Big mapping class groups with uncountable integral homology
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866912176896212992 |
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| author | Palmer, Martin Wu, Xiaolei |
| author_facet | Palmer, Martin Wu, Xiaolei |
| contents | We prove that, for any infinite-type surface $S$, the integral homology of the closure of the compactly-supported mapping class group $\overline{\mathrm{PMap}_c(S)}$ and of the Torelli group $\mathcal{T}(S)$ is uncountable in every positive degree. By our results in arXiv:2211.07470 and other known computations, such a statement cannot be true for the full mapping class group $\mathrm{Map}(S)$ for all infinite-type surfaces $S$. However, we are still able to prove that the integral homology of $\mathrm{Map}(S)$ is uncountable in all positive degrees for a large class of infinite-type surfaces $S$. The key property of this class of surfaces is, roughly, that the space of ends of the surface $S$ contains a limit point of topologically distinguished points. Our result includes in particular all finite-genus surfaces having countable end spaces with a unique point of maximal Cantor-Bendixson rank $α$, where $α$ is a successor ordinal. We also observe an order-$10$ element in the first homology of the pure mapping class group of any surface of genus $2$, answering a recent question of G. Domat. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_11942 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Big mapping class groups with uncountable integral homology Palmer, Martin Wu, Xiaolei Geometric Topology Algebraic Topology Group Theory 57K20, 20J06 We prove that, for any infinite-type surface $S$, the integral homology of the closure of the compactly-supported mapping class group $\overline{\mathrm{PMap}_c(S)}$ and of the Torelli group $\mathcal{T}(S)$ is uncountable in every positive degree. By our results in arXiv:2211.07470 and other known computations, such a statement cannot be true for the full mapping class group $\mathrm{Map}(S)$ for all infinite-type surfaces $S$. However, we are still able to prove that the integral homology of $\mathrm{Map}(S)$ is uncountable in all positive degrees for a large class of infinite-type surfaces $S$. The key property of this class of surfaces is, roughly, that the space of ends of the surface $S$ contains a limit point of topologically distinguished points. Our result includes in particular all finite-genus surfaces having countable end spaces with a unique point of maximal Cantor-Bendixson rank $α$, where $α$ is a successor ordinal. We also observe an order-$10$ element in the first homology of the pure mapping class group of any surface of genus $2$, answering a recent question of G. Domat. |
| title | Big mapping class groups with uncountable integral homology |
| topic | Geometric Topology Algebraic Topology Group Theory 57K20, 20J06 |
| url | https://arxiv.org/abs/2212.11942 |