Disordered arcs and Harer stability

TL;DR

This paper presents a new proof of homological stability for mapping class groups of surfaces, utilizing a novel categorical framework with Euler characteristic grading and a Yang-Baxter element.

Contribution

It introduces a new proof method using Euler characteristic instead of genus and identifies a Yang-Baxter element in a non-braided monoidal category for stability.

Findings

Established homological stability with optimal isomorphism range

Developed a framework using Euler characteristic as a grading

Identified a Yang-Baxter element in a non-braided setting

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.