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


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