#### Geometry & Topology, Vol. 9 (2005)
Paper no. 30, pages 1295--1336.

## Stabilization for the automorphisms of free groups with boundaries

### Allen Hatcher and Nathalie Wahl

**Abstract**.
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

**AMS subject classification**.
Primary: 20F28.
Secondary: 57M07.

**E-print:** `arXiv:math.GT/0406277`

**DOI:** 10.2140/gt.2005.9.1295

Submitted to GT on 15 July 2004.
(Revised 20 July 2005.)
Paper accepted 4 July 2005.
Paper published 26 July 2005.

Notes on file formats
Allen Hatcher, Nathalie Wahl

Mathematics Department, Cornell University, Ithaca NY 14853, USA

and

Aarhus University, 116 Ny Munkegade, 8000 Aarhus C, Denmark

Email: hatcher@math.cornell.edu, wahl@imf.au.dk

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