Homology stability for outer automorphism groups of free groups

Hatcher, Allen; Vogtmann, Karen

doi:10.2140/agt.2004.4.1253

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Homology stability for outer automorphism groups of free groups

Allen Hatcher and Karen Vogtmann

Algebraic & Geometric Topology 4 (2004) 1253–1272

DOI: 10.2140/agt.2004.4.1253

Erratum: Algebraic & Geometric Topology 6 (2006) 573–579

arXiv: math.GT/0406377

Abstract

We prove that the quotient map from $Aut (F_{n})$ to $Out (F_{n})$ induces an isomorphism on homology in dimension $i$ for $n$ at least $2 i + 4$ . This corrects an earlier proof by the first author and significantly improves the stability range. In the course of the proof, we also prove homology stability for a sequence of groups which are natural analogs of mapping class groups of surfaces with punctures. In particular, this leads to a slight improvement on the known stability range for $Aut (F_{n})$ , showing that its $i^{th}$ homology is independent of $n$ for $n$ at least $2 i + 2$ .

Keywords

automorphisms of free groups, homology stability

Mathematical Subject Classification 2000

Primary: 20F65

Secondary: 20F28, 57M07

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Publication

Received: 28 June 2004

Accepted: 7 December 2004

Published: 24 December 2004

Authors

Allen Hatcher
Department of Mathematics Cornell University Ithaca NY 14853-4201 USA

Karen Vogtmann
Department of Mathematics Cornell University Ithaca NY 14853-4201 USA

Volume 4, issue 2 (2004)