Homological stability for automorphism groups of free groups
type: Item
instance of: scholarly article
title: Homological stability for automorphism groups of free groups
zbMATH Open document ID: 0836.57003
DOI: 10.1007/BF02565999
published in: Commentarii Mathematici Helvetici
publication date: 1995-08-27T00:00:00Z
full work available at URL: https://eudml.org/doc/140361
review text: Let \(F_n\) be the free group of rank \(n\). An imbedding of \(\text{Aut}(F_n)\) into \(\text{Aut} (F_{n + 1})\) is defined by extending an automorphism of \(F_n\) to an automorphism of \(F_{n + 1}\) fixing the \((n + 1)\)st basis element. The main result of this paper is that the corresponding map \(H_i(\text{Aut} (F_n)) \to H_i (\text{Aut} (F_{n + 1}))\) is an isomorphism for \(n > i^2/4 + 2i - 1\), and the map \(H_i (\text{Aut} (F_n)) \to H_i (\text{Out} (F_n))\) is an isomorphism for \(n > i^2 + 5i/2\). Similar stability results are known for various groups, including symmetric groups, many linear groups, and mapping class groups of 2-manifolds. \(\text{Out} (F_n)\) is closely related to the mapping class group of the 3-manifold \(M_{n,1}\) which is a connected sum of \(n\) copies of \(S^2 \times S^1\), with one open ball removed. Precisely, this mapping class group contains a finite abelian normal subgroup (generated by homeomorphisms supported in neighborhoods of imbedded 2-spheres), whose quotient is \(\text{Aut} (F_n)\). The author's approach is analogous to Harer's proof of homological stability for mapping class groups of 2-manifolds, as refined by Ivanov. In the 2-dimensional version simplicial complexes are constructed whose \(k\)-simplices correspond to isotopy classes of \(k + 1\) disjoint simple arcs or closed curves. These complexes are often contractible, or if not have high connectivity, and the mapping class group acts on them in an obvious way. Analysis of this action using standard (but nontrivial) homological methods yields the desired information. In the 3-dimensional version, the author considers families of disjoint imbedded 2-spheres in \(M_{n,1}\) (none of which bounds a ball, is parallel to the boundary sphere of \(M_{n,1}\) and no two of which are isotopic). The simplicial complex \(S_{n,1}\) obtained from such families is proven to be contractible when \(n > 0\) (actually this is proven for the more general case \(S_{n,s}\) when the man…
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 791085
zbMATH Keywords: mapping class group of 3-manifold
zbMATH Keywords: Culler-Vogtmann space of actions on \(R\)-trees
zbMATH Keywords: outer automorphism group of free groups
zbMATH Keywords: free group of rank \(n\)
zbMATH Keywords: connected sum of \(n\) copies of \(S^ 2 \times S^ 1\)
zbMATH Keywords: homological stability
zbMATH Keywords: families of disjoint imbedded 2-spheres
author: Allen Hatcher
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 791085