Tethers and homology stability for surfaces

Abstract

Homological stability for sequences $G_n\to G_{n+1}\to \cdots$ of groups is often proved by studying the spectral sequence associated to the action of $G_n$ on a highly connected simplicial complex whose stabilizers are related to $G_k$ for $k<n$. When $G_n$ is the mapping class group of a manifold, suitable simplicial complexes can be made using isotopy classes of various geometric objects in the manifold. 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.