A Combinatorial Substitute for the Degree Theorem in Auter Space

Auter space An is contractible. A. Hatcher and K. Vogtmann constructd a stratification of An into subspaces An,k such that An,k is kconnected. Their argument that An,k is (k−1)-connected, the Degree Theorem and its proof, is somewhat global in nature. Here we present a combinatorial substitue for the Degree Theorem that uses only local considerations to show that An,k is (k − 1)-connected. LetR0 be a (topological) connected graph with one vertex and n edges and an identification π1(R0) ∼= Fn, the free group on n generators. If Γ is a metric graph with basepoint p, then a homotopy equivalence ρ : R0 → Γ sending the basepoint of R0 to p is called a marking on Γ. There is an equivalence relation on the set of markings where two markings are considered equivalent if there is a basepoint-preserving homotopy between them. The space of all marked graphs for which the underlying metric graph has fundamental group of rank n and edge lengths sum to 1 is denoted An. In this way we can identify Aut(Fn) with the group of basepoint-preserving homotopy classes of basepoint-preserving homotopy equivalences of R0. Thus there is a right action of Aut(Fn) on An as follows: If A ∈ Aut(Fn) and (Γ, p, ρ) is a point in An then A(Γ, p, ρ) = (Γ, p, ρ ◦A). The spine of Auter space, denoted here by Ln, is a deformation retract of An where the metric data is ignored and only the combinatorial data of the graph and the marking are considered. For a more complete description of the construction of the analogous space for the outer automorphism group of Fn see [1]. In [2], Hatcher and Vogtmann use a function denoted here by d0 to stratify the space An into subspaces An,k := {(Γ, p, ρ) | d0(Γ) ≤ k}. These subspaces are invariant under the action of Aut(Fn) since the action only affects markings. The main technical result of [2] is: Degree Theorem. A piecewise linear map f0 : D k → An is homotopic to a map f1 : D k → An,k by a homotopy ft during which d0 decreases monotonically, that is, if t1 < t2 then d0(ft1(s)) ≥ d0(ft2(s)) for all s ∈ D . Using the Degree Theorem, they show that the pair (An An,k) is kconnected. Since Auter space is contractible, this is equivalent to An,k being (k − 1)-connected. This result is used in the main theorems of [2] to show ranges for integral and rational homological stability. The proof of the Degree Theorem is of global nature. Here we present an alternate proof that Date: September 17, 2009.