Homological stability for automorphism groups of free groups
Let \(F_n\) be the free group of rank \(n\). An imbedding of \(\text{Aut}(F_n)\) into \(\text{Aut} (F_{n + 1})\) is defined by extending an automorphism of \(F_n\) to an automorphism of \(F_{n + 1}\) fixing the \((n + 1)\)st basis element. The main result of this paper is that the corresponding map \(H_i(\text{Aut} (F_n)) \to H_i (\text{Aut} (F_{n + 1}))\) is an isomorphism for \(n > i^2/4 + 2i - 1\), and the map \(H_i (\text{Aut} (F_n)) \to H_i (\text{Out} (F_n))\) is an isomorphism for \(n > i^2 + 5i/2\). Similar stability results are known for various groups, including symmetric groups, many linear groups, and mapping class groups of 2-manifolds. \(\text{Out} (F_n)\) is closely related to the mapping class group of the 3-manifold \(M_{n,1}\) which is a connected sum of \(n\) copies of \(S^2 \times S^1\), with one open ball removed. Precisely, this mapping class group contains a finite abelian normal subgroup (generated by homeomorphisms supported in neighborhoods of imbedded 2-spheres), whose quotient is \(\text{Aut} (F_n)\). The author's approach is analogous to Harer's proof of homological stability for mapping class groups of 2-manifolds, as refined by Ivanov. In the 2-dimensional version simplicial complexes are constructed whose \(k\)-simplices correspond to isotopy classes of \(k + 1\) disjoint simple arcs or closed curves. These complexes are often contractible, or if not have high connectivity, and the mapping class group acts on them in an obvious way. Analysis of this action using standard (but nontrivial) homological methods yields the desired information. In the 3-dimensional version, the author considers families of disjoint imbedded 2-spheres in \(M_{n,1}\) (none of which bounds a ball, is parallel to the boundary sphere of \(M_{n,1}\) and no two of which are isotopic). The simplicial complex \(S_{n,1}\) obtained from such families is proven to be contractible when \(n > 0\) (actually this is proven for the more general case \(S_{n,s}\) when the manifold has \(s \geq 0\) punctures). One of the key ideas is to move systems of 2- spheres into a normal form with respect to a decomposition of \(M_{n,s}\) into twice-punctured 3-balls. Also of interest is the identification of the Culler-Vogtmann space of actions of \(F_n\) on \(R\)-trees with a subcomplex of \(S_{n,0}\). The contraction of \(S_{n,0}\) given by the author's argument preserves this subcomplex, giving an alternative proof of its contractibility. This is used to deduce the virtual cohomological dimensions of \(\text{Aut} (F_n)\) and \(\text{Out} (F_n)\), first determined by Culler and Vogtmann.
Culler-Vogtmann space of actions on \(R\)-trees
connected sum of \(n\) copies of \(S^ 2 \times S^ 1\)
scientific article; zbMATH DE number 791085