Tethers and homology stability for surfaces

TL;DR

This paper introduces a refined geometric approach using tethered curve configurations to simplify homological stability proofs for surface mapping class groups, improving the spectral sequence analysis.

Contribution

It develops a new method employing tethered curve configurations to streamline the spectral sequence approach in homological stability proofs for surfaces.

Findings

Simplified stabilizer analysis for surface mapping class groups

Application of the method to braid groups and orientable surfaces

Enhanced understanding of geometric objects in homological stability

Abstract

Homological stability for sequences of groups is often proved by studying the spectral sequence associated to the action of a typical group in the sequence on a highly-connected simplicial complex whose stabilizers are related to previous groups in the sequence. In the case of mapping class groups of manifolds, suitable simplicial complexes can be made using isotopy classes of various geometric objects in the manifold. In this paper we focus on the case of surfaces and show that by using more refined geometric objects consisting of certain configurations of curves with arcs that tether these curves to the boundary, the stabilizers can be greatly simplified and consequently also the spectral sequence argument. We give a careful exposition of this program and its basic tools, then illustrate the method using braid groups before treating mapping class groups of orientable surfaces in full detail.