Big mapping class groups with uncountable integral homology

TL;DR

This paper demonstrates that the integral homology groups of certain infinite-type surface mapping class groups are uncountably infinite in all positive degrees, revealing complex algebraic structures in these groups.

Contribution

It establishes uncountability of the integral homology in positive degrees for the closure of compactly-supported and Torelli mapping class groups of infinite-type surfaces, and extends results to a large class of such surfaces.

Findings

Integral homology of the closure of compactly-supported mapping class groups is uncountable in all positive degrees.

Integral homology of the Torelli group is uncountable in all positive degrees.

Identifies an order-10 element in the first homology of the pure mapping class group of genus 2 surfaces.

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.