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Tethers and homology stability for surfaces

Allen Hatcher and Karen Vogtmann

Algebraic & Geometric Topology 17 (2017) 1871–1916

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. 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.

homology stability, mapping class group, curve complex
Mathematical Subject Classification 2010
Primary: 20J06, 57M07
Received: 1 November 2016
Revised: 23 January 2017
Accepted: 16 February 2017
Published: 17 July 2017
Allen Hatcher
Mathematics Department
Cornell University
Ithaca, NY 14853
United States
Karen Vogtmann
Mathematics Institute
University of Warwick
United Kingdom