Big mapping class groups with uncountable integral homology

  • Martin Palmer

    Institutul de Matematică Simion Stoilow al Academiei Române, Bucharest, Romania
  • Xiaolei Wu

    Fudan University, Shanghai, China
Big mapping class groups with uncountable integral homology cover
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Abstract

We prove that, for any infinite-type surface , the integral homology of the closure of the compactly-supported mapping class group and of the Torelli group 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 for all infinite-type surfaces . However, we are still able to prove that the integral homology of is uncountable in all positive degrees for a large class of infinite-type surfaces . The key property of this class of surfaces is, roughly, that the space of ends of the surface 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 , where is a successor ordinal. We also observe an order- element in the first homology of the pure mapping class group of any surface of genus , answering a recent question of G. Domat.

Cite this article

Martin Palmer, Xiaolei Wu, Big mapping class groups with uncountable integral homology. Doc. Math. 29 (2024), no. 1, pp. 159–189

DOI 10.4171/DM/938