First cohomologies and the Johnson homomorphisms of the automorphism group of a free group.
type: Item
instance of: scholarly article
title: First cohomologies and the Johnson homomorphisms of the automorphism group of a free group.
zbMATH Open document ID: 1286.20040
author: Takao Satoh
published in: Journal of Pure and Applied Algebra
publication date: 2013-04-17T00:00:00Z
review text: Let \(F_n\) be a free group of rank \(n\), let \(H\) be its Abelianisation, and let \(\Aut(F_n)\) be its automorphism group and \(\mathrm{Out}(F_n)\) be its outer automorphism group. Cohomology groups of \(\Aut(F_n)\) with coefficients in \(\mathbb Z\), \(\mathbb Q\), \(H\) and \(H^*\) (the dual of \(H\)) have been considered in the literature. This article considers the (first) cohomology group of \(\Aut(F_n)\) with coefficients in another \(\Aut(F_n)\)-module, \(V_L\), defined as follows. The kernel of the natural homomorphism \(\Aut(F_n)\to\Aut(H)\) induced from \(H\) is called the IA-automorphism group of \(F_n\) and is denoted \(IA_n\). Let \(V\) be the Abelianisation of \(IA_n\), let \(L\) be a principal ideal domain without 2-torsion, and set \(V_L=V\otimes_{\mathbb Z}L\). The main theorem is that for \(n\geq 5\) the first cohomology group \(H^1(\Aut(F_n),V_L)=L^{\oplus 2}\). Nielsen's presentation for \(\Aut(F_n)\) is central to the proof. Two applications are given (for \(n\geq 5\)). The first is that the first Johnson homomorphism \(\tau_1\colon IA_n\to V\) does not extend directly to \(\Aut(F_n)\) as a crossed homomorphism. The second is that \(H^1(\mathrm{Out}(F_n),V_L)=L\).
reviewed by: Gerald Williams
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 6155156
zbMATH Keywords: free groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: Johnson homomorphisms
zbMATH Keywords: cohomology groups
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.06.017
OpenAlex ID: W1983489351
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DOI: 10.1016/J.JPAA.2012.06.017
description: scientific article; zbMATH DE number 6155156