Marco Putman, Andrew Salter, Nick |
| contents | Putman and Wieland conjectured that if $\tildeΣ \rightarrow Σ$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of $H_1(\tildeΣ;\mathbb{Q})$ under the action of lifts to $\tildeΣ$ of mapping classes on $Σ$ are infinite. We prove that this holds if $H_1(\tildeΣ;\mathbb{Q})$ is generated by the homology classes of lifts of simple closed curves on $Σ$. We also prove that the subspace of $H_1(\tildeΣ;\mathbb{Q})$ spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that $H_1(\tildeΣ;\mathbb{Q})$ is generated by the homology classes of lifts of loops on $Σ$ lying on subsurfaces homeomorphic to 3-holed spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13109 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generating the homology of covers of surfaces Boggi, Marco Putman, Andrew Salter, Nick Geometric Topology Putman and Wieland conjectured that if $\tildeΣ \rightarrow Σ$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of $H_1(\tildeΣ;\mathbb{Q})$ under the action of lifts to $\tildeΣ$ of mapping classes on $Σ$ are infinite. We prove that this holds if $H_1(\tildeΣ;\mathbb{Q})$ is generated by the homology classes of lifts of simple closed curves on $Σ$. We also prove that the subspace of $H_1(\tildeΣ;\mathbb{Q})$ spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that $H_1(\tildeΣ;\mathbb{Q})$ is generated by the homology classes of lifts of loops on $Σ$ lying on subsurfaces homeomorphic to 3-holed spheres. |
| title | Generating the homology of covers of surfaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2305.13109 |