On the homology of big mapping class groups
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912176883630080 |
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| author | Palmer, Martin Wu, Xiaolei |
| author_facet | Palmer, Martin Wu, Xiaolei |
| contents | We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite-type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite-type surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_07470 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the homology of big mapping class groups Palmer, Martin Wu, Xiaolei Geometric Topology Algebraic Topology Group Theory 57K20, 20J06 We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite-type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite-type surfaces. |
| title | On the homology of big mapping class groups |
| topic | Geometric Topology Algebraic Topology Group Theory 57K20, 20J06 |
| url | https://arxiv.org/abs/2211.07470 |