Martin David Palmer -- Publication -- Big mapping class groups with uncountable integral homology

Abstract

We prove that, for any infinite-type surface S, the integral homology of the closure of the compactly-supported mapping class group PMap_c(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.