We prove that the quotient map from to induces an isomorphism on homology in dimension for at least . 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 , showing that its homology is independent of for at least .
"Homology stability for outer automorphism groups of free groups." Algebr. Geom. Topol. 4 (2) 1253 - 1272, 2004. https://doi.org/10.2140/agt.2004.4.1253