Stabilization for the automorphisms of free groups with boundaries

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

Mathematical Subject Classification 2000

Primary: 20F28

Secondary: 57M07

References

Forward citations

Publication

Received: 15 July 2004

Revised: 20 July 2005

Accepted: 4 July 2005

Published: 26 July 2005

Proposed: Martin Bridson

Seconded: Ralph Cohen, Thomas Goodwillie

Correction: 12 May 2008

Authors

Allen Hatcher
Mathematics Department Cornell University Ithaca New York 14853 USA

Nathalie Wahl
Aarhus University 116 Ny Munkegade 8000 Aarhus C Denmark

Volume 9, issue 3 (2005)

Allen Hatcher and Nathalie Wahl

Abstract