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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Aut</mo><mrow><mo class="MathClass-open">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Out</mo><mrow><mo class="MathClass-open">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow></math>$ induces an isomorphism on homology in dimension $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>i</mi></math>$ for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ at least $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>i</mi> <mo class="MathClass-bin">+</mo> <mn>4</mn></math>$ . 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Aut</mo><mrow><mo class="MathClass-open">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow></math>$ , showing that its $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>i</mi></mrow><mrow><mo class="qopname">th</mo></mrow></msup></math>$ homology is independent of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ at least $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>i</mi> <mo class="MathClass-bin">+</mo> <mn>2</mn></math>$ .

Keywords

automorphisms of free groups, homology stability

Mathematical Subject Classification 2000

Primary: 20F65

Secondary: 20F28, 57M07

References

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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)