Homological stability for automorphism groups of free groups

Let F, be a free group on n generators, Aut(F,) its group of automorphisms, and Out(F,) its outer automorphism group, the quotient of Aut(F,) by inner automorphisms. There has been much progress of late in the study of these groups via the one-dimensional model which arises from regarding F, as the fundamental group of a graph; see e.g., [BH] and [CV]. In this paper we return to a three-dimensional model first used by J. H. C. Whitehead in the 1930's, which involves looking at embedded 2-spheres in a connected sum of S 1 x S2's. Refining Whitehead's techniques and applying subsequent results of Laudenbach, we use this three-dimensional model to prove: