TETHERS AND HOMOLOGY STABILITY FOR SURFACES

Abstract. Homological stability for sequences Gn → Gn+1 → · · · of groups is often proved by studying the spectral sequence associated to the action of Gn on a highly-connected simplicial complex whose stabilizers are related to Gk for k < n. When Gn is the mapping class group of a manifold, 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 pairs of dual curves together with arcs that tether these pairs 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.