Homology Stability for Outer Automorphisms of Free Groups

  title={Homology Stability for Outer Automorphisms of Free Groups},
  author={Allen Hatcher and Karen Vogtmann},
In this paper we present a proof that the quotient map Aut(Fn)→Out(Fn) induces an isomorphism on the integral homology groups Hi for n ≥ 2i + 4. We also show that the natural map Aut(Fn)→Aut(Fn+1) induces an isomorphism on Hi for n ≥ 2i+ 2, a slight improvement on the result in [HV]. It follows that Hi(Out(Fn)) is independent of n for n ≥ 2i+ 4. These stable homology groups have a significant amount of nontrivial torsion, since they contain the homology of Ω∞S∞ as a direct summand, as observed… CONTINUE READING