Algebraic & Geometric Topology

Tethers and homology stability for surfaces

Allen Hatcher and Karen Vogtmann

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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.

Article information

Algebr. Geom. Topol., Volume 17, Number 3 (2017), 1871-1916.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20J06: Cohomology of groups 57M07: Topological methods in group theory

homology stability mapping class group curve complex


Hatcher, Allen; Vogtmann, Karen. Tethers and homology stability for surfaces. Algebr. Geom. Topol. 17 (2017), no. 3, 1871--1916. doi:10.2140/agt.2017.17.1871.

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