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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.
Erratum
See Geom. Topol. 12
(2008) 639–641 for a correction to this paper.
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