  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.
