Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the homology of big mapping class groups

Martin Palmer, Xiaolei Wu

TL;DR

The paper advances the understanding of homology for big mapping class groups by developing a general Mather-style infinite-iteration framework together with a robust homological stability mechanism for infinite-type surfaces. It introduces binary-tree surface constructions Gr$_ ext{B}(\\Sigma)$ and leverages a homogeneous-category approach to reduce homology questions to high-connectivity properties of tethered-curve complexes, ultimately proving acyclicity for the mapping class groups of the one-holed Cantor tree and related surfaces, and obtaining explicit homology calculations for the plane minus a Cantor set. Central technical contributions include the filtration of mapping-class groups via a homological-dissipation criterion, the reduction of stability to the connectivity of $W_n(A,X)_ullet$, and the detailed analysis of tethered-curve complexes across finite-type, telescoping, and binary-tree settings. Collectively, these results not only answer questions of acyclicity and homology in the infinite-type setting but also provide a versatile framework for future investigations into big mapping class groups and their homological invariants.

Abstract

We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite-type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite-type surfaces.

On the homology of big mapping class groups

TL;DR

The paper advances the understanding of homology for big mapping class groups by developing a general Mather-style infinite-iteration framework together with a robust homological stability mechanism for infinite-type surfaces. It introduces binary-tree surface constructions Gr and leverages a homogeneous-category approach to reduce homology questions to high-connectivity properties of tethered-curve complexes, ultimately proving acyclicity for the mapping class groups of the one-holed Cantor tree and related surfaces, and obtaining explicit homology calculations for the plane minus a Cantor set. Central technical contributions include the filtration of mapping-class groups via a homological-dissipation criterion, the reduction of stability to the connectivity of , and the detailed analysis of tethered-curve complexes across finite-type, telescoping, and binary-tree settings. Collectively, these results not only answer questions of acyclicity and homology in the infinite-type setting but also provide a versatile framework for future investigations into big mapping class groups and their homological invariants.

Abstract

We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite-type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite-type surfaces.
Paper Structure (20 sections, 37 theorems, 61 equations, 6 figures)

This paper contains 20 sections, 37 theorems, 61 equations, 6 figures.

Table of Contents

  1. Background
  2. Infinite-type surfaces
  3. End spaces of linear and binary tree surfaces.
  4. Homogeneous categories and homological stability
  5. High-connectivity
  6. Bad simplices arguments
  7. Complete joins
  8. Abstracting Mather's infinite iteration argument
  9. The general argument
  10. Application to binary tree surfaces
  11. Homological stability
  12. The homogeneous category for infinite-type surfaces
  13. Reduction to high-connectivity
  14. From semi-simplicial sets to simplicial complexes
  15. Simplifying the simplicial complexes
  16. ...and 5 more sections

Key Result

Theorem A

For each integer $i\geq 0$, we have

Figures (6)

  • Figure 1: The surfaces $D_\mathcal{C}$ and $B_\mathcal{C}$, whose mapping class groups are acyclic by Theorem \ref{['thm:disc-minus-Cantor']}. Capping off the boundary (the top circle) with a disc results in the Cantor tree surface$\mathrm{Gr}_\mathfrak{B}(\mathbb{S}^2)$ and the blooming Cantor tree surface$\mathrm{Gr}_\mathfrak{B}(T^2)$.
  • Figure 2: The Loch Ness monster surface $L = \mathrm{Gr}_\mathfrak{L}(T^2)$.
  • Figure 3: The surface $D_\mathcal{C}$, which is the disc minus the Cantor set (drawn in blue here). The subsurface $S$ is drawn in darker grey and the boundary is green.
  • Figure 5: The strategy of the high-connectivity proofs for $X=\mathfrak{L}(\Sigma)$ and for $X=\mathfrak{B}(\Sigma)$. The symbol $(*)$ on an inclusion indicates a bad simplex argument. The label $[n-2]$ on one of the inclusions indicates that only the $(n-2)$-skeleton includes into the larger complex.
  • Figure 6: The level structure of $D^2\natural \mathfrak{L}(T^2)^{\natural n}$, inherited from that of $\mathfrak{L}(T^2)$.
  • ...and 1 more figures

Theorems & Definitions (113)

  • Theorem A
  • Remark 1
  • Theorem B
  • Definition 2: Graph surfaces
  • Definition 3: Linear and binary tree surfaces
  • Example 5
  • Theorem C
  • Corollary D
  • proof
  • Remark 7
  • ...and 103 more