Cerf Theory for Graphs

Abstract

AWe develop a deformation theory for k-parameter families of pointed marked graphs with fixed fundamental group F n . Applications include a simple geometric proof of stability of the rational homology of Aut(F n ), computations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of F n are highly connected, and an improvement on the stability range for the integral homology of Aut(F n ).

Homology stability for outer automorphism groups of free groups

“…We also show that the natural map Aut(F n ) → Aut(F n+1 ) induces an isomorphism on H i for n ≥ 2i + 2, a slight improvement on the result in [6]. It follows that H i (Out(F n )) is independent of n for n ≥ 2i + 4.…”

“…For X n,s and its quotient this was shown in [4]. The connectivity of W n,s will follow from an easy generalization of the main technical result of [6], the Degree Theorem. The quotient W n,s /Γ n,s is combinatorially the same as X n,s /Γ n,s .…”

“…As in [4,6] which dealt with the cases s ≤ 1, there is a homeomorphism Φ : SS(M ) → A n,s , defined in the following way. Given a sphere system in SS(M ), thicken each sphere to S 2 × I , and also thicken the boundary spheres other than ∂ 0 .…”

“…We will need a variation of the Degree Theorem of [6]. In that paper, the degree of a basepointed graph of rank n was defined to be 2n − |v 0 |, where |v 0 | is the valence of the basepoint v 0 .…”

“…In contrast, it is conjectured that they are rationally trivial, though a few non-trivial unstable rational classes have been found [2,3,6]. One of these non-trivial classes, in H 7 (Aut(F 5 ); Q), becomes trivial under the map to H 7 (Out(F 5 ); Q), showing that this map is not always an isomorphism on homology [3].…”