Stabilization for the automorphisms of free groups with boundaries

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.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper30.abs.html

References in corpus (4)

• Homology stability for outer automorphism groups of free groups
• Erratum to: Homology stability for outer automorphism groups of free groups
• The stable moduli space of Riemann surfaces: Mumford's conjecture
• Automorphisms of free groups with boundaries

Cited by in corpus (19)

• Stabilization for mapping class groups of 3-manifolds
• Homology stability for outer automorphism groups of free groups
• Automorphisms of free groups with boundaries
• On string topology of classifying spaces
• A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
• Hodge decomposition for stable homology of automorphism groups of free groups
• A Birman exact sequence for Aut(F_n)
• Erratum to ``Stabilization for the automorphisms of free groups with boundaries''
• Twisted Morita-Mumford classes on braid groups
• The free factor complex and the dualizing module for the automorphism group of a free group
• Rational homological stability for groups of partially symmetric automorphisms of free groups
• First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group
• Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
• Cutting and Pasting in the Torelli subgroup of Out($F_n$)
• Stable cohomology of $Aut(F_n)$ with bivariant twisted coefficients
• The stable mapping class group of simply connected 4-manifolds
• On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone
• Topological Field Theories and Harrison Homology
• From mapping class groups to automorphism groups of free groups