Erratum to: Homology stability for outer automorphism groups of free groups

By A. Hatcher, Karen Vogtmann. 2004

94 citations recorded by the source corpus.

In Hatcher‐Vogtmann [2] a proof was presented that the homology of certain groups An;s is independent of both n and s for n sufficiently large. The groups An;s include Aut.Fn/ (sD 1) and Out.Fn/ (sD 0). In August of 2005 Nathalie Wahl discovered an error in the proof, and the purpose of this note is to fix that error. We assume the reader is familiar with [2], whose notation and conventions we will use here without further comment. The error occurs in the first part of the proof of Theorem 5, showing that the map W Hi.An;sC2/! Hi.AnC1;s/ is injective for n i and s 1. The argument used a diagram chase in the following diagram: Hi.An;sC2;An 1;sC4/ ! Hi 1.An 1;sC4/ ? y ? y Hi.AnC1;s;An;sC2/ ! Hi 1.An;sC2/ It was asserted that the top horizontal and right vertical arrows were successive maps in the long exact sequence of the pair .An;sC2;An 1;sC4/, and hence their composition was the zero map, but in fact the group An;sC2 in the lower right corner of the diagram is a different subgroup of AnC1;s from the An;sC2 in the upper left corner, so that is not induced by the inclusion map of the pair. It is in fact true that the composition is the zero map for n sufficiently large, but a proof seems to require the results proved in this correction. We correct the problem by giving a completely new proof of stability with respect to s for s 1, complementing the earlier proof of stability with respect to n. The new proof entirely avoids the diagram displayed above and instead focuses on the map W An;s!An;sC1 . In the range where is an isomorphism, the relation D 2

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5faa5d6a14c96d7c3f23d7763f57660239abe28

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Citations in the corpus

Stabilization for mapping class groups of 3-manifolds (2007, background): “To show independence of the number of disks in R, we follow the same strategy as in [ 20 ].”
Representation stability and finite linear groups (2014, methodology): “th basis element to vi−1. Also, deﬁne C = Tp i=0 Ci. Then η2(((v0,C0),...,(vp,Cp))) := (f,C) is an element of Hom Fr(Fp+1,Fn). 39 The space PB(Fn) is similar to a space introduced by Hatcher–Vogtmann [HatVo2]. The following is a variant on a theorem of theirs. Theorem 6.9. The space PB(Fn) is n−3 2 ”
Homological stability for automorphism groups (2014, background): “Alternatively, we could have shown that Sn(Z, e) is isomorphic to the complex V ± n,1 of [28] and use Proposition 2 of that paper.”
Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn) (2011, background): “In the final section of this paper we note that hairy Lie graph homology is related to the cohomology of mapping class groups of certain punctured 3-manifolds, as defined in [13].”
Hyperbolicity of the complex of free factors (2011, methodology): “They defined this complex in terms of sphere systems in #1S 1 × S2 and used variants in their work on homological stability [19, 21, 18].”
Problems on Mapping Class Groups And Related Topics (2006, background): “Thurston’s original construction of pseudo-Anosov (pA) mappings [37] (cf.”
Hodge‐type decomposition in the homology of long knots (2010, background): “Similar graph-complexes appear in the study of the homology of the outer spaces [11, 12, 17, 19, 24].”
Cohomological structure of the mapping class group and beyond (2005, background): “ping class group and OutFn which will be discussed at several places in this book. Here we would like to concentrate on the cohomological side of this problem. 10 SHIGEYUKIMORITA Hatcher and Vogtmann [37] (see also Hatcher [35]) proved that the homology of OutFn stabilizes in a certain stable range.”