On the homology of big mapping class groups

Martin Palmer; Xiaolei Wu

doi:10.1112/topo.12358

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arXiv · 2211.07470

On the homology of big mapping class groups

Martin Palmer·Xiaolei Wu

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Abstract

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.

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Martin Palmer, Xiaolei Wu. 2022-11-14. On the homology of big mapping class groups. https://doi.org/10.1112/topo.12358

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