Tethers and homology stability for surfaces

Hatcher, Allen¹ ; Vogtmann, Karen²

¹Mathematics Department, Cornell University, Ithaca, NY 14853, United States
²Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

Algebraic and Geometric Topology, Tome 17 (2017) no. 3, pp. 1871-1916

Résumé

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.

Détail

DOI : 10.2140/agt.2017.17.1871

Classification : 20J06, 57M07
Keywords: homology stability, mapping class group, curve complex

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