Erratum to ``Stabilization for the automorphisms of free groups with boundaries''

Erratum to: Homology stability for outer automorphism groups of free groups

We correct the proof of Theorem 5 of the paper Homology stability for outer automorphism groups of free groups, by the first two authors.

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 ma…

Stabilization for mapping class groups of 3-manifolds

We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures.

自由群外自同构群中的尼尔森类

Nielsen classes in outer automorphism groups of free groups

针对 $n\ge8$，研究自由群外自同构群中生成对的尼尔森类，构造了 $GL(n,\mathbb Z)$ 等的生成对族，推导关系模障碍，Evans例子验证其可区分稳定化后的尼尔森类。