Homological stability for sequences
of groups is often proved by studying the spectral sequence associated to the action
of
on a highly connected simplicial complex whose stabilizers are related to
for
.
When
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.
Keywords
homology stability, mapping class group, curve complex
