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 $\overline{{\mathrm{PMap}}_c(S)}$ and of the Torelli group $\mathcal{T}(S)$ 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 $\mathrm{Map}(S)$ for all infinite-type surfaces $S$. However, we are still able to prove that the integral homology of $\mathrm{Map}(S)$ 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 $\alpha$, where $\alpha$ 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.