The homology groups of the automorphism group of a free group are known to
stabilize as the number of generators of the free group goes to infinity, and this paper
relativizes this result to a family of groups that can be defined in terms of homotopy
equivalences of a graph fixing a subgraph. This is needed for the second author’s
recent work on the relationship between the infinite loop structures on the
classifying spaces of mapping class groups of surfaces and automorphism
groups of free groups, after stabilization and plus-construction. We show
more generally that the homology groups of mapping class groups of most
compact orientable 3–manifolds, modulo twists along 2–spheres, stabilize under
iterated connected sum with the product of a circle and a 2–sphere, and the
stable groups are invariant under connected sum with a solid torus or a ball.
These results are proved using complexes of disks and spheres in reducible
3–manifolds.
Keywords
automorphism groups of free groups, homological stability,
mapping class groups of 3–manifolds