THE TOPOLOGY AT INFINITY OF Out(F_{n})

MLADEN BESTVINA AND MARK FEIGHN

Abstract. We construct a bordification of Outer Space analogous to the Borel-Serre bordification of symmetric spaces. As an application, we show that Out(F_{n}) is (2n−5)-connected at infinity and that it is a virtual duality group of dimension 2n − 3.

Contents

1. Introduction 2 1.1. Absolute Neighborhood Retracts 5 1.2. Semi-algebraic sets 5 1.3. Z-sets 5 2. The Bordification of Outer Space 6 2.1. The Bordification of Σ_{G} 6 2.2. Local structure 11 2.3. Cell decomposition 14 2.4. Functoriality 16 2.5. The group Out(N, K) and its Outer Space 16 3. Morse theory 19 3.1. Hierarchy of lengths of a homotopy class 20 3.2. The behavior on cells 22 3.3. Critical points 26 4. Ascending links 27 4.1. Case |N| = p = 1 32 4.2. Necklaces and the inward normal 33 5. The topology at infinity 40 References 43

Date: October 1999.

Both authors gratefully acknowledge the support by the National Science Foundation.

1. Introduction

A locally compact space X is said to be r-connected at infinity provided for every compact set K ⊆ X there exists a compact set L ⊆ X such that L ⊇ K and any map α : S^{d} → X \ L with d ≤ r extends to a map α̃ : B^{d+1} → X \ K. For example, X is (−1)-connected at infinity iff it is non-compact, and it is 0-connected at infinity iff it has 1 end.

A discrete group Γ is said to be r-connected at infinity if it acts cocompactly, freely, and simplicially on an r-connected simplicial complex X which is rconnected at infinity. If Γ acts cocompactly, freely, and simplicially on another r-connected simplicial complex Y, then Y is necessarily also r-connected at infinity. For example, a group is (−1)-connected at infinity iff it is infinite. The aim of this paper is to prove:

Theorem 1.1. For n ≥ 2, Out(F_{n}) is (2n − 5)-connected at infinity.

There are several natural contractible spaces on which Out(F_{n}) acts, namely Culler-Vogtmann’s Outer Space [CV86] and its variants. Outer Space is to be regarded as the analog of a homogeneous space with the action of an arithmetic group or the Teichmüller space with the action of the mapping class group of a compact surface: the action is not quite free (point stabilizers are finite) and it is not quite cocompact. The first problem is not serious at all and is dealt with using the following observation:

Lemma 1.2. If Γ acts cocompactly, simplicially, and with finite stabilizers of points on an r-connected simplicial complex X which is r-connected at infinity, and if there is a homomorphism ϕ : Γ → Δ to a finite group with torsion-free kernel, then Γ is r-connected at infinity.

Proof. Δ acts freely on an r-connected finite simplicial complex Y. The diagonal action of Γ on X × Y (induced by ϕ in the second coordinate) is cocompact and free, and X × Y is r-connected and r-connected at infinity.

The group Out(F_{n}) satisfies the assumptions of Lemma 1.2. Indeed, the natural homomorphism Out(F_{n}) → GL_{n}(ℤ/3) has torsion-free kernel ^{1} [BT68].

There are several ways to resolve the second problem. There is a cocompact equivariant deformation retract of Outer Space, called the Spine of Outer Space [CV86]. Alternatively, we follow the constructions of Siegel [Sie59] for GL_{n}(ℤ) and Borel-Serre [BS73] for general arithmetic groups to produce the Bordification of Outer Space. The procedure consists of adding ideal points at infinity rather than “cutting off cusps”. Roughly speaking, a point of Outer Space is a marked metric graph (of rank n) of total volume 1 in which essential loops have positive length. At an ideal point, the metric is allowed to vanish on a subgraph which is not a forest, but a part of the structure at this ideal point is the “secondary” metric of volume 1 defined only on the core of the subgraph on which the primary metric vanishes. Similarly, the secondary metric is allowed to vanish non-trivially and there may be a “tertiary” metric, etc. Thus an ideal point is represented by a hierarchy of metrics defined on a decreasing sequence of subgraphs of a rank n marked metric graph. For example, imagine a sequence of metrics ℓ_{i}, i = 2, 3, ⋯ on the rose with loops a, b, c such that ℓ_{i}(a) = 1 − 1/i and ℓ_{i}(b) = ℓ_{i}(c) = 1/2i. The sequence converges to an ideal point with primary metric ℓ given by ℓ(a) = 1 and ℓ(b) = ℓ(c) = 0 and the secondary metric ℓ′ defined only on b and c by ℓ′(b) = ℓ′(c) = 1/2. The secondary metric is obtained by rescaling the restriction of ℓ_{i} to b ∪ c (the subgraph on which ℓ vanishes) and taking the limit. For another example, the sequence of metrics ℓ_{i} defined by ℓ_{i}(a) = 1 − 1/i − 1/i^{2}, ℓ_{i}(b) = 1/i, ℓ_{i}(c) = 1/i^{2} converges to the ideal point determined by a primary metric that assigns 1 to a and 0 to b and c, a secondary metric defined on b ∪ c that assigns 1 to b and 0 to c, and a tertiary metric defined on c and assigning 1 to it.

1More precisely, they show that Ker[Out(F_{n}) → GL_{n}(ℤ)] is torsion-free. The desired statement follows then from the standard fact (see [Bro82, p.40]) that Ker[GL_{n}(ℤ) → GL_{n}(ℤ/3)] is torsion-free.

The reason for working with bordification instead with the spine is that we analyze the topology of the underlying space using a Morse function, and a natural Morse function would have too many (canceling) critical points at the frontier of the cusps that were cut off. We succeed in understanding the topology at infinity because of the lucky event that a certain naturally defined Morse-like function on the bordified Outer Space has all critical points of the same index, so there is no need to worry about possible cancellations.

A model Morse theory argument for investigating the topology at infinity is:

Lemma 1.3. Suppose M^{n} is an open manifold and f : M → ℝ_{+} = [0, ∞) is a proper Morse function such that at all critical points the ascending sphere has dimension ≥ r + 1 (i.e. the index is ≤ n − r − 2). If M is r-connected, then it is r-connected at infinity.

Outer Space is not a manifold and we develop a Morse theory that applies to (bordified) Outer Space in the style of [BB97]. Roughly speaking, for any conjugacy class α in F_{n} the map “length of α” is a good candidate for a Morse function on the bordified Outer Space. Unfortunately, such a function does not have isolated critical points.

To remedy this, we work with an infinite collection (an “ample sequence”) of conjugacy classes α_{1}, α_{2}, ⋯ and consider a function that to a marked graph assigns the tuple of lengths of the α_{i}’s. Worse, the topology near the ideal points is not captured by ordinary length, and we are forced to consider higher order lengths as well. The target of our “Morse function” is not ℝ_{+} but an infinite product of ℝ_{+}’s, ordered lexicographically, so it makes sense to talk about sub- and super-level sets.

The analysis of the topology of “ascending links” is inductive, and we are forced to consider more general bordified Outer Spaces, allowing the underlying graphs to be disconnected and to have distinguished points.

As a corollary of our analysis of the topology of bordified Outer Space we obtain:

Theorem 1.4. For n ≥ 2, Out(F_{n}) is a virtual duality group of dimension 2n − 3.

It was shown by Culler and Vogtmann [CV86] that the virtual cohomological dimension (see [Bro82]) of Out(F_{n}) is 2n − 3. The notion of a duality group was introduced by Bieri and Eckmann [BE73] and it generalizes the concept of Poincaré duality groups. One definition (see [Bro82, p.220]) is that a group Γ that acts freely and cocompactly on a contractible cell complex X (by permuting cells) is a duality group of dimension n if H^{i}(Γ; ℤΓ) vanishes when i ≠ n and is torsion-free (as an abelian group) when i = n. The cohomology group H^{i}(Γ; ℤΓ) is isomorphic (see [Bro82, p.209]) to the compactly supported cohomology H_{c}^{i}(X; ℤ). From the Culler-Vogtmann result we have H^{i}(Γ; ℤΓ) = 0 for i > 2n − 3, and it is a theorem of Geoghegan-Mihalik [GM85] that if X as above is k-connected at infinity, then H_{c}^{i}(X; ℤ) vanishes for i ≤ k + 1 and is torsion-free for i = k + 2. Applying this result with k = 2n − 5 gives a proof of Theorem 1.4. In fact, our analysis gives a concrete description of H_{c}^{2n−3}(X; ℤ) from which it is clear that it is torsion-free (and moreover a free abelian group).

When n = 2 we don’t say anything new. Vogtmann [Vog95] showed that Out(F_{n}) has one end for n ≥ 3 and that it is 1-connected at infinity for n ≥ 5, and she also showed [Vog90] that Outer Space and its variants are Cohen-Macauley complexes, thus providing evidence for Theorem 1.4. Rickert [Ric] proved that Out(F_{4}) is 1-connected at infinity. The statement that Out(F_{3}) is 1-connected at infinity is new. For arithmetic groups, the equivalents of Theorems 1.1 and 1.4 were proved by Borel and Serre [BS73], and for mapping class groups of compact surfaces by Harer [Har86].

The layout of the paper is as follows. In Section 2 we give the definition of Outer Space, its bordification, and some variants, and analyze the local topology of these spaces. In particular, the bordified Outer Space can be thought of as a cell-complex with each cell a convex cell in Euclidean space, much like Siegel’s bordification of GL_{n}(ℝ)/O_{n}(ℝ). As with many other things in this paper, this is not technically how things go, and we only have large charts with convex polyhedral image. In fact, the “cells” are only that modulo the Poincaré Conjecture.

In Section 3 we define our Morse function and show that its restriction to each “cell” is standard (empty or contractible super-level sets). An immediate consequence of this is that each critical point is a vertex (so the critical points are isolated).

In Section 4 we study the change in the topology of super-level sets that occurs after passing through a vertex. In the language of our Morse theory, we study the topology of ascending links, and show that they are homotopy equivalent to the wedge of spheres of dimension 2n − 4.

Section 5 deals with the last remaining problem – that our Morse function didn’t go to ℝ_{+} and wasn’t continuous (much less proper). Theorem 5.3 is the model Morse theory statement that replaces Lemma 1.3 in this setting. Theorem 1.1 is restated (and proved) as Theorem 5.6. The trick consists of replacing Bordified Outer Space by a simplicial model, and replacing the Morse function by another one which is proper and has ascending links homotopy equivalent to the ascending links of the original function.

We wish to thank Noel Brady, Vincent Guirardel, Gilbert Levitt, and the referee for pointing out some mistakes in earlier versions of the manuscript.

We end this introductory section by reviewing some concepts we need in this paper.

1.1. Absolute Neighborhood Retracts. All spaces will be separable and will admit a complete metric. (The targets of our Morse functions will be totally ordered sets, but we will not topologize them.) A space X is said to be an ANR if whenever embedded in a locally convex topological vector space as a closed subset, it admits a neighborhood retraction. The standard reference is [Bor67]. A finite-dimensional space is an ANR iff it is locally contractible. If A, B ⊆ X are closed subsets such that A, B, A ∩ B are ANR’s, then A ∪ B is an ANR. A map between connected ANR’s that induces an isomorphism between homotopy groups is a homotopy equivalence. A contractible ANR is called an absolute retract (AR).

1.2. Semi-algebraic sets. A subset V of ℝ^{n} is semi-algebraic if it is the union of finitely many subsets each of which is cut out from ℝ^{n} by a finite number of polynomial equations f(x) = 0 and inequalities f(x) > 0. Every semi-algebraic set is an ANR (i.e. locally contractible). For semi-analytic sets this was first proved by Lojasiewicz in [Loj64]. The semi-algebraic case is easier, and a simple proof is presented by Hironaka in [Hir75]. Moreover, for any x ∈ V there is an analytic change of coordinates h from a neighborhood U of x in ℝ^{n} to ℝ^{n} such that h(x) = 0 and h(U ∩ V) is a cone. In particular, points of V have cone neighborhoods in V. In fact, semi-algebraic sets are triangulable.

1.3. Z-sets. A closed subset A ⊂ X of an ANR X is a Z-set if for any open set U ⊆ X inclusion U \ A ↪ U is a homotopy equivalence. Equivalently, there is a homotopy H : X × [0, 1] → X with H(x, 0) = x and H(x, t) ∈ X \ A for t > 0. For example, any closed subset of the boundary of a manifold is a Z-set in the manifold. The standard reference is [Hen75]. We find it useful to slightly extend the language of Z-sets. We say that a closed subset A of an ANR X is locally Z at a ∈ A if there is a neighborhood V of a in X and a homotopy H : V × [0, 1] → X such that H(x, 0) = x and H(x, t) ∈ X \ A for t > 0. It can be shown with the methods of [Hen75] that if a closed set is locally Z at each point then it is a Z-set.

A Z-sandwich is a triple X ⊇ Y ⊇ Z such that X is an ANR and X \ Z is a Z-set in X. It follows that inclusions Z ↪ Y ↪ X are homotopy equivalences.

2. The Bordification of Outer Space

Definitions 2.1. A finite (possibly disconnected) graph is said to be a core graph if it does not have vertices of valence 0 or 1, and none of its components has a separating edge. Equivalently, a core graph is one that can be represented as the union of finitely many circles. Note that every finite graph H which is not a forest contains a unique nonempty maximal subgraph which is a core graph. We call this subgraph the core of H, denoted core(H). By C(G) we denote the collection of all nonempty subgraphs of a graph G which are core graphs.

An ℝ-metric on a graph G is a function ℓ that assigns a nonnegative number ℓ(e) to each edge e of G. We refer to ℓ(e) as the length of e (with respect to ℓ). The volume of G with respect to ℓ is the sum of the lengths of all edges of G. More generally, the volume of a subgraph is the sum of the lengths of edges in the subgraph. When G is a finite graph, we denote by Σ_{G} the simplex of all ℝ-metrics on G of volume 1. An ℝ-metric on G is degenerate if a nonempty core subgraph has volume 0; otherwise it is non-degenerate. We denote by ND(G) the subset of Σ_{G} consisting of non-degenerate metrics.

2.1. The Bordification of Σ_{G}. We now construct a space Σ̂_{G}. It will be homeomorphic to a closed cell (at least modulo the Poincaré Conjecture) and it will naturally map onto Σ_{G}. Further, this map will be a homeomorphism over the non-degenerate ℝ-metrics. Thus Σ̂_{G} can be viewed as a more refined compactification of the space of non-degenerate ℝ-metrics on G of volume 1.

We assume that G is a core graph. For every core subgraph C ∈ C(G) define the coordinate restriction map r_{C} : Int Σ_{G} → Int Σ_{C} ⊂ Σ_{C} by restricting the given ℝ-metric to C and then rescaling so the volume is 1. The product map

∏_{C∈C(G)} r_{C} : Int Σ_{G} → ∏_{C∈C(G)} Σ_{C}

is an embedding, as one of the coordinates is inclusion Int Σ_{G} ↪ Σ_{G}. Define Σ̂_{G} as the closure of the image of this product map. By definition, Σ̂_{G} is a compactification of Int Σ_{G}. The natural map

π : Σ̂_{G} → Σ_{G}

is defined by restricting the projection map. Equivalently, the definition can be made with ND(G) replacing Int Σ_{G}. It will sometimes be convenient to take this point of view.

Example 1. When G is the circle with n vertices and edges, Σ̂_{G} = Σ_{G} is the (n − 1)-simplex.

Example 2. When G is the bouquet of two circles, Σ̂_{G} = Σ_{G} is an arc.

Example 3. When G is the theta graph (the bipartite graph with 2 vertices and 3 edges), Σ̂_{G} is a hexagon.

Example 4. When G is the bouquet of three circles, Σ̂_{G} is a hexagon, and π : Σ̂_{G} → Σ_{G} collapses every other face to a point. The hexagon is depicted in the figure. A representative point in each open face is indicated by a decorated bouquet. A point is determined by the ℝ-metrics associated to each of its core subgraphs. In the figure, solid edges have non-zero ℓ_{G}-length. Dashed edges have non-zero length with respect to the metric on the core graph of non-solid edges. Finally, dotted edges have non-zero length with respect to the metric on the core graph of dotted edges. The metrics on the other core graphs can be reconstructed from these. This will be made more precise in the remainder of this section starting with Definition 2.7. For example, if a is the edge of G displayed with vertical symmetry, and b and c are the other edges in counterclockwise order then at the topmost point of the hexagon, ℓ_{G}(a) = 1, ℓ_{G}(b) = 0, and ℓ_{G}(c) = 0. If H_{1} is the core graph b ∪ c then at this same point ℓ_{H_{1}}(b) = 1 and ℓ_{H_{1}}(c) = 0. If H_{2} is the core graph with only edge c then ℓ_{H_{2}}(c) = 1. At the point immediately to the right of this one ℓ_{H_{1}}(b) = ℓ_{H_{1}}(c) = 1/2. At the central point ℓ_{G}(a) = ℓ_{G}(b) = ℓ_{G}(c) = 1/3. This example will be expanded as we progress through the paper.

Figure 1

Remark 2.2. An interesting variation of the above construction results if one drops the requirement that core graphs have no separating edges. The reader may verify that in that case the bordification Σ̂_{G} for the case when G consists of two circles joined by an arc is a quadrilateral and the projection collapses one face to a point. We choose to include the non-separation requirement as the resulting space is then somewhat more efficient and our ‘Morse function’ will have fewer critical points.

A point in the product ∏_{C∈C(G)} Σ_{C} assigns an ℝ-metric ℓ_{C} of volume 1 to every nonempty core subgraph C of G. We will often refer to ℓ_{G} as the topmost metric.

Lemma 2.3. A point {ℓ_{C}}_{C∈C(G)} belongs to Σ̂_{G} iff the following compatibility condition holds. (∗) If C_{1} ⊂ C_{2} are two core subgraphs and ℓ_{C_{2}} does not vanish on all edges of C_{1}, then ℓ_{C_{1}} is the rescaled restriction of ℓ_{C_{2}}.

Proof. Condition (∗) is a closed condition and it clearly holds on the image of Int Σ_{G} (and also ND(G)). Thus (∗) holds for every point in Σ̂_{G}.

Conversely, assume {ℓ_{C}}_{C∈C(G)} satisfies (∗). Define a decreasing sequence of core subgraphs of G as follows. Set G_{0} = G. If the collection of edges of ℓ_{G}- length 0 forms a forest, then stop. Otherwise, let G_{1} be the maximal nonempty core graph on which ℓ_{G} vanishes. If the collection of edges on which ℓ_{G_{1}} vanishes forms a forest, stop; otherwise define G_{2} as the core of the subgraph of G_{1} on which ℓ_{G_{1}} vanishes. Continuing in this fashion, we obtain a finite sequence G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{m} of core graphs, and ℓ_{G_{m}} vanishes on a forest.

Claim 2.4. In the presence of the compatibility condition (∗), metrics ℓ_{G_{i}} uniquely determine all ℓ_{C} for C ∈ C(G).

[Indeed, let i be the largest index such that C ⊆ G_{i}. Then ℓ_{G_{i}} doesn’t vanish on every edge of C (or else C ⊆ G_{i+1}), and hence ℓ_{C} is determined by ℓ_{G_{i}} via (∗).]

For ϵ > 0 define the ℝ-metric ℓ_{ϵ} on G by

ℓ_{ϵ}(e) = ϵ^{j}ℓ_{G_{j}}(e)

where j is the largest index such that e ⊂ G_{j}. After rescaling, ℓ_{ϵ} represents a point in ND(G), and as ϵ → 0, the corresponding point converges to {ℓ_{C}}_{C∈C(G)} ∈ ∏_{C∈C(G)} Σ_{C}. The last assertion is immediate on the coordinates of the form Σ_{G_{i}} and follows from Claim 2.4 for all others.

Corollary 2.5. Σ̂_{G} is a semi-algebraic set, hence an ANR.

Proposition 2.6. Σ̂_{G} is contractible and π : Σ̂_{G} → Σ_{G} has point inverses which are AR’s (in particular, contractible).

Proof. We argue by induction on the number of edges of G. First observe that the second statement implies the first, since proper maps with contractible point inverses between ANR’s are homotopy equivalences [Sma57] (see also [Lac77]). A point inverse of π : Σ̂_{G} → Σ_{G} consists of those points in Σ̂_{G} that have a fixed ℓ_{G}. In view of Claim 2.4, the point inverse can be identified with Σ̂_{H} where H is the core of the subgraph of G on which ℓ_{G} vanishes (if H is empty, the point inverse is a single point). By induction, it follows that the point inverses are AR’s.

Definition 2.7. The hierarchy of scales associated to a point {ℓ_{C}}_{C∈C(G)} ∈ Σ̂_{G} is the sequence of core graphs G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{m} constructed in the proof of Lemma 2.3.

Intuitively, a point of Σ̂_{G} represents the limit of a sequence of non-degenerate ℝ-metrics on G of volume 1, where we keep track of ratios of lengths on different scales.

It will also be necessary to understand a neighborhood of the point inverse of a vertex under the map π above. Let v be a vertex of Σ_{G}. For 0 < ϵ < 1 let Q_{ϵ} (resp. Q_{≤ϵ}) be the set of points in the simplex Σ_{G} whose barycentric coordinate associated to v is 1 − ϵ (resp. ≥ 1 − ϵ), thus Q_{ϵ} is obtained from the face opposite to v by a homotethy with scaling factor ϵ centered at v (and Q_{≤ϵ} is the closed neighborhood of v determined by Q_{ϵ}). Let Q̂_{ϵ} = π^{−1}(Q_{ϵ}) and similarly Q̂_{≤ϵ} = π^{−1}(Q_{≤ϵ}). Fix some ϵ ∈ (0, 1) and define a path α_{x}(s) that joins a point x ∈ Q̂_{ϵ} to a point in π^{−1}(v). For a point y ∈ Σ̂_{G} we denote by ℓ_{C}^{y} the metric on the core graph C ⊂ G associated with y. Let e_{0} be the edge of G of length 1 under the metric given by v. We then define:

ℓ_{C}^{α_{x}(s)} = ℓ_{C}^{x} if e_{0} ⊈ C

and otherwise, when e_{0} ⊆ C,

\begin{matrix}ℓ_{C}^{α_{x}(s)}(e_{0}) = 1 − ϵ + sϵ \\ ℓ_{C}^{α_{x}(s)}(e) = (1 − s)ℓ_{C}^{x}(e), if e ≠ e_{0}. \end{matrix} (1)

Of course, in view of the compatibility condition, we could have stated the definition for ℓ_{G}^{α_{x}(s)} only, instead of the general core graph C with e_{0} ⊆ C. The path α_{x} maps by π to the straight line segment joining π(x) to v.

Q̂_{ϵ} π^{−1}(v)  π  Q_{ϵ} v

Proposition 2.8. The paths α_{x} define a mapping cylinder structure on Q̂_{≤ϵ}. The point inverses of the associated map ρ : Q̂_{ϵ} → π^{−1}(v) are simplices.

Proof. Each α_{x} is continuous in s since by definition each ℓ_{C} changes continuously with s, and for the same reason the paths α_{x} vary continuously in x. If x ≠ x′, the paths α_{x} and α_{x′} can intersect only at the terminal endpoint, since x can be recovered from α_{x}(s) for every s < 1. Each path is contained in Q̂_{≤ϵ} and each point of Q̂_{≤ϵ} is on some such path. Consequently, Q̂_{≤ϵ} has the structure of the mapping cylinder of a map ρ : Q̂_{ϵ} → π^{−1}(v). To describe the point inverses of ρ, note that ℓ_{C}^{x} = ℓ_{C}^{ρ(x)} when e_{0} ⊈ C, so the only indeterminacy is ℓ_{G}. In particular, π is injective on the point inverses of ρ. The only restrictions on ℓ_{G} come from the volume normalization and the compatibility conditions. Since ℓ_{G}^{x}(e_{0}) = 1 − ϵ for every x ∈ Q̂_{ϵ}, ℓ_{G}^{x} is uniquely determined by the ℓ_{G}-volume of H = core(G \ e_{0}) and the ℓ_{G}-lengths of the edges in G \ (e_{0} ∪ H), subject to the condition that the sum of these numbers is ϵ. Thus the point inverses are simplices.

2.2. Local structure. A core subgraph C_{0} fails to occur in the hierarchy of scales for {ℓ_{C}}_{C∈C(G)} if and only if there is a core subgraph C_{1} that properly contains C_{0} such that ℓ_{C_{1}} doesn’t vanish on C_{0}. In particular, ‘not occurring in the hierarchy of scales’ is an open condition.

If S is a collection of edges in G, we denote by Δ_{S} the simplex {(t_{s})_{s∈S}|t_{s} ≥ 0, ∑ t_{s} ≤ 1}, and by Σ_{S} the simplex {(t_{s})_{s∈S}|t_{s} ≥ 0, ∑ t_{s} = 1}.

Definition 2.9. Let \underline{F} = (G = C_{0} ⊃ C_{1} ⊃ ⋯ ⊃ C_{q}) be a sequence of nonempty core graphs. It is sometimes convenient to put C_{q+1} = ∅. The chart U\underline{F} corresponding to \underline{F} is the open set of points of Σ̂_{G} with the property that each graph in the hierarchy of scales is one of the C_{i}’s. The chart map

Φ = Φ\underline{F} : U = U\underline{F} → Δ = Δ\underline{F} = Δ_{C_{0}\C_{1}} × Δ_{C_{1}\C_{2}} × ⋯ × Δ_{C_{q−1}\C_{q}} × Σ_{C_{q}}

is defined by

{ℓ_{C}}_{C∈C(G)} ↦ (ℓ_{C_{0}}|_{C_{0}\C_{1}}, ℓ_{C_{1}}|_{C_{1}\C_{2}}, ⋯, ℓ_{C_{q−1}}|_{C_{q−1}\C_{q}}, ℓ_{C_{q}}).

We view Δ as a product polytope. In particular, by a face of Δ we mean the product of faces of the coordinate simplices.

The same formula defines an extension Φ̂ : Σ̂_{G} → Δ of Φ.

Proposition 2.10. Φ̂ has AR’s as its point inverses. Moreover, a single point {ℓ_{C}} ∈ Σ̂_{G} is a point inverse if each graph in the hierarchy of scales for {ℓ_{C}} is one of C_{i}’s.

Proof. We induct on the number of edges of G. Fix a point (x_{0}, x_{1}, ⋯, x_{q}) ∈ Δ_{C_{0}\C_{1}} ×Δ_{C_{1}\C_{2}} ×⋯×Σ_{C_{q}}. We first note that all points in Φ̂^{−1}(x_{0}, x_{1}, ⋯, x_{q}) have the same ℓ_{G}. Indeed, x_{0} determines ℓ_{G} on edges in C_{0} \C_{1}. If x_{0} ∈ Σ_{C_{0}\C_{1}}, then ℓ_{G} vanishes on C_{1}, and otherwise x_{1} determines ℓ_{G} on edges in C_{1} \ C_{2}, by rescaling. If x_{1} ∈ Σ_{C_{1}\C_{2}} then ℓ_{G} vanishes on C_{2}, and otherwise x_{2} determines ℓ_{G} on C_{2} \ C_{3}, etc.

Denote by H the core of the subgraph of G on which ℓ_{G} vanishes (if ℓ_{G} vanishes on a forest, there is nothing left to prove). If H = C_{i} for some i, then x_{0}, x_{1}, ⋯, x_{i−1} are determined by ℓ_{G} by rescaling and carry no additional information, so we can identify Φ̂^{−1}(x_{0}, x_{1}, ⋯, x_{q}) with Φ̂_{\underline{F}′}^{−1}(x_{i}, x_{i+1}, ⋯, x_{q}),

where Φ̂\underline{F}′ : Σ̂_{C_{i}} → Δ_{C_{i}\C_{i+1}} × Δ_{C_{i+1}\C_{i+2}} × ⋯ × Σ_{C_{q}} is the similarly defined map and \underline{F}′ = (H = C_{i} ⊃ C_{i+1} ⊃ ⋯ ⊃ C_{q}). The proposition now follows from the inductive assumption applied to Φ̂\underline{F}′.

Now suppose H ≠ C_{i} for any i. In this situation we need to argue that the point inverse is an AR. Note that if C_{i} ⊈ H, then ℓ_{G} determines x_{i} by rescaling. Let i be the smallest index with H ⊃ C_{i} (if there is no such i then the point inverse can be identified Σ̂_{H} and we are done). Then the point inverse under consideration can be identified with the inverse of (x_{i}, x_{i+1}, ⋯, x_{q}) under the composition

Σ̂_{H} →^{Φ̂_{\underline{F} ′}}Δ_{H\C_{i}} × Δ_{C_{i}\C_{i+1}} × ⋯ × Σ_{C_{q}} → Δ_{C_{i}\C_{i+1}} × ⋯ × Σ_{C_{q}}

with the second map the projection and \underline{F}′ = (H ⊃ C_{i} ⊃ C_{i+1} ⊃ ⋯ ⊃ C_{q}). Thus this point inverse maps onto the simplex Δ_{H\C_{i}} with AR point inverses, and it is an ANR (it is a semi-algebraic set), so by [Sma57] it is an AR.

Remark 2.11. There are other degenerate point inverses, e.g. when H is a single edge (loop) that does not equal to any C_{i}.

Proposition 2.12. 1. Φ : U → Δ is 1-1.

2. Φ is a homeomorphism onto its image.

3. The image of Φ is the complement of the union of a collection of faces of Δ.

4. Σ̂_{G} is a compact contractible manifold whose interior is Int Σ_{G}.

Proof. It follows from Proposition 2.10 that Φ is injective and that for open V ⊂ U, Σ̂_{G} = Φ̂(V) ∐ Φ̂(Σ̂_{G} \ V). In particular, being the complement of a compact set, Φ(V) = Φ̂(V) is open and Φ is an open map. So, Φ is a homeomorphism onto its image. We need to argue that Φ(U) is the complement of the union of some faces of Δ. Let 0 = n_{0} < n_{1} < n_{2} < ⋯ < n_{k} ≤ q and consider the subset X of Σ̂_{G} consisting of those points whose hierarchy of scales is G ⊃ C_{n_{1}} ⊃ C_{n_{2}} ⊃ ⋯ ⊃ C_{n_{k}}. We now argue that Φ(X) is the union of some faces. For each point in X denote by D_{i} the subgraph of G consisting of edges on which ℓ_{C_{n_{i}}} vanishes, so that core(D_{i}) = C_{n_{i+1}}. Thus X can be viewed as the union over all different possibilities for the D_{i}’s. The Φ-image of the subset of X with the fixed choice of the D_{i}’s is the union of faces of Δ given by the equations that the sum of the coordinates of x_{n_{i}} ∈ Δ_{C_{n_{i}−1}\C_{n_{i}}} is 1 for all i, and that the coordinates of x_{n_{i}}, x_{n_{i}+1}, ⋯, x_{n_{i+1}−1} corresponding to the edges of D_{i} \ C_{n_{i+1}} vanish. To prove (4) observe that Int Σ_{G} ⊂ U, that Φ(Int Σ_{G}) = Int Δ and that the charts cover Σ̂_{G}.

Remark 2.13. One can show that Σ̂_{G} has the structure of a real analytic manifold with corners (see [BS73]). Indeed, note that each {ℓ_{C}}_{C∈C(G)} is contained in a unique minimal chart, given by the hierarchy of scales of {ℓ_{C}}. To study the transition maps, it suffices to consider the case of inclusion U\underline{F}′ ⊂ U\underline{F}, where \underline{F}′ is obtained from \underline{F} by omitting one of the core graphs, say C_{i}. For concreteness assume i ≠ m. Define the following map I on the complement of a face of Δ\underline{F}′ into Δ\underline{F}. The map is the projection to the coordinates other than Δ_{C_{i−1}\C_{i}} × Δ_{C_{i}\C_{i+1}}. If the Δ_{C_{i−1}\C_{i+1}}-coordinate of x ∈ Δ\underline{F}′ is (t_{1}, t_{2}, ⋯, t_{k}, s_{1}, s_{2}, ⋯, s_{l}) (with k edges in C_{i−1} \ C_{i} and l edges in C_{i} \ C_{i+1}), then the Δ_{C_{i−1}\C_{i}} × Δ_{C_{i}\C_{i+1}}-coordinates of I(x) are

((t_{1}, ⋯, t_{k}), (\frac{s_{1}}{1 − t_{1} − t_{2} − ⋯ − t_{k}} , ⋯, \frac{s_{l}}{1 − t_{1} − t_{2} − ⋯ − t_{k}})).

The inverse map J is defined on all of Δ\underline{F}, and the formula in interesting coordinates is given by

((t_{1}, ⋯, t_{k}), (s_{1}, ⋯, s_{l})) ↦ (t_{1}, ⋯, t_{k}, (1−t_{1}−⋯−t_{k})s_{1}, ⋯, (1−t_{1}−⋯−t_{k})s_{l})

Remark 2.14. It follows from the Poincaré Conjecture that Σ̂_{G} is homeomorphic to the closed ball when its dimension is not 4.

Question. Is Σ̂_{G} a PL ball in all dimensions (with the PL structure induced from the analytic manifold with corners structure described above)?

As we are interested in homotopy-theoretic issues, we will not pursue this.

2.3. Cell decomposition. Fix a descending sequence

\underline{K} = (G = K_{0} ⊃ K_{1} ⊃ K_{2} ⊃ ⋯ ⊃ K_{m+1})

of subgraphs of G such that K_{i+1} is a proper subgraph of the core of K_{i} (K_{m+1} is allowed to be empty). The associated open cell is the subset W (\underline{K}) of Σ̂_{G} consisting of those points whose hierarchy of scales (G_{i}) consists of the cores G_{i} = core(K_{i}) (i ≤ m), and further, the metric ℓ_{G_{i}} vanishes exactly on K_{i+1}. Each W (\underline{K}) is visibly an open cell, canonically identified with the interior of Σ_{K_{0}\K_{1}} × Σ_{core(K_{1})\K_{2}} × ⋯ × Σ_{core(K_{m})\K_{m+1}} and these cells form a partition of Σ̂_{G}.

Example 4 continued. When G is a bouquet of three circles, Σ̂_{G} is a hexagon, and the partition consists of 13 cells: the vertices, the interiors of the sides, and the interior of the hexagon. Referring back to Figure 1, the sequence \underline{K} is obtained from a displayed bouquet as follows. K_{0} = G, If K_{1} is nonempty then it is the union of the dashed and dotted edges, and if K_{2} is non-empty then it is the union of the dotted edges. A vertex corresponds to \underline{K} = (G, 2 edges, 1 edge, ∅). There are two types of 1-cells, namely \underline{K} = (G, 2 edges, ∅) and \underline{K} = (G, 1 edge, ∅). The 2-cell corresponds to \underline{K} = (G, ∅).

Note that the chart U\underline{F} with \underline{F} = (G = C_{0} ⊃ C_{1} ⊃ ⋯ ⊃ C_{q}) contains the open cell W (\underline{K}) with \underline{K} = (G = K_{0} ⊃ K_{1} ⊃ K_{2} ⊃ ⋯ ⊃ K_{m+1}) if and only if it intersects it if and only if each core(K_{i}) equals one of the C_{j}’s, say C_{j(i)} = core(K_{i}). In that case, the image

Φ\underline{F} (W (\underline{K})) ⊂ Δ\underline{F} = Δ_{C_{0}\C_{1}} × Δ_{C_{1}\C_{2}} × ⋯ × Δ_{C_{q−1}\C_{q}} × Σ_{C_{q}}

is the open cell Int Δ_{C_{0}\(C_{1}∪K_{1})} × Int Δ_{C_{1}\(C_{2}∪K_{1})} × ⋯ × Int Σ_{C_{j(1)−1}\(C_{j(1)}∪K_{1})}

× Int Δ_{C_{j(1)}\(C_{j(1)+1}∪K_{2})} × Int Δ_{C_{j(1)+1}\(C_{j(1)+2}∪K_{2})}
× ⋯ × Int Σ_{C_{j(2)−1}\(C_{j(2)}∪K_{2})}
× ⋯ × Int Δ_{C_{j(m)}\(C_{j(m)+1}∪K_{m+1})} × Int Δ_{C_{j(m)+1}\(C_{j(m)+2}∪K_{m+1})}
× ⋯ × Int Σ_{C_{q}\(C_{q−1}∪K_{m+1})}.

(2)

In particular, the topological frontier of each open cell is the union of open cells of smaller dimension and the closure \overline{W} (\underline{K}) of W (\underline{K}) is a compact topological manifold whose interior is homeomorphic to an open ball. Thus Σ̂_{G} is equipped with a homotopy-cell decomposition, and the restriction of this decomposition to a chart coincides with the decomposition into the faces. We will refer to the \overline{W} (\underline{K})’s as the ‘(closed) cells’ in this decomposition.

For a fixed \underline{K} = (G = K_{0} ⊃ K_{1} ⊃ K_{2} ⊃ ⋯ ⊃ K_{m+1}) it is sometimes convenient to consider the collection of subgraphs K ⊂ G such that core(K_{i}) ⊆ K ⊆ K_{i} for some i. Call these \underline{K}-null subgraphs. They are characterized by the property that for every core graph C ⊈ K the metric ℓ_{C}^{x} vanishes on K for each point x ∈ W (\underline{K}). An open cell W (\underline{K}′) is contained in \overline{W} (\underline{K}) if and only if every \underline{K}-null subgraph is a \underline{K}′-null subgraph. For graphs K ⊃ L, let K/L denote the graph obtained by collapsing each component of L to a point.

Lemma 2.15. \overline{W} (\underline{K}) is canonically identified with ∏_{i=0}^{m} Σ̂_{Core(K_{i})/K_{i+1}}.

Proof. Let \{ℓ\overline{C}\}_{\overline{C}∈C(Core(K_{i})/K_{i+1})} ∈ Σ̂_{Core(K_{i})/K_{i+1}}. If C is core graph of G such that Core(K_{i+1}) ⊄ C ⊂ Core(K_{i}) then set ℓ_{C}(e) = 0 if e ⊂ K_{i+1}. Otherwise, set ℓ_{C}(e) = ℓ\overline{C}(\overline{e}) where \overline{C} = C/K_{i+1} and \overline{e} = e/K_{i+1}. It is easy to verify that the induced map ∏_{i=0}^{m} Σ̂_{Core(K_{i})/K_{i+1}} → ∏_{C∈C(G)} Σ_{C} gives a homeomorphism onto \overline{W} (\underline{K}).

Question. Are the \overline{W} (\underline{K})’s closed PL cells?

A slight generalization of Proposition 2.6 will be needed in the sequel:

Proposition 2.16. The restriction π′ : \overline{W} (\underline{K}) → π(\overline{W} (\underline{K})) of π : Σ̂_{G} → Σ_{G} has point inverses which are AR’s.

Proof. Fix a point ℓ_{G} of Σ_{G} and consider the point inverse π^{−1}(ℓ_{G}). If H is the core of the subgraph of G on which ℓ_{G} vanishes, then π^{−1}(ℓ_{G}) can be canonically identified with Σ̂_{H}. The point inverse intersects the cell W (\underline{K}) if and only if K_{1} is the subgraph on which ℓ_{G} vanishes, and in that case the intersection π^{−1}(ℓ_{G}) ∩ W (\underline{K}) is canonically identified with the cell W (H ⊃ K_{2} ⊃ K_{3} ⊃ ⋯ ⊃ K_{m+1}). Therefore, intersections with the cells of Σ̂_{G} induces a cell decomposition of the point inverse π^{−1}(ℓ_{G}) that coincides with the cell decomposition of Σ̂_{H}. In particular, the nonempty intersections of the point inverse with the closed cells in Σ̂_{G} can be identified with the closed cells in Σ̂_{H}, and they are AR’s.

Remark 2.17. The mapping cylinder lines α_{x} of Proposition 2.8 have the property that with the exception of the terminal endpoint they are contained in a single open cell. Therefore, the mapping cylinder neighborhood restricts to mapping cylinder neighborhoods in each closed cell. More precisely, if π′ is the map of Proposition 2.16, then π′^{−1}(v) has a mapping cylinder neighborhood in \overline{W} (\underline{K}) whose frontier is Q̂_{ϵ} ∩ \overline{W} (\underline{K}). The associated map ρ′ : Q̂_{ϵ} ∩ \overline{W} (\underline{K}) → π′^{−1}(v) is the restriction of ρ : Q̂_{ϵ} → π^{−1}(v). Its point inverses are simplices as well: a point in ρ′^{−1}(y) is uniquely determined by specifying ℓ_{G}-lengths of edges outside K_{1} ∪ core(G \ e_{0}) ∪ e_{0} and the ℓ_{G}-volume of core(G \ e_{0}), subject to the condition that these numbers add to ϵ.

2.4. Functoriality. Suppose G′ is obtained from G by collapsing a forest, and let π : G → G′ be the collapsing map. Any ℝ-metric on G′ induces an ℝ-metric on G by the requirement that π be an isometry: explicitly, the collapsed edges have length 0, and the other edges have the same length as in G′. If C is a core graph in G, then π(C) is a core graph in G′, and an ℝ-metric on π(C) determines one on C. It is easy to see that if the metrics {ℓ_{C′}}_{C′∈C(G′)} satisfy the compatibility condition, then so do the pulled back metrics on the core graphs in G. We thus have a natural map π^{∗} : Σ̂_{G′} → Σ̂_{G}.

It is immediate that π^{∗} is injective. Further, the intersection of a chart U\underline{F} with π^{∗}(Σ̂_{G′}) is sent by Φ\underline{F} to the intersection of Φ(U\underline{F}) with the face of Δ\underline{F} given by the condition that the coordinates corresponding to the edges in the collapsed forest are 0. We also see that π^{∗} respects the cell decompositions. We summarize this discussion. If K′ is a subgraph of G′, define the corresponding subgraph K of G to consist of the edges of K′ pulled back to G together with the edges in the collapsed forest.

Lemma 2.18. π^{∗} is injective, and it sends each pair (\overline{W} (\underline{K}′), W (\underline{K}′)) homeomorphically to the corresponding pair (\overline{W} (\underline{K}), W (\underline{K})), where \underline{K} is obtained from \underline{K}′ by applying the construction described in the sentence preceding this lemma to each subgraph.

2.5. The group Out(N, K) and its Outer Space. Our arguments will be inductive and we are led to consider more general versions of Culler-Vogtmann’s Outer Space [CV86]. Our graphs will have distinguished points and may be disconnected.

Fix an integer p > 0 (the number of components), a vector of positive integers N = (n_{1}, n_{2}, ⋯, n_{p}) (ranks of components), and another vector of non-negative integers K = (k_{1}, k_{2}, ⋯, k_{p}) (numbers of distinguished points in each component). Let G be a core graph with p components G(1), G(2), ⋯, G(p) such that the rank of G(i) is n_{i}. A labeling of G(i) is a map {1, 2, ⋯, k_{i}} → G(i) whose image is in the vertex set of G(i). Elements of the domain of a labeling are labels; elements of the image are labeled points. A labeling of G is a labeling of each G(i). We say that a core graph G equipped with a labeling has type (N, K) if each valence 2 vertex is in the image of the labeling map. To avoid silly issues, we always assume that each component of rank 1 contains at least two labels, i.e. if n_{i} = 1 then k_{i} ≥ 2. By analogy with cutting a surface along essential simple closed curves, this assumption corresponds to the usual requirement that all components have negative Euler characteristic. A map f : G → G′ between two graphs of the same type is label-preserving if it sends G(i) to G′(i) for each i and the labeling of G′(i) is obtained from the labeling of G(i) by composing with f|G(i).

Fix a model graph G = G(N, K) of type (N, K) in which the labeling of each component is injective (so the labeled points are all distinct). A marking of a graph G of type (N, K) is a homotopy equivalence G → G which is label-preserving, modulo homotopy through label-preserving maps. The group Out(N, K) is defined to be the group of markings G → G. In the case that the number of components p = 1, we suppress the parentheses in the vectors N and K. For example:

Out(n, 0) ≅ Out(F_{n})
Out(n, 1) ≅ Aut(F_{n})
Out(1, k) ≅ ℤ^{k−1} ⋊ ℤ/2

where in the last case the action of ℤ/2 on ℤ^{k−1} is given by x ↦ −x.

This family of groups was first considered by Hatcher [Hat95] in his work on the homological stability of Out(F_{n}), where it arose naturally in the course of an inductive argument. This will be the case here.

There is an important alternative description of marked labeled core graphs. For simplicity, we restrict to the case of one component, say of rank n and with k labels; in general the discussion is applied componentwise. First, let T be the universal cover of the model graph G, together with a choice of lifts of the labeled points – so T is a labeled minimal free simplicial F_{n}-tree. Let G be any labeled core graph with marking f : G → G, and let f̃ : T → T be a lift of f to the universal covers. This lift is unique up to post-composition by deck transformations of T. Define the labeling of T so that f̃ is label-preserving. Conversely, if T is a labeled free simplicial F_{n}-tree, there is a label-preserving F_{n}-equivariant map T → T, unique up to homotopy through such maps, and it projects to a marking G → G = T/F_{n}. Of course, the quotient graph G may not be a core graph, and we see that there is a 1-1 correspondence between marked labeled core graphs and labeled minimal free simplicial F_{n}-trees with quotient graph a core graph.

A point {ℓ_{C}} of Σ̂_{G} with hierarchy of scales G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{m} can be viewed as giving the structure of a Λ-tree (see [AB87]) to the associated tree T, where Λ is the ordered abelian group ⊕_{i=0}^{m}ℝ (with the lexicographic order), and the Λ-length of a lift of an edge e ∈ G_{i} \ G_{i+1} is the element of Λ with i^{th} coordinate ℓ_{G_{i}}(e) and all other coordinates 0. All edges of T with Λ-length 0 are collapsed.

An element of Out(n, k) with k > 0 can, from this point of view (after choosing the first labeled point as the basepoint in G), be regarded as an isomorphism α : F_{n} → F_{n} together with an α-equivariant map f̃: T → T that preserves the first label, and sends every other label to a translate of itself by a group element, modulo homotopies of f̃ through such maps. Thus, an element of Out(n, k) for k > 0 is a k-tuple (α, g_{2}, ⋯, g_{k}) with α ∈ Aut(F_{n}) and g_{2}, ⋯, g_{k} ∈ F_{n}, and we see that Out(n, k) in this case is the semi-direct product

Out(n, k) = F_{n}^{k−1} ⋊ Aut(F_{n})

with the diagonal action of Aut(F_{n}) on F_{n}^{k−1}.

Define the category C(N, K) as follows. Objects are marked graphs f : G → G of type (N, K) modulo the equivalence relation: f : G → G is equivalent to f′ : G → G′ if there is a label-preserving homeomorphism h : G → G′ such that hf is homotopic to f′ through label-preserving maps. Morphisms are induced by collapses of forests π : G → G′ (where the labeling and marking of G′ are induced by π from the labeling and the marking of G).

The group Out(N, K) acts on the right on the category C(N, K) via precomposition. Object stabilizers are finite.

The Spine of Culler-Vogtmann’s Outer Space SCV (N, K) for Out(N, K) is defined to be the geometric realization of the category C(N, K). That is to say, SCV (N, K) is the (underlying space of the) simplicial complex with a vertex for every object of C(N, K) and a simplex for every chain of morphisms. A point in SCV (N, K) can be thought of as a marked graph of type (N, K) equipped with a non-degenerate ℝ-metric in which all edges have length ≤ 1, and the edges of length < 1 form a forest. The natural (right, simplicial) action of Out(N, K) on SCV (N, K) is cocompact and has finite stabilizers.

Definition 2.19. The Bordification of Culler-Vogtmann’s Outer Space is the colimit BCV (N, K) of the contravariant functor that to an object G → G in C(N, K) assigns Σ̂_{G} and to a morphism the corresponding inclusion.

In plain English, start with the disjoint union of Σ̂_{G}’s over marked graphs G of type (N, K), and then identify Σ̂_{G′} with the appropriate subcomplex of Σ̂_{G} whenever G′ is obtained from G by collapsing a forest.

In this language, Culler-Vogtmann’s Outer Space CV (N, K) for Out(N, K) is the colimit of the contravariant functor that to an object G → G in C(N, K) assigns the open subset of Σ_{G} that consists of non-degenerate ℝ-metrics.

It is sometimes advantageous to think about BCV (n, k) as a space of Λtrees. For example, BCV (1, k) for k > 0 can be thought of as the space of labelings of ℝ by k labels, with two labelings considered equivalent if there is a translation that takes one labeling to the other. Thus BCV (1, k) is identified with ℝ^{k}/(a, a, ⋯, a) ≅ ℝ^{k−1}.

The natural action of Out(N, K) on CV (N, K) and BCV (N, K) is properly discontinuous. It is cocompact on the latter (but not on the former).

Observe that the 3 spaces CV (N, K), SCV (N, K), and BCV (N, K) are homotopy equivalent to each other – the latter two are (equivariantly) proper homotopy equivalent. This is a consequence of the following fact.

Proposition 2.20. [BS73, Theorem 8.2.1] Let X be a metrizable space and F = {F_{i}}_{i∈Λ} a locally finite closed covering of X which is closed under the operation of taking finite non-empty intersections. Further assume that

• the order of the cover is finite,

• each F_{i} is a non-empty absolute retract.

Then X is homotopy equivalent to the (geometric realization of the) poset of the cover. Further, if each F_{i} is compact, then X and the poset are proper homotopy equivalent.

The poset of the cover is the simplicial complex whose vertices are the elements of the cover and whose simplices are chains of the F_{i}’s.

Theorem 2.21. CV (N, K), SCV (N, K), and BCV (N, K) are contractible.

Proof. It suffices to prove this for the case of a single component, as it is clear that SCV (N, K) is the product ∏_{i} SCV (n_{i}, k_{i}). When there are no labels, this is the theorem of Culler-Vogtmann [CV86]. In the presence of labels, one can either imitate Culler-Vogtmann’s argument, or one of the other proofs of contractibility of Outer Space, or else consider the “forget the labels” map CV (n, k) → CV (n, 0) and apply Proposition 2.20 to the cover of CV (n, 0) by intersections with Σ_{G}’s and its pull-back.

Remark 2.22. The set BCV (N, K) \ CV (N, K) is a Z-set in BCV (N, K). We will not use this fact, but the proof is straightforward. In the cases of arithmetic groups [BS73] and mapping class groups of compact surfaces [Har86] the bordification is a manifold and the set of ideal points forms the boundary. The notion of a Z-set is a homotopy-theoretic replacement for a “boundary” in a non-manifold setting. See the introduction for definitions.

Question. Is there is an equivariant homeomorphism between bordified Outer Space BCV (N, K) and the regular neighborhood of the spine SCV (N, K) in Outer Space CV (N, K)?

3. Morse theory

We now aim to understand the topology at infinity of BCV (N, K) via Morse theory, as explained in the introduction.

Definition 3.1. A homotopy class in a graph G of type (N, K) is a map f : J → G from a compact 1-manifold J that sends ∂J to the labeled points and is essential on circle components, up to homotopy rel ∂J.

Via the marking, any homotopy class in the model graph G = G(N, K) induces a homotopy class in a marked graph of type (N, K). We say that f : J → G is an immersion if it is either locally injective or constant on each component, and further, all components on which f is constant are arcs.

Any homotopy class can be represented by an immersion (uniquely up to reparametrization).

The simplest function one could define on BCV (N, K) would be the length (with respect to ℓ_{G}) of a particular homotopy class. Such a function would not be “Morse” in that it would be constant on large subcomplexes. We are thus led to a refinement of this idea as follows.

3.1. Hierarchy of lengths of a homotopy class. Let α : J → G be an immersion representing a homotopy class, where G is a marked graph of type (N, K). Suppose {ℓ_{C}}_{C∈C(G)} is a point in Σ̂_{G} ⊂ BCV (N, K), and say its hierarchy of scales is G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{m}. Note that m is smaller than the total rank |N| = n_{1} + n_{2} + ⋯ + n_{p}. For i = 0, 1, ⋯, m define l_{i}^{α}({ℓ_{C}}) to be the total ℓ_{G_{i}}-length of edges of G_{i} crossed by α, counted with multiplicity. For m < i < |N| define l_{i}^{α}({ℓ_{C}}) = 0. Note that l_{0}^{α} : BCV (N, K) → ℝ_{+} = [0, ∞) is continuous (and factors through π : Σ̂_{G} → Σ_{G}), while l_{i}^{α} are generally discontinuous for i > 0.

Fix a (possibly finite) sequence A = (α_{1}, α_{2}, ⋯) of homotopy classes and define

L = L_{A} : BCV (N, K) → (ℝ_{+} × ℝ_{+} × ⋯)^{|N|}

by

L = L_{0} × L_{1} × ⋯ × L_{|N|−1}

where

L_{j} = l_{j}^{α_{1}} × l_{j}^{α_{2}} × ⋯

We will use the more suggestive notation ℝ_{+}^{A} for ℝ_{+} × ℝ_{+} × · · ·.

Definition 3.2. A sequence A of homotopy classes in the model graph G = G(N, K) is ample provided that L_{A} is injective.

Note that when A is ample, L_{0} already determines ℓ_{G}.

Lemma 3.3. If every homotopy class occurs in A at least once, then A is ample.

Proof. From the point of view of Λ-trees, these are [AB87, Theorem 7.13] and [AB87, 3.14] (for the case when there is only one component and at most one labeled point). The general case easily follows from these.

Remark 3.4. That finite ample sequences do not exist follows from [SV92]. Say that a sequence of homotopy classes is weakly ample if for every marked graph G and any two edges in G there is a homotopy class in the sequence that crosses the two edges with distinct multiplicities. The existence of a finite weakly ample sequence would simplify the proof somewhat.

Question. Do finite weakly ample sequences exist?

With respect to the natural (product) topology on (ℝ_{+}^{A})^{|N|}, L is discontinuous. However, we view L as a “Morse function” and study the super-level sets in the following sense. We order ℝ_{+}^{A} and then (ℝ_{+}^{A})^{|N|} lexicographically. For t ∈ (ℝ_{+}^{A})^{|N|} consider the “intervals” (t, ·) = {s ∈ (ℝ_{+}^{A})^{|N|}|s > t} and [t, ·) = {s ∈ (ℝ_{+}^{A})^{|N|}|s ≥ t}. We are interested in understanding the change in the homotopy type of L^{−1}(t, ·) and L^{−1}[t, ·) as t changes.

Define the weight of an edge e ⊂ G as the A-tuple of non-negative integers

w(e) = (μ_{e}(α_{1}), μ_{e}(α_{2}), ⋯)

where μ_{e}(α) is the number of times the immersion representing α crosses e. We order the weights lexicographically.

Example 4 continued. Let G be a bouquet of three circles, and consider the subset Σ̂_{G} of BCV (3, 0) corresponding to the identity marking (we are taking G for the model graph). Let A consist of the single homotopy class a^{2}b^{4}c^{6}. Of course, this A is not ample. In Figure 3, the values of L at the representative points are given.

Figure 3 3.2. The behavior on cells. The intersections with \overline{W} (\underline{K}) with super-level sets. The following lemma will be used to show these sets are semi-algebraic. Lemma 3.5. Let f : ℝ^{n} → (ℝ^{ℕ})^{M} with each coordinate (f_{i})_{j} : ℝ^{n} → ℝ polynomial. Order ℝ^{ℕ} and then (ℝ^{ℕ})^{M} lexicographically, and for t ∈ (ℝ^{ℕ})^{M} let (t, ·), respectively [t, ·), denote the subset of (ℝ^{ℕ})^{M} consisting of elements > t, respectively ≥ t. Then f^{−1}(t), f^{−1}(t, ·), and f^{−1}[t, ·) are semi-algebraic. Proof. We do the case M = 1, the general case follows easily. The set f^{−1}(t) is algebraic by Hilbert’s Basis Theorem, see for example [AM69]. In particular, there is a C such that for all x ∈ ℝ^{ℕ} if f_{j}(x) = t_{j} for j ≤ C then f_{i}(x) = t_{i} for all i ∈ ℕ.

Let X_{i} denote ∩_{j=1}^{i}f_{j}^{−1}(t_{j}). We have

f^{−1}(t, ·) = ⋃_{k=1}^{∞} (f_{k}^{−1}(t_{k}, ∞) ∩ X_{k−1}).

By the above comment, f_{k}^{−1}(t_{k}, ∞)∩X_{k−1} is empty for large k. Hence, f^{−1}(t, ·) is a finite union of semi-algebraic sets and so is itself semi-algebraic. The case of f^{−1}[t, ·) is similar. Proposition 3.6. Let Φ = Φ\underline{F} : U = U\underline{F} → Δ = Δ\underline{F} be a chart map as in Definition 2.9. For every t ∈ (ℝ_{+}^{A})^{|N|} the sets

Φ(L^{−1}(t, ·) ∩ U) and Φ(L^{−1}[t, ·) ∩ U)

are semi-algebraic (viewed as subsets of Δ, which in turn is viewed as a polyhedral subset of Euclidean space). In particular, the super-level sets intersect the cells \overline{W} (\underline{K}) in sets which are locally semi-algebraic (and hence they are ANR’s). Proof. Say two points of U are equivalent if they have exactly the same hierarchy of scales. Let E be the equivalence class corresponding to \underline{K} = (G = C′_{0} ⊃ C′_{1} ⊃ ⋯ ⊃ C′_{q′}) (which is a subsequence of \underline{F}). Since the union of a finite collection of semi-algebraic sets is semi-algebraic, it suffices to show that Φ(L^{−1}(t, ·) ∩ E) and Φ(L^{−1}[t, ·) ∩ E) are semi-algebraic.

Note that E is the set of points in U\underline{K} with Φ\underline{K}-image in Σ\underline{K} = Σ_{C′_{0}\C′_{1}} × ⋯ × Σ_{C′_{q′−1}\C′_{q′}} × Σ_{C′_{q′}} and that each component of the restriction of LΦ−1 ; \underline{K} to Φ\underline{K}(E) extends to a linear function on Δ\underline{K}. From Proposition 2.12 and Lemma 3.5 it follows that Φ\underline{K}(L^{−1}(t, ·) ∩ E) and Φ\underline{K}(L^{−1}[t, ·) ∩ E) are semi-algebraic subsets of Δ\underline{K}. The proposition now follows from Proposition 2.12 because the transition map Δ\underline{F} → Δ\underline{K} is polynomial (see Remark 2.13).

Theorem 3.7. Fix a (homotopy-)cell \overline{W} (\underline{K}) ⊆ Σ̂_{G} and assume that A is ample. The intersections \overline{W} (\underline{K}) ∩ L^{−1}(t, ·) and \overline{W} (\underline{K}) ∩ L^{−1}[t, ·) are either contractible or empty.

Example 4 continued. Before proving the theorem, we continue with our example with A = {a^{2}b^{4}c^{6}}. Recall that this A is not ample, but it illustrates the point. The super-level sets in this case are: empty, single point (vertex where the function attains the maximum on Σ̂_{G}), arc (possibly with an endpoint missing), 2-cell (possibly with an arc in the boundary missing) and the wedge of a 2-cell (possibly with a boundary arc missing) and an arc along boundary points. The set L^{−1}(t, ·) is pictured below with t = (4, 0, 0).

Figure 4

Proof of Theorem 3.7. Denote either one of the two sets by W_{t}.

We argue by induction on the total rank |N|. If |N| = 1, then G is a circle with k labeled points, Σ̂_{G} = Σ_{G} is the (k − 1)-simplex, and each component function of L is linear, so the result in this case is immediate.

Now assume |N| > 1. Let π : Σ̂_{G} → Σ_{G} be the natural projection to the simplex Σ_{G}. The image of \overline{W} (\underline{K}) under π is a face of Σ_{G}. Denote by π′ : \overline{W} (\underline{K}) → π(\overline{W} (\underline{K})) the restriction of π.

Write t = (t_{0}, t_{1}, ⋯, t_{|N|−1}) and observe that L_{0} factors through π(\overline{W} (\underline{K})):

\overline{W} (\underline{K})  ⟶^{L_{0}}  ℝ_{+}^{A}
π′ ↘  ↗ ℓ
π(\overline{W} (\underline{K}))

and we have

π′^{−1}(ℓ^{−1}(t_{0}, ·)) ⊆ W_{t} ⊆ π′^{−1}(ℓ^{−1}[t_{0}, ·))

The difference between the two sets that sandwich W_{t} is π′^{−1}ℓ^{−1}(t_{0}). Note that ℓ^{−1}(t_{0}) is at most a point (by the ampleness assumption).

The proof of the theorem will be broken up into a series of claims.

Claim 1. Both sets sandwiching W_{t} are contractible (or empty) ANR’s.

Indeed, it follows from Proposition 3.6 that they are ANR’s. Also, they are the preimages of convex sets under a proper PL map (see [Hir75]) with contractible point-inverses (by Proposition 2.16), and the claim follows from [Sma57].

We show that this sandwich is generically a Z-sandwich (see the introduction for definitions).

Claim 2. If ℓ^{−1}(t_{0}) is not a vertex, then π′^{−1}ℓ^{−1}(t_{0}) is a Z-set in π′^{−1}ℓ^{−1}[t_{0}, ·).

We show that π′^{−1}ℓ^{−1}(t_{0}) is locally Z at all points. Indeed, fix a point z ∈ π′^{−1}ℓ^{−1}(t_{0}). Say e_{1} and e_{2} are distinct edges of G with nonzero ℓ_{G}-length (at z). For concreteness, say w(e_{1}) > w(e_{2}) (ampleness guarantees that they are not equal). For a small s ≥ 0 and a point {ℓ_{C}} ∈ π′^{−1}ℓ^{−1}[t_{0}, ·) near z define F_{s}({ℓ_{C}}) by adding s to the ℓ_{G}-length of e_{1} and subtracting s from the ℓ_{G}-length of e_{2}. This gives an instant deformation of a neighborhood of z into π′^{−1}ℓ^{−1}(t_{0}, ·) and proves the claim.

If ℓ^{−1}(t_{0}) is not a vertex, the theorem follows from the Z-sandwich fact. Assume that ℓ^{−1}(t_{0}) is a vertex v. If W_{t} ∩ π′^{−1}(v) = ∅ then W_{t} = π′^{−1}ℓ^{−1}(t_{0}, ·) and the theorem again follows.

Now assume that ℓ^{−1}(t_{0}) is a vertex v and that W_{t} ∩ π′^{−1}(v) ≠ ∅. Denote by e_{0} the unique edge of nonzero length (length 1) in the ℝ-metric corresponding to v.

Claim 3. Suppose that there is an edge e of G such that e ⊈ K_{1} ∪ core(G \ e_{0}) and w(e) > w(e_{0}) (K_{1} is a graph in \underline{K} that defines the cell W (\underline{K})). Then π′^{−1}(v) is a Z-set in π′^{−1}ℓ^{−1}[t_{0}, ·).

The desired instant deformation of a neighborhood of π′^{−1}(v) is obtained by decreasing ℓ_{G}(e_{0}) and increasing ℓ_{G}(e) by s ≥ 0. This deformation is welldefined since e ⊈ core(G \ e_{0}) and we leave ℓ_{C} unchanged for C ⊆ core(G \ e_{0}). The condition e ⊈ K_{1} guarantees that the deformation takes place in \overline{W} (\underline{K}), and finally w(e) > w(e_{0}) means that the function L_{0} increases during the deformation, so that the deformation takes place inside π′^{−1}(ℓ^{−1}[t_{0}, ·)) and proves the claim

Under the hypotheses of the Claim 3, the theorem follows from the Zsandwich as before.

We now finish the proof of the theorem under the assumption W_{t}∩π′^{−1}(v) ≠ ∅ plus the assumption that there are no edges e as in Claim 3. We may assume that W_{t} is not contained in a proper sub-cell of \overline{W} (\underline{K}), since otherwise the statement of the theorem follows by induction on the dimension of the cell. In particular, there are interior points of π(\overline{W} (\underline{K})) with larger L_{0}-value than v. It is convenient to represent W_{t} as the union of two open sets π′^{−1}(ℓ^{−1}(t_{0}, ·)) and W_{t} ∩ Q̂_{<ϵ}, where Q̂_{<ϵ} = Q̂_{≤ϵ} \ Q̂_{ϵ} (see Proposition 2.8). The first set is contractible by Proposition 2.16 and [Sma57], and so is the intersection π′^{−1}(ℓ^{−1}(t_{0}, ·)∩Q_{<ϵ}) where Q_{<ϵ} = Q_{≤ϵ}\Q_{ϵ}. So it remains to prove the following claim.

Claim 4. W_{t} ∩ Q̂_{<ϵ} is contractible.

Denote by Q_{<ϵ}^{0} the intersection of Q_{<ϵ} with the face of Σ_{G} where the ℓ_{G}- lengths of all edges e outside e_{0} ∪ core(G \ e_{0}) is 0. Likewise, let Q̂_{<ϵ}^{0} be the intersection of Q̂_{<ϵ} with the preimage of that face under π.

We first note that there is a deformation retraction of W_{t}∩Q̂_{<ϵ} onto W_{t}∩Q̂_{<ϵ}^{0}. The deformation does not change ℓ_{C} when C is a core graph that does not contain e_{0}. It suffices to specify the deformation on ℓ_{G}. On ℓ_{G}, the deformation is given by decreasing the lengths of edges e outside e_{0} ∪ core(G \ e_{0}) with constant speed and increasing ℓ_{G}(e_{0}) to insure that the volume is 1. During this deformation, the value of L_{0} can only increase due to our assumption that w(e) < w(e_{0}) for all affected edges e. Note also that if a point moves during this deformation, it does not land into π′^{−1}(v) (points in that set have minimal L_{0}).

It remains to argue that W_{t} ∩ Q̂_{<ϵ}^{0} is contractible. Consider the subset E of π′^{−1}(v) consisting of the points x with L_{1}(x) > L_{0}(x) (these are elements of the ordered abelian group ℝ^{A}). Since L_{0} is constant on π′^{−1}(v) (and equal to w(e_{0})), the set E is a super-level set in the cell π′^{−1}(v), and is thus contractible or empty by induction on the dimension of the cell. The intersection of W_{t} with π′^{−1}(v) is also a super-level set in π′^{−1}(v), and we consider two cases:

Case 1. E ⊆ W_{t} ∩ π′^{−1}(v).

We construct a deformation of W_{t} ∩ Q̂_{<ϵ}^{0} onto W_{t} ∩ π′^{−1}(v) (which is contractible). The deformation consists of following the mapping cylinder lines of Proposition 2.8. This amounts to increasing the length of e_{0} to 1 with linear speed, and decreasing the ℓ_{G}-lengths of other edges with linear speed, keeping the volume equal to 1. It remains to argue that all points that move during this deformation land into E. Let x be a landing point, and x_{δ} the point where ℓ_{G}(e_{0}) = 1 − δ, volume(core(G \ e_{0})) = δ, and which lands to x. Then

L_{0}(x_{δ}) = (1 − δ)w(e_{0}) + δL_{1}(x).

Since L_{0}(x) = w(e_{0}) and L_{0}(x) ≤ L_{0}(x_{δ}), it follows that L_{0}(x) ≤ L_{1}(x). Equality cannot hold since the collection A is ample.

Case 2. W_{t} ∩ π′^{−1}(v) ⊆ E.

In this case, we will construct an instant deformation of W_{t} ∩ Q̂_{<ϵ}^{0} into

π′^{−1}(ℓ^{−1}(t_{0}, ·)) ∩ Q̂_{<ϵ}^{0} = (W_{t} ∩ Q̂_{<ϵ}^{0}) \ π′^{−1}(v).

Since the latter set is contractible (either by appealing to [Sma57] again, or noting that it is a deformation retract of the contractible set (W_{t} ∩ Q̂_{<ϵ}) \ π′^{−1}(v)), it will follow that W_{t} ∩ Q̂_{<ϵ}^{0} is contractible.

The deformation is the reverse of the deformation of Case 1. Choose an instant deformation h_{s} of [0, ϵ) into (0, ϵ) that moves all points in the positive direction. The deformation we are defining does not change ℓ_{C} when e_{0} ⊈ C, and ℓ_{G} changes according to h_{s}: at time s, volume(core(G \ e_{0})) is equal to h_{s} applied to the volume of the same graph at the starting point, and ℓ_{G}(e_{0}) = 1−volume(core(G\e_{0})) at all points. The calculation in Case 1 combined with the assumption W_{t} ∩ π′^{−1}(v) ⊆ E shows that the L_{0}-value increases during the deformation, and so this deformation takes place in W_{t} ∩ Q̂_{<ϵ}^{0}. If core(G \ e_{0}) ⊈ K_{1}, then ℓ_{core(G\e_{0})} vanishes on K_{1} for all points in \overline{W} (\underline{K}), and as a result ℓ_{G} vanishes on K_{1} ∩ core(G \ e_{0}) throughout the deformation, so that the deformation takes place in the closed cell. The last case to consider is that core(G \ e_{0}) ⊆ K_{1}. In that case W_{t} ∩ Q̂_{<ϵ}^{0} is equal to W_{t} ∩ π′^{−1}(v), which is contractible by induction.

3.3. Critical points. Denote either of the super-level sets L^{−1}(t, ·) or L^{−1}[t, ·) by B_{t}. Consider the closed cover U(B_{t}) of B_{t} by the nonempty intersections of B_{t} with the cells \overline{W} (\underline{K}) of BCV (N, K). Denote by P(B_{t}) the poset of this cover.

Proposition 3.8. B_{t} is homotopy equivalent to P(B_{t}).

Proof. This follows from Proposition 2.20 and Theorem 3.7

Proposition 3.9. If B_{t} and B_{t′} are two super-level sets such that B_{t} ⊂ B_{t′} but B_{t′} \ B_{t} contains no vertices, then B_{t} and B_{t′} are homotopy equivalent.

Proof. The Morse function attains its maximum on every cell at a vertex. It then follows that a cell intersects B_{t} if and only if it intersects B_{t′}, so that the posets P(B_{t}) and P(B_{t′}) are isomorphic.

Proposition 3.9 can be interpreted as saying that the vertices are the only critical points of L.

4. Ascending links

We next turn to the study of the change of the homotopy type of a super-level set as it passes through a vertex v. Intuitively, one expects that the subspace corresponding to the set of directions from v in which L increases gets coned off. This is what we now make precise.

Fix a cell \overline{W} (\underline{K}) ⊆ Σ̂_{G} ⊂ BCV (N, K) and a vertex v of \overline{W} (\underline{K}). An ample sequence A and the corresponding Morse function L are fixed as well.

The chart associated with the hierarchy of scales at v identifies a neighborhood of v in \overline{W} (\underline{K}) ⊂ Σ̂_{G} ⊂ BCV (N, K) with a neighborhood of a vertex of a polytope. We define Lk(v, \overline{W} (\underline{K})) to be the space of germs of unit speed straight line segments leaving the vertex into the polytope. Thus Lk(v, \overline{W} (\underline{K})) is naturally identified with a spherical simplex in the unit sphere (and so is an AR if nonempty). The construction is natural with respect to inclusions of cells, and we define Lk(v, BCV (N, K)) to be the union of Lk(v, \overline{W} (\underline{K}))’s with identifications induced by inclusions of cells. Thus Lk(v, BCV (N, K)) is a spherical cell complex (and hence an ANR). We also refer to Lk(v, BCV (N, K)) as the set of directions at v.

Definition 4.1. Fix a Morse function L : BCV (N, K) → (ℝ_{+}^{A})^{|N|} coming from an ample sequence A of relative homotopy classes. For a vertex v ∈ \overline{W} (\underline{K}) ⊆ Σ̂_{G} ⊂ BCV (N, K) define the ascending link Lk_{>}(v, \overline{W} (\underline{K})) to be the subset of Lk(v, \overline{W} (\underline{K})) consisting of the germs of directions d at v with the property that the first coordinate function l_{i}^{α_{j}} which is not constant along d is monotonically increasing along d (in a small neighborhood of v). We define Lk_{>}(v, BCV (N, K)) as the subset of Lk(v, BCV (N, K)) similarly.

We also define the descending links Lk_{<}(v, \overline{W} (\underline{K})) and Lk_{<}(v, BCV ( N, K)) analogously (replacing ‘increasing’ by ‘decreasing’).

At a vertex v ∈ Σ̂_{G}, the metric ℓ_{G_{i}} is determined by edges e_{i}, 0 ≤ i < |N|, such that ℓ_{G_{i}}(e_{i}) = 1. There is a unique chart containing v determined by \underline{F} = (G = C_{0} ⊃ C_{1} ⊃ ⋯ ⊃ C_{|N|−1}) where C_{i+1} = core(C_{i} \ e_{i}) for i ≥ 0, this is the chart at v. Let x = (x_{0}, x_{1}, ⋯, x_{|N|−1}) ∈ Δ_{v} = Δ_{G\C_{1}} × Δ_{C_{1}\C_{2}} × ⋯ × Σ_{C_{|N|−1}} denote the coordinates at v. We may assume that the first coordinate x_{i,1} of x_{i} ∈ Δ_{C_{i}\C_{i+1}} corresponds to the edge of ℓ_{C_{i}}-length 1 at v (so all other coordinates of x_{i} are 0 at v).

The local coordinate chart Δ_{v} at v has a form that naturally lends itself to inductive arguments. Namely, Δ_{v} = Δ_{G\C_{1}} × Δ_{w} where Δ_{w} is the local chart of the vertex w of Σ̂_{C_{1}} with hierarchy of scales C_{1} ⊃ C_{2} ⊃ ⋯ ⊃ C_{|N|−1}. The coordinate map Φ_{v} at v is a product with second coordinate Φ_{w}.

If α is a homotopy class, then the function l_{0}^{α} can be written in the local coordinates Δ_{G\C_{1}} × Δ_{w} as

l_{0}^{α}(x_{0}, y) = x_{0} · a_{0} + (1 − S(x_{0}))l_{0}^{[α]}(y) (3)

where a_{i} is the vector (a_{i,j}) of multiplicities with which α crosses the edges of C_{i} \ C_{i+1}, · is the dot product, S(x_{i}) denotes the sum of the coordinates of the point x_{i}, and l_{0}^{[α]} the function on Δ_{w} corresponding to the homotopy class [α] obtained from α by deleting edges in G \ C_{1}.

More generally, if V is a subset containing v of the chart at v where L_{j} is constant for j < i then the restriction of l_{i}^{α} is given in the coordinates Δ_{G\C_{1}} × Δ_{C_{1}\C_{2}} × ⋯ × Δ_{C_{i−1}\C_{i}} × Δ_{w^{i}} by

l_{i}^{α}|_{V} (x_{0}, x_{1}, ⋯, x_{i−1}, y) = x_{i} · a_{i} + (1 − S(x_{i}))l_{i}^{[α]}(y) (4)

where now w^{i} is the vertex with hierarchy of scales C_{i} ⊃ C_{i+1} ⊃ ⋯ ⊃ C_{|N|−1} and [α] is obtained from α by deleting edges in G \ C_{i}.

The following lemma is easily proved inductively from Formulas 3 and 4.

Lemma 4.2. 1. Each coordinate of L_{0} is a polynomial in x.

2. Each coordinate of L_{i} is a polynomial function on each straight line d leaving the vertex v provided L_{j} is constant on d in a neighborhood of v for j < i.

3. l_{0}^{α}(v) = a_{0,1}.

4. The derivative of l_{0}^{α} at v is the dot product (a_{0,1} − a_{1,1}, a_{0,2} − a_{1,1}, a_{0,3} − a_{1,1}, ⋯) · x_{0}.

Corollary 4.3. Lk(v, BCV (N, K)) is the disjoint union of the descending link Lk_{<}(v, BCV (N, K)) and of the ascending link Lk_{>}(v, BCV (N, K)).

Proof. Since A is ample, there is a minimal i and α such that the restriction of l_{i}^{α} to d is not constant. By Lemma 4.2(2), l_{i}^{α}|_{d} is a polynomial. So, l_{i}^{α} and hence L is increasing or decreasing along d, and d is accordingly either in the ascending or descending link at v.

We focus on ascending links, as only those are needed for understanding of the topology at infinity. The next lemma will be used to show that they are ANR’s.

Lemma 4.4. Let P : ℝ^{n} → ℝ be a polynomial with P_{i} its homogeneous part of degree i. Let d be a direction at 0 ∈ ℝ^{n} represented by the unit vector x. Then P is increasing (respectively constant, decreasing) in the direction d iff (P_{1}(x), P_{2}(x), ⋯) is greater than 0 (respectively = 0, < 0) in lexicographically ordered ℝ^{ℕ}.

Proof. We have P (tx) − P (0) = Σ_{i>0}P_{i}(tx) = Σ_{i>0}t^{i}P_{i}(x) which is 0 if the vector (P_{1}(x), P_{2}(x), ⋯) = 0. Otherwise, let k be minimal greater than 0 such that P_{k}(x) ≠ 0. Then P (tx) − P (0) = Σ_{i≥k}t^{i}P_{i}(x) has a non-zero leading term of degree k which dominates for small t.

Corollary 4.5. Let f : ℝ^{n} → (ℝ^{ℕ})^{M} be as Lemma 3.5. Then the set of directions at 0 ∈ ℝ^{n} (as a subset of the unit sphere) along which f is increasing is semi-algebraic.

Proof. Let ((f_{k})_{j})_{i} denote the homogeneous part of degree i of the component ((f_{k})_{j} of f. According to Lemma 4.4, f is increasing in the direction given by the unit vector x iff the vector (((f_{k})_{j})_{i}(x)) is greater than 0 in lexicographically ordered ((ℝ^{ℕ})^{ℕ})^{M}. The result now follows as in Lemma 3.5.

Corollary 4.6. Lk_{>}(v, \overline{W} (\underline{K})) and Lk_{>}(v, BCV (N, K)) are ANR’s.

Proof. Since unions of ANR’s over ANR’s are ANR’s (see Section 1.1), it is enough to show that Lk_{>}(v, \overline{W} (\underline{K})) is an ANR. As in Proposition 3.6, it is enough to show that Lk_{>}(v, W (\underline{K})) is semi-algebraic, and this follows as in Proposition 3.6 from Corollary 4.5.

We will need the following Z-set lemma which follows quickly from definitions.

Lemma 4.7. Let A, B, and A′ be ANRs with A′ ⊂ A.

1. If A is contractible then ∅ ∗ B is a Z-set in A ∗ B.

2. If A′ and A differ by a Z-set in A then A′ ∗ B and A ∗ B differ by Z-set in A ∗ B.

3. If both A is contractible and A′ and A differ by a Z-set in A then (A′ ∗ B) \ (∅ ∗ B) and A ∗ B differ by a Z-set in A ∗ B.

Proposition 4.8. 1. If L|\overline{W} (\underline{K}) attains its unique maximum at v, then

Lk_{>}(v, \overline{W} (\underline{K})) = ∅.

2. Otherwise, there are points in \overline{W} (\underline{K}) where L has larger value than at v and Lk_{>}(v, \overline{W} (\underline{K})) is contractible (and so nonempty). Moreover, there is a vertex of \overline{W} (\underline{K}) with larger L-value that spans a 1-cell with v (and the corresponding edge determines a direction d in Lk_{>}(v, \overline{W} (\underline{K}))).

Proof. By Lemma 2.15, \overline{W} (\underline{K}) has the form ∏_{i} Σ̂_{H_{i}} where each H_{i} is a quotient of a subgraph of G. The map L can be identified with ∏_{i} L_{i} where the target is ordered lexicographically, and L_{i} is the Morse function on Σ̂_{H_{i}} obtained by restricting the elements of A in the obvious way. The vertex v is identified with a vertex (v_{1}, v_{2}, ⋯) ∈ ∏_{i} Σ̂_{H_{i}}. Set v′ = (v_{2}, v_{3}, ⋯).

With these identifications, we also have

Lk(v, \overline{W} (\underline{K})) = Lk(v_{1}, Σ̂_{H_{1}}) ∗ Lk(v′, ∏_{i>1} Σ̂_{H_{i}})

and

Lk_{>}(v, \overline{W} (\underline{K})) = (Lk_{>}(v_{1}, Σ̂_{H_{1}}) ∗ Lk(v′, ∏_{i>1} Σ̂_{H_{i}})) \ (∅ ∗ Lk_{<}(v′, ∏_{i>1} Σ̂_{H_{i}})).

Let k and α be minimal so that l_{k}^{α} is not constant on a neighborhood of v in the cell \overline{W} (\underline{K}) (ampleness). Let Lk_{>}(l_{k}^{α}), respectively Lk_{≥}(l_{k}^{α}), denote the subset of Lk(v, \overline{W} (\underline{K})) along which l_{k}^{α} is increasing, respectively non-decreasing. We have a sandwich

Lk_{>}(l_{k}^{α}) ⊂ Lk_{>}(v, \overline{W} (\underline{K})) ⊂ Lk_{≥}(l_{k}^{α}).

Claim:

1. If Lk_{>}(l_{k}^{α}) is non-empty then the above sandwich is a Z-sandwich of contractible sets, and there is a vertex in \overline{W} (\underline{K}) that spans an edge with v and with larger l_{k}^{α}-value than v.

2. If Lk_{>}(l_{k}^{α}) is empty, then Lk_{>}(v, \overline{W} (\underline{K})) is contained in a proper subcell of Lk(v, \overline{W} (\underline{K})).

To prove the claim, we argue by induction on the dimension of \overline{W} (\underline{K}) and the number of edges in G. The claim is clear if the dimension is 1.

If the ℓ_{G}-metric is the same for all points in the cell then by ampleness L_{0} is constant on the cell, and the claim follows by induction after replacing G by C_{1}. Hence we may assume that k = 0. With this assumption, the restriction of l_{0}^{α} to ∏_{i>1} Σ̂_{H_{i}} is zero.

It follows that the restriction l_{0}^{[α]} of l_{0}^{α} to Σ̂_{H_{1}} is not constant and that Lk_{>}(l_{0}^{α}), respectively Lk_{≥}(l_{0}^{α}), is identified with

(Lk_{>}(l_{0}^{[α]}) ∗ Lk(v′, ∏_{i>1} Σ̂_{H_{i}})) \ (∅ ∗ Lk(v′, ∏_{i>1} Σ̂_{H_{i}})),

respectively Lk_{≥}(l_{0}^{[α]}) ∗ Lk(v′, ∏_{i>1} Σ̂_{H_{i}}).

By our induction hypothesis, Lemma 4.7, Corollary 4.6, and the discussion above, it is enough prove the claim in the case that \overline{W} (\underline{K}) = Σ̂_{G}, i.e. \underline{K} is the hierarchy of scales at v. We will use notation as in Lemma 4.2.

Case 1. The derivative D of l_{0}^{α} is non-zero on some direction in Lk(v, \overline{W} (\underline{K})).

Let D_{>}, respectively D_{≥}, denote the subset of Lk(v, \overline{W} (\underline{K})) where D is positive, respectively non-negative. We can enlarge our sandwich

D_{>} ⊂ Lk_{>}(l_{0}^{α}) ⊂ Lk_{>}(v, \overline{W} (\underline{K})) ⊂ Lk_{≥}(l_{0}^{α}) ⊂ D_{≥}.

Since D is linear, the outermost sets are visibly convex subsets of Lk(v, \overline{W} (\underline{K})) and the sandwich is a Z-sandwich unless D_{>} is empty. The latter occurs only if the set of directions where D = 0 (and hence Lk_{>}(v, \overline{W} (\underline{K}))) is contained in a proper face of the cell, and we are done by induction on the dimension of the cell.

If D is positive on some direction in Lk(v, \overline{W} (\underline{K})) then by Lemma 4.2(4) either a_{0,1}−a_{1,1} < 0 and the direction obtained by scaling up x_{1,1} at the expense of x_{0,1} is in Lk_{>}(v, \overline{W} (\underline{K})) or one of a_{0,j} − a_{1,1} > 0 for some j > 0 and the direction obtained by scaling up x_{0,j} at the expense of x_{0,1} is in Lk_{>}(v, \overline{W} (\underline{K})). (Recall that x_{0,1} doesn’t increase along a direction at v whereas the other x_{0,j} don’t decrease). In the first case, the desired vertex is obtained from v by switching the roles of the edges e_{0,1} and e_{1,1} and in the latter case by switching the roles of e_{0,1} and e_{0,j}. This finishes the first case.

Case 2. The derivative D is zero on Lk(v, \overline{W} (\underline{K})).

It follows from Lemma 4.2(4) and Formula 3 that a_{1,1} = a_{0,j} for all j. Let C be this common value. Then,

l_{0}^{α}(x) = C + (1 − S(x_{0}))(l_{0}^{[α]}(y) − C).

So,

Lk_{>}(l_{0}^{α}) = (Lk(x_{0}, Δ_{G\C_{1}}) ∗ Lk_{>}(l_{0}^{[α]})) \ (Lk(x_{0}, Δ_{G\C_{1}}) ∗ ∅)

and

Lk_{≥}(l_{0}^{α}) = Lk(x_{0}, Δ_{G\C_{1}}) ∗ Lk_{≥}(l_{0}^{[α]}).

If Lk_{>}(l_{0}^{α}) is non-empty then Lk_{>}(l_{0}^{[α]}) is non-empty in this case. By induction, Lk_{>}(l_{0}^{[α]}) and Lk_{≥}(l_{0}^{[α]}) are contractible and differ by a Z-set in Lk_{≥}(l_{0}^{[α]}). By Lemma 4.7(3), Lk_{>}(l_{0}^{α}) and Lk_{≥}(l_{0}^{α}) differ by a Z-set. Since Lk_{≥}(l_{0}^{[α]}) is contractible so is Lk_{≥}(l_{0}^{α}) by Lemma 4.7(1). By induction, the desired edge direction can be found in ∅ ∗ Lk_{>}(l_{0}^{[α]}).

If Lk_{>}(l_{0}^{α}) is empty, then Lk_{>}(l_{0}^{[α]}) is empty. By induction, Lk_{>}(w, \overline{W} (\underline{K}′)) where \underline{K}′ = (C_{1}, C_{2}, ⋯) is contained in a proper face \overline{W}′ of \overline{W} (\underline{K}′). Then Lk_{>}(v, \overline{W} (\underline{K})) is contained in the proper face Lk(x_{1}, Δ_{G\C_{1}}) ∗ Lk(w, \overline{W}′). This completes the proof of the claim.

To finish the proof of the proposition, we now show that the proposition follows from the claim. Again we argue by induction on the number of edges in G and the dimension of the cell. If Lk_{>}(l_{k}^{α}) is non-empty we are done by the claim. If Lk_{>}(l_{k}^{α}) is empty then Lk_{>}(v, \overline{W} (\underline{K})) is contained in a proper face and we are done by induction.

If v is a vertex, denote by L(v) the poset of the cover of Lk(v, BCV (N, K)) by closed simplices Lk(v, \overline{W} (\underline{K})), and by L_{>}(v) the poset of the subcollection consisting of those Lk(v, \overline{W} (\underline{K})) with the property that Lk_{>}(v, \overline{W} (\underline{K})) is nonempty.

Notice that L sends each vertex to a point of (ℝ_{+}^{A})^{|N|} with all coordinates integral.

Proposition 4.9. Suppose v is a vertex of BCV (N, K) and L(v) = t ∈ (ℝ_{+}^{A})^{|N|}. 1. Lk_{>}(v, BCV (N, K)) is homotopy equivalent to L_{>}(v).

2. P(L^{−1}[t, ·)) is isomorphic to P(L^{−1}(t, ·)) with a subcomplex {P_{v}} coned off. The subcomplex P_{v} is isomorphic to L_{>}(v).

Proof. 1. This follows directly from Propositions 2.20 and 4.8.

2. We can regard P(L^{−1}(t, ·)) as a subcomplex of P(L^{−1}[t, ·)). L_{>}(v) naturally embeds as a subcomplex P_{v} of P(L^{−1}(t, ·)). The vertex v gives rise to a vertex c_{v} of P(L^{−1}[t, ·)) not contained in P(L^{−1}(t, ·)). Now observe that the cone c_{v} ∗ P_{v} is a subcomplex of P(L^{−1}[t, ·)) and that

P(L^{−1}[t, ·)) = P(L^{−1}(t, ·)) ∪ (c_{v} ∗ L(v))

which collapses onto P(L^{−1}(t, ·)) ∪ (c_{v} ∗ P_{v}).

Theorem 4.10. Each ascending link Lk_{>}(v, BCV (N, K)) is homotopy equivalent to the wedge of finitely many spheres of dimension 2|N| + |K| − 3p − 1 (where p is the number of components of the model graph).

By the wedge of 0 spheres we mean a point, so the ascending link is possibly contractible. However, it is never empty (recall that we are assuming that each rank 1 component contains at least two labels, so the above integer is non-negative).

4.1. Case |N| = p = 1. We start by verifying the theorem in case |N| = p = 1. Recall from Section 2.5 that BCV (1, k) is identified with ℝ^{k}/span(1, 1, ⋯, 1) ≅ ℝ^{k−1}. We claim that Lk_{<}(v, BCV (1, k)) is convex subset of Lk(v, BCV (1, k)) ≅ S^{k−2} that contains no antipodal points. It follows that the complement Lk_{>}(v, BCV (1, k)) of Lk_{<}(v, BCV (1, k)) is either S^{k−2} or contractible. Since ℝ^{k−1} is contractible, it further follows from standard Morse theory that Lk_{>}(v, BCV (1, k)) is contractible except at the unique minimum. We now prove the claim.

Let θ_{j} be the k^{th} root of unity e^{2πj/k}. Take for the model graph G = G(1, k) the unit circle with labeling {1, ⋯, k} → G given by j ↦ θ_{j}. Let β_{j} : I → G the homotopy class with immersed image a circular arc from θ_{j−1} to θ_{j} (subscripts taken mod k). Then l_{0}^{β_{j}}(x) = |x_{j} − x_{j−1}|. For general α ∈ A, l_{0}^{α} is a finite sum of such functions. Hence, each coordinate of the Morse function L (which equals L_{0} in this case) is convex.

Let L_{<} denote the subset of points of BCV (1, k) with smaller L-value than v, i.e. the set of x ∈ BCV (1, k) such that L(x) < L(v). Since each coordinate of L is convex, L_{<} is convex. Since v ∉ L_{<}, Lk_{<}(v, BCV (1, k)) contains no points that are antipodal. The claim follows.

Remark 4.11. The set L_{<}, and hence also Lk_{<}(v, BCV (1, k)) and Lk_{>}(v, BCV (1, k)), need be neither open nor closed.

4.2. Necklaces and the inward normal. Consider a vertex v ∈ BCV (N, K). There is a unique minimal Σ̂_{H} ⊂ BCV (N, K) that contains v: each component of H is a rose. This Σ̂_{H} is contained in all other Σ̂_{G} that contain v.

Let e_{0} be the unique edge of H with ℓ_{H}(e_{0}) ≠ 0, and let e_{1} be the unique edge with ℓ_{H_{1}}(e_{1}) ≠ 0. Consider the 1-cell \overline{W} (\underline{K}) determined by \underline{K} = (H, H_{2}, H_{3}, ⋯) (only H_{1} is omitted from the hierarchy of scales of v). This 1-cell determines a direction in Lk(v, BCV (N, K)), called the inward normal at v.

We say that an ℝ-graph G is a necklace if the graph obtained from G by collapsing all edges of length 0 is homeomorphic to the circle plus a collection of isolated points. By Neck(Σ̂_{G}) denote the subset of Σ̂_{G} consisting of those points that are taken by π : Σ̂_{G} → Σ_{G} to a necklace. This is a subcomplex of Σ̂_{G}. Let Neck = Neck(N, K) denote the union of Neck(Σ̂_{G}) over all marked graphs G of type (N, K). Then Neck is a subcomplex of BCV (N, K) that contains all vertices. In particular, Neck, Lk(v, Neck), Lk_{>}(v, Neck), and Lk_{<}(v, Neck) are all ANR’s. Recall that for a point {ℓ_{C}}_{C∈C(G)} ∈ Σ̂_{G} we refer to ℓ_{G} as the topmost metric.

Lemma 4.12. Denote by β_{1}, ⋯, β_{|N|−1} the homotopy classes of the embedded loops at v of topmost length 0. A point of BCV (N, K) near v is a necklace iff each β_{i} has topmost length 0.

Proof. Let G be a graph obtained from v by blowing up a forest. Homotopy classes β_{i} are represented by embedded loops.

Suppose {ℓ_{C}}_{C} ∈ Σ̂_{G} represents a necklace. If ℓ_{G}(β_{i}) > 0, then ℓ_{G}(β_{i}) = 1, and this is impossible in a small neighborhood of v.

Conversely, suppose {ℓ_{C}}_{C} ∈ Σ̂_{G} does not represent a necklace. Then there are two homology classes γ_{1} and γ_{2} in H_{1}(G; ℚ) such that no nontrivial linear combination t_{1}γ_{1} + t_{2}γ_{2} is represented by a cycle of ℓ_{G}-length 0. By β_{0} denote the homotopy class represented by an embedded loop in v of length 1. Since β_{0}, β_{1}, ⋯, β_{|N|−1} form a basis in homology, there is a nontrivial linear combination t_{1}γ_{1} + t_{2}γ_{2} which has the form s_{1}β_{1} + ⋯ + s_{|N|−1}β_{|N|−1}, and so not all β_{i} (for i > 0) can have length 0.

Proposition 4.13. 1. If the inward normal is in Lk_{>}(v, BCV (N, K)) (i.e. w(e_{1}) > w(e_{0})), then Lk_{>}(v, BCV (N, K)) is contractible.

2. If the inward normal is not in Lk_{>}(v, BCV (N, K)) (i.e. w(e_{1}) < w(e_{0})), then Lk_{>}(v, BCV (N, K)) is homotopy equivalent to Lk_{>}(v, Neck).

Proof. 1. The intersection-closed cover of Lk_{>}(v, BCV (N, K)) by the sets of the form Lk_{>}(v, Σ̂_{G}) has contractible elements and its poset is a cone (there is a unique minimal element corresponding to G being the disjoint union of roses).

2. Consider the intersection-closed cover of Lk_{>}(v, BCV (N, K)) by nonempty sets of the form Lk_{>}(v, Σ̂_{G}) and the induced intersection-closed cover of the ascending link Lk_{>}(v, Neck). First notice that both covers consist of contractible sets. For the first cover this follows from Proposition 4.8. As for the second, it follows from Lemma 4.12 that if G is obtained from v by blowing up a forest, then there is a cell \overline{W} (\underline{K}) (consisting of those points in Σ̂_{G} where β_{1}, ⋯, β_{|N|−1} have topmost length 0) such that U ∩Σ̂_{G} ∩Neck = U ∩\overline{W} (\underline{K}) for a small neighborhood U of v. Hence Proposition 4.8 can be applied again.

It remains to show that the two covers have the same poset. To this end, we prove that if Lk_{>}(v, Σ̂_{G}) is nonempty, then Lk_{>}(v, Neck∩Σ̂_{G}) is also nonempty. As usual, denote by e_{0} the edge of G whose topmost length at v is 1.

Case 1. There is an edge e ⊆ G such that w(e) > w(e_{0}) and e ⊈ core(G\e_{0}). Then Lk_{>}(v, Neck ∩ Σ̂_{G}) is nonempty: the direction determined by ℓ_{G}(e_{0}) =

1 − t, ℓ_{G}(e) = t for t ≥ 0 (and ℓ_{G_{i}}(e′) constant in all other cases) is ascending and is contained in Neck.

Case 2. There is no such e.

It then follows from Proposition 4.8(2) that all directions in which the topmost metric ℓ_{G} at v changes are descending. In particular, all ascending directions are contained in the necklace space, and so in this case Lk_{>}(v, Σ̂_{G}) = Lk_{>}(v, Neck ∩ Σ̂_{G}).

Proposition 4.14. Assume that Theorem 4.10 holds for total rank < |N|. Then Lk_{>}(v, Neck(N, K)) is homotopy equivalent to the wedge of finitely many spheres of dimension 2|N| + |K| − 3p − 1.

Theorem 4.10 (for total rank |N|) follows from Proposition 4.13 and Proposition 4.14. Our next goal is to formulate a proposition (Proposition 4.17) from which Proposition 4.14 will follow. After reordering the components we may assume that e_{0} is in the first component. Case: The first component is a circle. If the first component is the circle, then the component of Neck containing v can be identified with

ℝ^{k_{1}−1} × BCV ((n_{2}, n_{3}, ⋯, n_{p}), (k_{2}, k_{3}, ⋯, k_{p}))

with v corresponding to (x, v′). For convenience, we denote

BCV ((n_{2}, n_{3}, ⋯, n_{p}), (k_{2}, k_{3}, ⋯, k_{p}))

by BCV′. The function L induces a Morse function on BCV′ by removing L_{0} and the components of the homotopy classes contained in the first component. It is with respect to this function that we write e.g. Lk_{>}(v′, BCV′). The link Lk(v, Neck) can be identified with the join S^{k_{1}−2} ∗ Lk(v′, BCV′). The first component L_{0} of the Morse function factors through ℝ^{k_{1}−1}. If x corresponds to the absolute minimum of L_{0}, then the ascending link Lk_{>}(v, Neck) is the complement

(S^{k_{1}−2} ∗ Lk(v′, BCV′)) \ (∅ ∗ Lk_{<}(v′, BCV′)).

That this complement is homotopy equivalent to S^{k_{1}−2} ∗Lk_{>}(v′, BCV′) follows from the next two lemmas.

Lemma 4.15. Let A be an ANR, Z be a Z-set in A, and let Y ⊂ Z. Then A \ Y is an ANR.

Proof. This follows from [Han51]. For the sake of completeness, we give a proof in the case that A is finite dimensional. Recall that a finite dimensional space is an ANR iff it is locally contractible. Since A\Y ⊂ A, A\Y is finite dimensional. Since A \ Z ⊂ A \ Y ⊂ A is a Z-sandwich, it follows from the Z-sandwich fact in Section 1.3 that A \ Y is also locally contractible.

Lemma 4.16. Let X and F be as in Proposition 2.20. Suppose also that X is finite dimensional. Then

Y = (X ∗ Lk(v, BCV (N, K))) \ (∅ ∗ Lk_{<}(v, BCV (N, K)))

is homotopy equivalent to Z = X ∗ Lk_{>}(v, BCV (N, K)).

Proof. The covers of Y and Z induced by sets of the form F_{i} ∗ Lk(v, \overline{W} (\underline{K})), F_{i} ∗ ∅, and ∅ ∗ Lk(v, \overline{W} (\underline{K})) are finite intersection closed, have finite order, have identical posets, and by Proposition 4.8 have contractible elements. Thus Y and Z are homotopy equivalent by Proposition 2.20 if the elements of the covers are ANRs. That this is the case follows from Corollary 4.6 and Lemmas 4.7(1) and 4.15.

Recalling that k_{1} ≥ 2 in this case and using Theorem 4.10 for BCV′, we conclude that Lk_{>}(v, Neck) is homotopy equivalent to the wedge of finitely many spheres of dimension

1 + (k_{1} − 2) + 2(|N| − 1) + (|K| − k_{1}) − 3(p − 1) − 1 = 2|N| + |K| − 3p − 1.

We saw in Section 4.1 that if x does not correspond to the minimum, the ascending link D at x with respect to the function L_{0} is contractible. The set Lk_{>}(v, Neck) can be identified with the complement in D ∗ Lk(v′, BCV′) of ∅ ∗ Lk_{<}(v′, BCV′). By Lemma 4.16, the latter set is homotopy equivalent to D ∗ Lk_{>}(v′, BCV′), and is therefore contractible.

General case. For the rest of the argument, we assume that the first component has rank > 1.

We start with an informal discussion. Let G ∈ Neck be near v. The metric ℓ_{G} sees a circle with a certain number of distinguished points (“beads”) plus some additional isolated points (“spare beads”). Some of the beads come from labels in G; these we color blue and retain the labels. Others come from collapsed edges of G; these are colored red. Each β_{i} of Lemma 3.5 that is contained in the first component determines a red bead; we label this red bead i. We thus have k_{1} labeled blue beads and n_{1} − 1 red beads on the circle (some beads may coincide). One of the edges in the circle has length close to 1; this edge can be identified with e_{0}. If we choose an orientation on e_{0}, then the circle is naturally oriented. Removing e_{0} produces n_{1} + k_{1} − 1 points on an (oriented) line.

Each red bead represents a core graph of total rank < |N|. There are 2 additional labels in this core graph indicating where the corresponding positive length edges are to be attached.

We are thus led to the following definition. Let

R = ℝ^{n_{1}−1} × ℝ^{k_{1}}/span (1, 1, ⋯, 1) ≅ ℝ^{k_{1}+n_{1}−2}

be the “configuration space” (as usual we mod out by simultaneous translation of all points).

Consider the partition of ℝ^{n_{1}−1} × ℝ^{k_{1}}, and the induced partition of R, into cones, as follows. Two points

(x_{1}, x_{2}, ⋯, x_{n_{1}−1}, y_{1}, ⋯, y_{k_{1}}) and (x′_{1}, x′_{2}, ⋯, x′_{n_{1}−1}, y′_{1}, ⋯, y′_{k_{1}})

belong to the same cone in the partition provided 1. x_{i} < x_{j} ⟺ x′_{i} < x′_{j} for all i, j = 1, 2, ⋯, n_{1} − 1,

2. x_{i} < y_{j} ⟺ x′_{i} < y′_{j} for all i = 1, 2, ⋯, n_{1} − 1 and all j = 1, 2, ⋯, k_{1}, and

3. x_{i} > y_{j} ⟺ x′_{i} > y′_{j} for all i = 1, 2, ⋯, n_{1} − 1 and all j = 1, 2, ⋯, k_{1}. Informally, one configuration is in the same cone as another if it can be

transformed to the other by moving the points around and not allowing the red points to collide or uncollide with any other points. Blue points are allowed to pass through each other freely.

There is a natural map ϕ from a neighborhood of v in Neck to R, described above (collapse all edges of topmost length 0 and remove e_{0}). Pulling back by ϕ induces a partition of this neighborhood. We study the preimages of higher and higher skeleta inductively. The map ϕ is affine on closed cells (with respect to the structure induced by the chart at v). In particular, ϕ induces a partition of the ascending link at v. We show:

Proposition 4.17. Assume that Theorem 4.10 is true when the total rank is < |N|. If R_{0} is a nonempty closed subset of R which is a union of cones in the partition, then the ascending link at v with respect to ϕ^{−1}(R_{0}) is homotopy equivalent to the wedge of spheres of dimension 2|N| + |K| − 3p − 1.

Proposition 4.14 then follows by taking R_{0} = R. The rest of this section is devoted to the proof of Proposition 4.17. Proof of Proposition 4.17. We proceed by induction on the number of cones in R_{0}. Consider first the subspace Neck_{0} of Neck consisting of those graphs where there is a unique edge of length 1 (with respect to the topmost metric). We claim that Lk_{>}(v, Neck_{0}) is homotopy equivalent to the wedge of finitely many spheres of dimension 2|N| + |K| − 3p − 1.

Indeed, (a neighborhood of v in) Neck_{0} can be identified with (a corresponding neighborhood in)

BCV ((n_{1} − 1, n_{2}, ⋯, n_{p}), (k_{1} + 2, k_{2}, ⋯, k_{p}))

and Theorem 4.10 applied to this situation says that Lk_{>}(v, Neck_{0}) is homotopy equivalent to the wedge of finitely many spheres of dimension 2(|N|−1)+(|K|+ 2) − 3p − 1 = 2|N| + |K| − 3p − 1. Thus the proposition is verified for the case R_{0} equals the cone point.

To illustrate the inductive step in the next easiest case, we next prove the proposition for the case when R_{0} is a ray.

A generic point on a ray has exactly two distinct coordinates. The ray r(A, B) is given by a partition of the k_{1} + n_{1} − 1 coordinates into two nonempty subsets A and B and consists of equivalence classes of points in R where all coordinates in A coincide and are less than or equal to all coordinates in B, and these also coincide. The ray r(A, B) is in the partition if and only if either each A and B contain a red coordinate or else one of A, B consists of a single blue coordinate.

Consider the former case first.

The preimage under ϕ of the ray minus the origin is the subset consisting of those elements whose first component from the perspective of the topmost metric looks like a circle of length 1 with points a and b close to each other, with a preceding b (with respect to the orientation of the circle induced by e_{0}), and with all loops and labeled points in A being at a and all loops and labeled points in B being at b. Denote by Neck(A, B) the closure of the preimage of the open ray. There is a map Neck(A, B) → [0, ϵ) that measures the distance between a and b. There is also a “connection” (homeomorphism between fibers of Neck(A, B) → [0, ϵ)) that changes the lengths of e_{0} and of the short edge connecting a and b, and this induces a product structure W (A, B) × [0, ϵ) in a neighborhood of v in Neck(A, B). Say A contains k_{A} labeled points and n_{A} loops, and k_{B}, n_{B} are defined similarly for B. Thus n_{A} + n_{B} = n_{1} − 1, k_{A} + k_{B} = k_{1}, and n_{A} > 0, n_{B} > 0.

A neighborhood of v in W (A, B)×0 can be identified with a neighborhood of v(A, B) (= H \e_{0} and cut open according to A and B, i.e. the graph underlying v(A, B) is obtained from the disjoint union of roses representing v by removing the topmost edge in the first component and then dividing the rest of the first component into two components according to A and B) in

BCV ((n_{A}, n_{B}, n_{2}, ⋯, n_{p}), (k_{A} + 2, k_{B} + 2, k_{2}, ⋯, k_{p})).

(The extra 4 points keep track of where the long and the short edges are attached.) By induction, the ascending link Lk_{>}(v(A, B), W (A, B) × {0}) is homotopy equivalent to the wedge of spheres of dimension

2(|N| − 1) + (|K| + 4) − 3(p + 1) − 1 = 2|N| + |K| − 3p − 2.

If the “inward normal” at v corresponding to the connection above (i.e. the direction from v in which the length of e_{0} goes down at the expense of the distance between a and b) is not in the ascending link at v, then the ascending link at v with respect to the preimage of the closed ray is homotopy equivalent to the ascending link of v in the preimage of the origin, and this is homotopy equivalent to the wedge of spheres of dimension 2|N| + |K| − 3p − 1, as seen above. If this inward normal is in the ascending link, then the ascending link at v with respect to the preimage of the ray is obtained from the ascending link with respect to the preimage of the origin by coning off a subset which is homotopy equivalent to the wedge of spheres of one less dimension, and so is homotopy equivalent to the wedge of spheres of dimension 2|N| + |K| − 3p − 1.

Now consider the case where B consists of a single blue coordinate. A product structure W (A, B) × [0, ϵ) in a neighborhood of v is defined similarly. Then a neighborhood of v in W (A, B) × 0 can be identified with a neighborhood of v(A, B) (= H \ e_{0}) in

BCV ((n_{1} − 1, n_{2}, ⋯, n_{p}), (k_{1} + 1, k_{2}, ⋯, k_{p})).

The arithmetic now is

2(|N| − 1) + (|K| + 1) − 3p − 1 = 2|N| + |K| − 3p − 2.

The argument in general is similar. Suppose dim R_{0} = l.

Choose an l-dimensional open cone C in R_{0}. By induction, Proposition 4.17 holds for R_{0} \ C. By V (C) denote the closure of ϕ^{−1}(C). The map ϕ induces a map V (C) → \overline{C}, which in turn induces a product structure W (C) × \overline{C} in a neighborhood of v in ϕ^{−1}(R_{0}).

A generic point in C has l + 1 distinct coordinates and each equivalence class either contains a red bead, or else consists of a single blue bead. Say there are l(r) classes of the first kind, and l(b) of the second, so that l(r) + l(b) = l + 1. Denote by n(j) and k(j) the number of red and blue beads respectively in the j^{th} equivalence class of the first kind, j = 1, 2, ⋯, l(r).

A neighborhood of v in W (C) × {0} can be identified with a neighborhood of a certain graph v(C) (obtained from H \ e_{0} by cutting open and assigning labeled points according to C) in

BCV ((n(1), ⋯, n(l(r)), n_{2}, ⋯, n_{p}), (k(1) + 2, ⋯, k(l(r)) + 2, k_{2}, ⋯, k_{p})).

The ascending link with respect to W (C) × 0 is therefore homotopy equivalent to the wedge of spheres of dimension

\begin{matrix}2(|N| − 1) + (|K| − l(b) + 2l(r)) − 3(p − 1 + l(r)) − 1 = \\ 2|N| + |K| − 3p − l(b) − l(r) = 2|N| + |K| − 3p − 1 − l. \end{matrix}

Consider the commutative diagram

W (C) × \overline{C} ⟶^{L} (ℝ_{+}^{A})^{|N|}
↓  ↓
\overline{C}  ⟶^{ℓ} ℝ_{+}^{A}

in which the vertical arrows are projections (the right one projects to the first A coordinates). The function ℓ : \overline{C} → ℝ_{+}^{A} measures the length of α_{i}’s with respect to the topmost metric.

We consider two cases.

Case 1. The subset Lk_{<}(0, C) of Lk(0, C) of directions at 0 in C along which ℓ is not strictly increasing is not empty.

As in Section 4.1, Lk_{<}(0, C) is convex and contractible. It follows that in this case the radial projection (with respect to the spherical metric) from a direction in Lk_{<}(0, C) induces a strong deformation retraction from Lk_{>}(v, W (C) × \overline{C}) onto Lk_{>}(v, W (C) × (\overline{C} \ C)), and thus from Lk_{>}(v, ϕ^{−1}(R_{0})) onto Lk_{>}(v, ϕ^{−1}(R_{0} \ C)), and this proves the proposition in this case.

Case 2. Lk_{>}(0, C) = Lk(0, C).

We first claim that Lk_{>}(v, W (C) × \overline{C}) is contractible. Indeed, consider inclusions

\begin{matrix}(Lk(v(C), W (C)) ∗ Lk(0, C)) \ (Lk(v(C), W (C)) ∗ ∅) \\ ⊆ Lk_{>}(v, W (C) × \overline{C}) ⊆ Lk(v(C), W (C)) ∗ Lk(0, \overline{C}). \end{matrix}

By Lemma 4.7(1), since Lk(0, C) is contractible, Lk(v(C), W (C)) ∗ ∅ is a Z-set in

Lk(v(C), W (C)) ∗ Lk(0, C).

Also, since Lk(0, C) and Lk(0, \overline{C}) differ by a Z-set, the above sequence of inclusions is a Z-sandwich by Lemma 4.7(3). Hence we see that Lk_{>}(v, W (C)× \overline{C}) is contractible.

Secondly, we claim that inclusion

Lk_{>}(v(C), W (C)) ∗ Lk_{>}(0, \overline{C} \ C) ↪ Lk_{>}(v, W (C) × (\overline{C} \ C))

is a homotopy equivalence. Indeed,

\begin{matrix}Lk_{>}(v, W (C) × (\overline{C} \ C)) \\ = (Lk(v(C), W (C)) ∗ Lk_{>}(0, \overline{C} \ C)) \ (Lk_{<}(v(C), W (C)) ∗ ∅) \end{matrix}

which by Lemma 4.16 is homotopy equivalent to

Lk_{>}(v(C), W (C)) ∗ Lk_{>}(0, \overline{C} \ C).

As in Section 4.1, from the convexity of ℓ it follows that Lk_{>}(0, \overline{C} \ C) is either contractible (and nonempty!) or the whole Lk(0, \overline{C} \ C) = S^{l−2}. In the first case, Lk_{>}(v, W (C) × (\overline{C} \ C)) is contractible, and in the second it is homotopy equivalent to the wedge of finitely many spheres of dimension 2|N| + |K| − 3p − 1 − l + (l − 2) + 1 = 2|N| + |K| − 3p − 2. Now Lk_{>}(v, ϕ^{−1}(R_{0})) is obtained from Lk_{>}(v, ϕ^{−1}(R_{0} \ C)) (which is homotopy equivalent to the wedge of 2|N| + |K| − 3p − 1-spheres) by attaching Lk_{>}(v, W (C) × \overline{C}) (which is contractible) along Lk_{>}(v, W (C) × (\overline{C} \ C)) (which is homotopy equivalent to the wedge of (2|N|+|K|−3p−2)-spheres). The proposition finally follows.

5. The topology at infinity

To study the topology at infinity of BCV (N, K) it is more convenient to replace BCV (N, K) by the simplicial complex S(N, K), which is proper homotopy equivalent to BCV (N, K).

Definition 5.1. The simplicial complex S(N, K) is defined to have the same vertex set as BCV (N, K), and a collection v_{0}, v_{1}, ⋯, v_{k} of vertices spans a simplex iff they are contained in some Σ̂_{G} ⊂ BCV (N, K).

Proposition 5.2. 1. S(N, K) is proper homotopy equivalent to

BCV (N, K).

2. Let v ∈ S(N, K) be a vertex and by Lk_{↑}(v, S(N, K)) denote the link of v in the subcomplex of S(N, K) spanned by the vertices w with L(w) ≥ L(v). Then Lk_{↑}(v, S(N, K)) is homotopy equivalent to Lk_{>}(v, BCV (N, K)), and thus to the wedge of finitely many spheres of dimension 2|N|+|K|−3p−1.

Proof. The function that to each Σ̂_{G} assigns the simplex in S(N, K) spanned by the vertices in Σ̂_{G} induces an isomorphism between the posets of intersectionclosed covers by compact sets, and thus (1) follows from Proposition 2.20.

The proof of (2) is similar: Consider the covering of Lk_{>}(v, BCV (N, K)) by nonempty sets of the form Lk_{>}(v, Σ̂_{G}). To each such set assign the simplex in Lk_{↑}(v, S(N, K)) whose vertices are those w ∈ Σ̂_{G} with L(w) > L(v). That this correspondence induces a poset isomorphism follows from Proposition 4.8, and the claim then follows from Proposition 4.9.

The following theorem is the analog of Lemma 1.3 in our situation.

Theorem 5.3. Suppose S is a locally finite r-connected simplicial complex such that

1. There is an injective function L : S^{(0)} → Λ from the vertex set of S to a totally ordered set Λ.

2. There is a collection Δ of simplices in S, called “distinguished”, that is closed under intersections and such that every simplex in S is contained in a distinguished simplex.

3. There is a function μ : S^{(0)} → ℕ = {1, 2, 3, ⋯ } which is “proper”, i.e. μ^{−1}(n) is finite for all n ∈ ℕ, and compatible with L in the following sense: If σ is a distinguished simplex and v the vertex of σ with maximal L-value, then μ(v) ≥ μ(w) for all vertices w of σ.

4. For all vertices v ∈ S^{(0)} the ascending link Lk_{↑}(v, S) is r-connected (and nonempty). Here the ascending link is the link of v in the subcomplex of S spanned by the vertices w with L(w) ≥ L(v).

Then S is r-connected at infinity.

Proof. The first step is to reduce to the case where Λ = ℕ (and then μ can be taken to be L). Consider μ × L : S^{(0)} → ℕ × Λ where we order ℕ × Λ lexicographically. This map is injective and properness of μ implies that the image (μ × L)(S^{(0)}) ⊆ ℕ × Λ is order isomorphic to ℕ (assuming without loss of generality that it is infinite). Denote by Lk′_{↑}(v, S) the ascending link with respect to μ×L, i.e. the link of v in the subcomplex of S spanned by the vertices w with (μ×L)(w) ≥ (μ×L)(v). We will argue that Lk_{↑}(v, S) and Lk′_{↑}(v, S) are homotopy equivalent, and our first step will then be complete. Note that the compatibility assumption says that L and μ × L attain maximal value among the vertices of a distinguished simplex at the same vertex. Thus, if σ is a distinguished simplex that contains v, then Lk(v, σ) intersects Lk_{↑}(v, S) if and only if it intersects Lk′_{↑}(v, S). If these intersections are nonempty, then they are contractible (they are simplices). Therefore, the closed covers of Lk_{↑}(v, S) and Lk′_{↑}(v, S) consisting of nonempty intersections with Lk(v, σ) (as σ ranges over the distinguished simplices that contain v) are both intersection-closed and consist of AR’s. Since they have the same poset, it follows from Proposition 2.20 that Lk_{↑}(v, S) ≃ Lk′_{↑}(v, S).

For the rest of the proof we assume that Λ = ℕ and that L : S^{(0)} → ℕ is a bijection. For m ∈ ℕ denote by S_{m} the subcomplex of S spanned by the vertices w with L(w) ≥ m. Thus S_{0} = S and S_{m} is obtained from S_{m+1} by coning off Lk_{↑}(v, S). From the Mayer-Vietoris sequence we see that S_{m} and S_{m+1} have isomorphic homology groups in dimensions ≤ r. Since S_{0} = S is r-connected, it follows that homology groups of S_{m} vanish in dimensions ≤ r for all m. If r ≥ 1 we see similarly from van Kampen’s theorem that each S_{m} is simply connected. Therefore, Hurewicz theorem implies that each S_{m} is r-connected. Since the closure of S \ S_{m} is compact for all m and every compact subset of S is disjoint from S_{m} for large m it follows that S is r-connected at infinity.

Proposition 5.4. Under the hypotheses of Theorem 5.3 and assuming in addition that S is contractible and that each ascending link is the wedge of (r +1)- spheres, we have that compactly supported cohomology

H_{c}^{i}(S) ≅ \begin{cases}0 & if i ≠ r + 2 \\ ⨁_{v∈S^{(0)}} H̃^{r+1}(Lk_{↑}(v, S)) & if i = r + 2 \end{cases}

Proof. The group H_{c}^{i}(S) is by definition the direct limit of the sequence H^{i}(S, S_{m}) ≅ H̃^{i−1}(S_{m}). Argue by induction on m that

H̃^{j}(S_{m}) ≅ \begin{cases}0 & if j ≠ r + 1 \\ ⨁_{v∉S_{m}^{(0)}} H̃^{r+1}(Lk_{↑}(v, S)) & if j = r + 1 \end{cases}

The inductive step consists of writing down the Mayer-Vietoris sequence for

S_{m} = S_{m+1} ∪_{Lk_{↑}(v,S)} c(Lk_{↑}(v, S))

and noticing that the portion

0 → H̃^{r+1}(S_{m}) → H̃^{r+1}(S_{m+1}) → H̃^{r+1}(Lk_{↑}(v, S)) → 0

splits.

Lemma 5.5. There are finitely many homotopy classes B(N, K) in G(N, K) such that if some element α ∈ Out(N, K) fixes each of them then α is the identity.

Proof. In the case Out(F_{n} =< x_{1}, ⋯, x_{n} >) = Out(n, 0), the set B(n, 0) = {x_{1}, ⋯, x_{n}, x_{1}x_{2}, x_{1}x_{3}, ⋯, x_{1}x_{n}, x_{2}x_{3}, x_{2}x_{4}, ⋯, x_{2}x_{n}} suffices. (Proof: Let f : F_{n} → F_{n} be an automorphism that fixes all these elements up to conjugacy. By conjugating f, we may assume f(x_{1}) = x_{1}. After a further conjugation by a power of x_{1}, we may assume that f(x_{2}) = γx_{2}γ^{−1} is a reduced word that does not start with a nontrivial power of x_{1}. Then f(x_{1}x_{2}) = x_{1}γx_{2}γ^{−1} and x_{1}x_{2} can be fixed up to conjugacy only if γ = 1. Now if i > 2 and f(x_{i}) = γ_{i}x_{i}γ_{i}^{−1} is a reduced word, then γ_{i} does not start either with x_{1}^{±1} or x_{2}^{±1}. In the former case, f(x_{1}x_{i}) = x_{1}γ_{i}x_{i}γ_{i}^{−1} and x_{1}x_{i} can be fixed up to conjugacy only if γ_{i} = 1. In the latter case, we consider x_{2}x_{i}. Therefore, f = id.)

In the case of Aut(F_{n}) = Out(n, 1) we can take B(n, 1) = {x_{1}, ⋯, x_{n}}. For Out(n, k) take B(n, k) = B(n, 1) ∪ {y_{2}, ⋯, y_{k}} where y_{i} is a path from the first to the i^{th} labeled point. The general case follows by choosing the classes as above in each component.

Theorem 5.6. S(N, K) is (2|N| + |K| − 3p − 2)-connected at infinity. In particular, Out(F_{n}) is (2n − 5)-connected at infinity. Moreover,

H_{c}^{i}(S(N, K)) ≅ \begin{cases}0 & if i ≠ 2|N| + |K| − 3p \\ free abelian & if i = 2|N| + |K| − 3p \end{cases}

Proof. We apply Theorem 5.3 with S = S(N, K) and r = 2|N| + |K| − 3p − 2. The function L is the Morse function with respect to an ample sequence with the first homotopy class α_{1} equal to the disjoint union of the homotopy classes from Lemma 5.5. The function μ is the multiplicity with respect to α_{1}, i.e. μ(v) is the maximal number of times the class α_{1} crosses an edge of the underlying rose when represented by an immersion. Properness of μ is a consequence of Lemma 5.5, and assumption (4) of Theorem 5.3 comes from Proposition 5.2. A simplex of S is “distinguished” if its vertices form the entire vertex set of some Σ̂_{G}. To see that the compatibility condition (3) of Theorem 5.3 is satisfied note that the very first coordinate of L(v) (where v is a vertex of Σ̂_{G}) is the multiplicity with which α_{1} crosses the unique edge of length 1 (topmost edge) in v. Thus the largest value of L among the vertices of Σ̂_{G} will occur at a vertex v where this multiplicity equals the largest multiplicity with which α_{1} crosses an edge of G, and in particular μ(v) ≥ μ(w) for all other vertices w. Note also that the compatibility condition fails for arbitrary simplices in S(N, K) since the maximal multiplicity might occur on an edge that is never topmost at the vertices of a non-distinguished simplex. The other properties are immediate. The last sentence follows from Proposition 5.4.

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Department of Mathematics, University of Utah, Salt Lake City, UT 84112

   E-mail address: bestvinamath.utah.edu

Department of Mathematics, Rutgers University, Newark, New Jersey 07102

   E-mail address: feighnandromeda.rutgers.edu