Geodesic
Big mapping class groups with uncountable integral homology
Documenta mathematica, Tome 29 (2024) no. 1, pp. 159-189

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We prove that, for any infinite-type surface S, the integral homology of the closure of the compactly-supported mapping class group PMapc​(S)​ and of the Torelli group 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 Map(S) for all infinite-type surfaces S. However, we are still able to prove that the integral homology of 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.
DOI : 10.4171/dm/938
Classification : 57K20, 20J06
Mots-clés : Big mapping class groups, pure mapping class groups, Torelli groups, group homology, torsion 
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Martin Palmer; Xiaolei Wu. Big mapping class groups with uncountable integral homology. Documenta mathematica, Tome 29 (2024) no. 1, pp. 159-189. doi: 10.4171/dm/938

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