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Disordered arcs and Harer stability

Oscar Harr, Max Vistrup, Nathalie Wahl

TL;DR

We address homological stability for mapping class groups $\Gamma(S_{g,r}^s)$ by embedding disk stabilization into the monoidal category ${\mathbf M}_2$ of bidecorated surfaces and using Euler characteristic as a grading. The main method combines a Yang–Baxter operator with Krannich stability theory to obtain the best known ranges $i \le \frac{2g}{3}$ for isomorphisms (and related bounds for two stabilization maps) and twisted-stability for coefficient systems of degree $k$. The work also clarifies that full braiding is not required: the inverse Dehn twist $T^{-1}$ yields a braid group action that suffices for stability, while not arising from a braiding on the subcategory. These results unify previous proofs and provide a flexible framework for homological stability in nonbraided settings.

Abstract

We give a new proof of homological stability with the best known isomorphism range for mapping class groups of surfaces with respect to genus. The proof uses the framework of Randal-Williams-Wahl and Krannich applied to disk stabilization in the category of bidecorated surfaces, using the Euler characteristic instead of the genus as a grading. The monoidal category of bidecorated surfaces does not admit a braiding, distinguishing it from previously known settings for homological stability. Nevertheless, we find that it admits a suitable Yang-Baxter element, which we show is sufficient structure for homological stability arguments.

Disordered arcs and Harer stability

TL;DR

We address homological stability for mapping class groups by embedding disk stabilization into the monoidal category of bidecorated surfaces and using Euler characteristic as a grading. The main method combines a Yang–Baxter operator with Krannich stability theory to obtain the best known ranges for isomorphisms (and related bounds for two stabilization maps) and twisted-stability for coefficient systems of degree . The work also clarifies that full braiding is not required: the inverse Dehn twist yields a braid group action that suffices for stability, while not arising from a braiding on the subcategory. These results unify previous proofs and provide a flexible framework for homological stability in nonbraided settings.

Abstract

We give a new proof of homological stability with the best known isomorphism range for mapping class groups of surfaces with respect to genus. The proof uses the framework of Randal-Williams-Wahl and Krannich applied to disk stabilization in the category of bidecorated surfaces, using the Euler characteristic instead of the genus as a grading. The monoidal category of bidecorated surfaces does not admit a braiding, distinguishing it from previously known settings for homological stability. Nevertheless, we find that it admits a suitable Yang-Baxter element, which we show is sufficient structure for homological stability arguments.
Paper Structure (15 sections, 16 theorems, 85 equations, 9 figures)

This paper contains 15 sections, 16 theorems, 85 equations, 9 figures.

Table of Contents

  1. Introduction
  2. High connectivity of the disordered arc complex
  3. Case 1: $\nu=1$
  4. Case 2: $\nu=2$
  5. The monoidal category of bidecorated surfaces
  6. Bidecorated surfaces and the monoidal structure
  7. Braided action
  8. Homological stability
  9. Disk destabilizations and disordered arcs
  10. Coefficient systems
  11. The stability theorem
  12. Braiding and homological stability for groups
  13. Yang--Baxter operators and braid groupoid actions
  14. Homological stability from Yang--Baxter elements
  15. Braidings and bidecorated surfaces

Key Result

Theorem 1

Let $S_{g,b}^s$ be a surface of genus $g\ge 0$, with $r\ge 1$ marked boundary components and $s\ge 0$ punctures, and let $\Gamma(S_{g,r}^s)=\pi_0{\operatorname{Homeo}} (S_{g,r}^s\textrm{ rel }\partial S)$ denote its mapping class group. The map induced by gluing a pair or pants along one boundary component is always injective, and an isomorphism when $i\leq\frac{2g}{3}$, and the map induced by g

Figures (9)

  • Figure 1: Maximal regular bad simplex $\{a_0,\dots,a_p\}$ and simplex $\{a'_0,\dots,a'_q\}$ in its link.
  • Figure 2: Regular bad 1-simplex with $\nu=1$.
  • Figure 3: Gluing a disk $X_1 = D^2$ to a bidecorated surface $S$
  • Figure 4: The curve $a_i$ in $D_i\mathbin{\text{\normalfont \#}} D_{i+1}$
  • Figure 5: Intersection of $a_i$ (blue) and $a_{i+1}$ (green) in the underlying surface of $D_i\mathbin{\text{\normalfont \#}} D_{i+1}\mathbin{\text{\normalfont \#}} D_{i+2}$.
  • ...and 4 more figures

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Theorem 3
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • ...and 37 more