Tethers and homology stability for surfaces

Hatcher, Allen; Vogtmann, Karen

doi:10.2140/agt.2017.17.1871

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Tethers and homology stability for surfaces

Allen Hatcher and Karen Vogtmann

Algebraic & Geometric Topology 17 (2017) 1871–1916

DOI: 10.2140/agt.2017.17.1871

Abstract

Homological stability for sequences $G_{n} \to G_{n + 1} \to \dots$ of groups is often proved by studying the spectral sequence associated to the action of $G_{n}$ on a highly connected simplicial complex whose stabilizers are related to $G_{k}$ for $k < n$ . When $G_{n}$ 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

Mathematical Subject Classification 2010

Primary: 20J06, 57M07

References

Publication

Received: 1 November 2016

Revised: 23 January 2017

Accepted: 16 February 2017

Published: 17 July 2017

Authors

Allen Hatcher
Mathematics Department Cornell University Ithaca, NY 14853 United States
https://www.math.cornell.edu/~hatcher/
Karen Vogtmann
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom
http://www2.warwick.ac.uk/fac/sci/maths/people/staff/karen\_vogtmann/

Volume 17, issue 3 (2017)