Big mapping class groups with uncountable integral homology

TL;DR

The paper proves that for every infinite-type surface S, the integral homology H_i( overline{PMap_c(S)} ) and H_i( T(S) ) are uncountable for all i ≥ 1, by combining a lifting lemma with Domat’s escaping-sequence technology and a double-branched-cover strategy to transfer uncountability from Loch Ness-type constructions to general S. It also shows that Map(S) has uncountable H_i in a large class of infinite-type surfaces whose end spaces contain a limit point of topologically distinguished points, extending the uncountability to many finite-genus cases with countable end spaces but a distinguished end. A torsion element of order 10 is exhibited in H1 for genus-2 surfaces, and auxiliary appendices develop essential abelian-group tools and Freudenthal-extension techniques. Overall, the results significantly advance understanding of the homology of big mapping class groups, highlighting a dichotomy between uncountability and acyclicity depending on end-space structure. The methods blend geometric-topological constructions with algebraic embeddings, offering a versatile framework for exploring higher-degree homology in infinite-type settings.

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 $α$, 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.

Figures (5)

• Figure 2.1: The $3$-valent vertices of this graph are globally topologically distinguished but not topologically distinguished, since they are similar (but not globally similar) to each other.
• Figure 3.1: The once-punctured Loch Ness monster surface equipped with a sequence $\{\gamma_i\}_{i\in\mathbb{N}}$ of simple closed curves that is a well-spaced, escaping sequence in the sense of Definition \ref{['defn:escaping-sequence']}. The fact that it is well-spaced is witnessed by the associated sequence of simple closed curves $\{\gamma'_i\}_{i\in\mathbb{N}}$ given by $\gamma'_i = T_{\alpha_i}(\gamma_i)$.
• Figure 4.1: A surface with $n$ non-planar ends $e_1,\ldots,e_n$ for $2\leq n<\infty$. The top and bottom edges are identified to obtain a sphere, then the points $e_1,\ldots,e_n$ (together with a set of planar ends, which is not pictured) are removed, then we take a connected sum with a torus along each of the (infinitely many) small grey discs. The planar ends (not pictured) may have some or all of the non-planar ends $e_1,\ldots,e_n$ as limit points, but in any case lie outside of the subsurfaces $Y_1,\ldots,Y_{n-1}$, which support the handle shifts $h_1,\ldots,h_{n-1}$. The curves $\gamma_1,\gamma_2,\gamma_3,\ldots$ are chosen as illustrated such that the handle shift $h_1$ sends $\gamma_i$ to $\gamma_{i+1}$ (up to isotopy).
• Figure 5.1: The branched double covering \ref{['eq:branched-covering']}. After removing the subset marked in red (which includes the branch points), this restricts to the (genuine) double covering \ref{['eq:double-covering']}.
• Figure 5.2: A modification of the branched double covering depicted in Figure \ref{['fig:double-covering']}.

Theorems & Definitions (82)

• Theorem A: Corollary \ref{['cor:closure-of-compactly-supported-mcg']} and Theorem \ref{['thm:torelli']}
• Remark 1
• Definition 2
• Definition 3
• Theorem B
• Remark 4
• Remark 5
• Remark 7
• Remark 8
• Theorem C