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2017 Tethers and homology stability for surfaces
Allen Hatcher, Karen Vogtmann
Algebr. Geom. Topol. 17(3): 1871-1916 (2017). DOI: 10.2140/agt.2017.17.1871

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. 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.

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Allen Hatcher. Karen Vogtmann. "Tethers and homology stability for surfaces." Algebr. Geom. Topol. 17 (3) 1871 - 1916, 2017. https://doi.org/10.2140/agt.2017.17.1871

Information

Received: 1 November 2016; Revised: 23 January 2017; Accepted: 16 February 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06762603
MathSciNet: MR3677942
Digital Object Identifier: 10.2140/agt.2017.17.1871

Subjects:
Primary: 20J06, 57M07

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.17 • No. 3 • 2017
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