Homotopy type of the complex of free factors of a free group

We show that the complex of free factors of a free group of rank n⩾2 is homotopy equivalent to a wedge of spheres of dimension n−2 . We also prove that for n⩾2 , the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n−2) ‐connected. In addition, we show that for every non‐trivial free factor system of a free group, the corresponding relative free splitting complex is contractible.


Introduction
Let F be the free group of finite rank n.A free factor of F is a subgroup A such that F = A * B for some subgroup B of F. Let [.] denote the conjugacy class of a subgroup of F. Define F n to be the partially ordered set (poset) of conjugacy classes of proper, non-trivial free factors of F where [A] ≤ [B] if for suitable representatives, one has A ⊆ B. We will call the order complex (geometric realisation) of this poset the complex of free factors or free factor complex and denote it also by F n .Since a maximal nested chain of conjugacy classes of free factors has length n − 2 (see Section 2 for the notational conventions we use), F n is (n − 2)-dimensional.Note that for n = 2, our definition differs from the usual one: commonly, two conjugacy classes of free factors of F 2 are connected by an edge in F 2 if they have representatives that span a basis.There is a natural action of the group of outer automorphisms of F, denoted Out(F), on F n .The geometry of this complex has been studied very well in recent years and it was used to improve the understanding of Out(F).Most notably, Bestvina and Feighn in [BF14] showed that F n is Gromov-hyperbolic, in analogy to Masur-Minsky's hyperbolicity result for the curve complex of a surface [MM99].In this paper, we investigate the topology of F n .Our main result is as follows: Theorem A. For n ≥ 2, the free factor complex F n is homotopy equivalent to a wedge of spheres of dimension n − 2.
In [HV98], Hatcher and Vogtmann showed that the geometric realisation of the poset of proper free factors in F is homotopy equivalent to a wedge of spheres of dimension n−2.Note that Hatcher and Vogtmann's complex is different from the free factor complex F n in that its vertices are proper free factors and not conjugacy classes of proper free factors.Since F n comes with a natural action of Out(F) instead of Aut(F), the focus has shifted more towards this version over the years.
Motivation.The motivation for describing the homotopy type of this and similar factor complexes comes from the analogy with the rational Tits building, ∆(n, Q), associated to SL n (Z).The definition of these Out(F)-simplicial complexes is similar to ∆(n, Q).By the Solomon-Tits theorem ( [Sol69]), the rational Tits building is homotopy equivalent to a wedge of spheres of dimension n − 2. In [BS73], Borel and Serre used this to show that the dualising module of any torsion free finite index subgroup Γ of SL n (Z) is H n−2 (∆(n, Q), Z) =: D, that is, for any Γ-module M , for all i > 0, and d equal to the virtual cohomological dimension of SL n (Z).
This relationship between SL n (Z) and ∆(n, Q) has been successfully extended to the mapping class group of an orientable surface of genus g and p punctures MCG(Σ g,p ) and the associated curve complex C(Σ g,p ).Harvey ([Har81]) defined the curve complex and showed that it is homotopy equivalent to a wedge of spheres.Harer ([Har86] for punctured surfaces) and Ivanov ([Iva91] for closed surfaces) showed that the top dimensional reduced homology of the curve complex is the dualising module for the mapping class group.
The rational Tits building and the curve complex capture the asymptotic geometry of the symmetric space for SL n (Z) and Teichmüller space for MCG(Σ), respectively.In this paper, we also obtain some partial results about the asymptotic geometry of Culler-Vogtmann's Outer space CV n .
Asymptotic geometry of Outer space.Let CV n be Culler-Vogtmann's Outer space and L be its spine.We denote by CV r n reduced Outer space which has spine K. Let FS n be the free splitting complex.For the definitions, see Section 3. We view L, K and FS n as partially ordered sets and use the same notation for both the poset and its order complex.As explained in Section 3, FS n is the simplicial completion of CV n and L is a subposet of FS n .Analogously, there is a natural subposet FS r n of FS n which is the simplicial completion of CV r n and contains K. Before we state our next theorem, we consider another poset FF n (defined in [HM14]), whose order complex is called the complex of free factor systems (also denoted by FF n ).A free factor system of F is a finite collection of the form A = {[A 1 ], . . ., [A k ]}, where k > 0, each A i is a proper, non-trivial free factor of F, such that there exists a free factorisation There is a partial ordering on the set of free factor systems given as follows: A A if for every [A i ] ∈ A there exists [A j ] ∈ A such that A i ⊆ A j up to conjugation.This poset of free factor systems is denoted by FF n .The poset F n of free factors is a subposet of FF n .In fact, F n and FF n are quasi-isometric to each other by [HM14,Proposition 6.3].However, Theorem A and the following result show that they are not homotopy equivalent.
Theorem B. FS n \ L and FF n are homotopy equivalent.Moreover, for n ≥ 2, they are (n − 2)connected.
In order to establish the homotopy equivalence FS n \ L FF n , we are lead to study relative versions of the free splitting complex: Whenever one has a free splitting S of F, the set of conjugacy classes of non-trivial vertex stabilisers forms a free factor system V(S).Now for a free factor system A in F, the poset of free splittings of F relative to A, denoted FS n (A), is the subposet of FS n consisting of all free splittings S ∈ FS n such that A V(S).Its realisation is the relative free splitting complex studied in [HM14].We show: Theorem C. For any free factor system A of F, the relative free splitting complex FS n (A) is contractible.
In [BSV18], Bux, Smilie and Vogtmann introduced an equivariant deformation retract of CV r n called jewel space, denoted by J n .They showed that J n is homeomorphic to the bordification of Outer space defined by Bestvina and Feighn in [BF00] and asked what the homotopy type of its boundary ∂J n is.We mention the following result of Vogtmann to contrast the currently known results about the connectivity of the simplicial boundaries of unreduced and reduced Outer space and also because our methods for establishing Theorem A and Theorem B give an alternate proof (albeit longer) of the (n − 3)-connectivity of FS r n \ K. Theorem D. [Vog18] FS r n \ K and ∂J n are homotopy equivalent.Moreover, for n ≥ 3, they are (n − 3)-connected.
The partial results in Theorem B and D lead to the following question.
Question 1.1.What are the homotopy types of FS n \ L, FF n and FS r n \ K? Unfortunately, we cannot answer this.The main difficulty we are faced with is that FF n is (2n − 3)-dimensional [HM14, Proposition 6.1] and our method cannot be pushed to get higher connectivity results or lower the dimension.Note that the curve complex of a closed surface of genus g is (3g − 4)-dimensional but nevertheless it is homotopy equivalent to a wedge of spheres of dimension 2g − 2. For further comments on this, see Section 8.3.Methods of proof.Various methods have been used to determine the homotopy type of some of the complexes mentioned in this introduction: Shelling orders for ∆(n, Q), flow arguments for FS n , spectral sequences for Hatcher-Vogtmann's complex of free factors, Morse theory for the curve complex.
In this paper, we view all our simplicial complexes as order complexes of posets and use various Quillen type fibre lemmas (see Section 2 for details) to get the desired results.In particular, the following poset version of the Vietoris-Begle theorem (see [GV09, Corollary 2.4]) is the main tool we use.
Lemma 2.1 ([Bjö03, Theorem 2]).Let P and Q be connected posets and f : P → Q a poset map.Suppose that the fibre f −1 (Q ≤x ) is k-connected for all x ∈ Q.Then the induced map on homotopy groups f * j : π j (P ) → π j (Q) is an isomorphism for all j ≤ k.Using Whitehead's theorem, one gets in particular: If f −1 (Q ≤x ) is contractible for all x ∈ Q, then f induces a homotopy equivalence on geometric realisations.
Paper Outline.In Section 2, we set the notation for posets, state the various fibre lemmas and mention some results from algebraic topology that will be used later.In Section 3, we define (un-) reduced Culler-Vogtmann Outer space, its spine and the free splitting complex.We also explain the relationship between these spaces.Section 4 can be read independently of the rest of the paper.It establishes the homotopy type of posets of certain subgraphs of a fixed graph.In Section 5, we show contractibility of the relative free splitting complexes (Theorem C).This result is used in Section 6 to prove the homotopy equivalence of FS n \ L and FF n (the first statement of Theorem B).Also in Section 6, we show that F n is homotopy equivalent to the subposet of FS n , denoted FS 1 , given by free splittings with exactly one non-trivial vertex group.Finally, in Section 7, we prove Theorem B, Theorem A and the second statement of Theorem D. We close this article in Section 8 with some remarks concerning the complex of sphere systems and relative versions of our results and give an illustration of our considerations in the case where n = 2.
Proof outline for Theorem A. We now describe a brief outline for the proof of Theorem A, which also sheds some light on the structure of the paper.See Section 7 for the detailed proof.We first establish in Proposition 6.5 that FS 1 and F n are homotopy equivalent.Consider the pair of posets (X, Y ) where X = L and Y = FS 1 .In Section 7, we define a particular subposet Z of X × Y with projection maps p 1 : Z → X and p 2 : Z → Y .We then show that the fibres of the map p 2 (in the sense of Lemma 2.1) are contractible and the fibres of p 1 are given by posets of subgraphs which are (n − 3)-connected.Applying Lemma 2.1 twice then gives us that FS 1 , equivalently F n , is (n − 3)-connected.Since F n is (n − 2)-dimensional, we obtain the desired result.
For the proof of the second statement of Theorem B (resp.Theorem D), we consider the pair ( Acknowledgements.This project has benefited greatly from the discussions the second author had with Mladen Bestvina while she was a graduate student under his supervision at the University of Utah.We would like to thank Thomas Goller, Nir Lazarovich and Derrick Wigglesworth for helpful conversations.We are also grateful to Karen Vogtmann for enlightening conversations and thank Kai-Uwe Bux for reading drafts of this paper and in particular for helping to improve the exposition.We thank the organizers of the conference 'Geometry of outer space and outer automorphism groups', Warwick 2018 which gave us the opportunity to start this collaboration.The first-named author was supported by the grant BU 1224/2-1 within the Priority Programme 2026 "Geometry at infinity" of the German Science Foundation (DFG).The second-named author was supported by the Israel Science Foundation (grant 1026/15).

Preliminaries on poset topology
Let P = (P, ≤) be a poset (partially ordered set).If x ∈ P , the sets P ≤x and P ≥x are defined by A chain of length l in P is a totally ordered subset x 0 < x 1 < . . .< x l .For each poset P = (P, ≤), one has an associated simplicial complex ∆(P ) called the order complex of P .Its vertices are the elements of P and higher dimensional simplices are given by the chains of P .When we speak about the realisation of the poset P , we mean the geometric realisations of its order complex and denote this space by P := ∆(P ) .With an abuse of notation, we will attribute topological properties (e.g.homotopy groups and connectivity properties) to a poset when we mean that its realisation has these properties.
A map f : P → Q between two posets is called a poset map if x ≤ y implies f (x) ≤ f (y).Such a poset map induces a simplicial map from ∆(P ) to ∆(Q) and hence a continuous map on the realisations of the posets.It will be denoted by f or just by f if what is meant is clear from the context.
The direct product P ×Q of two posets P and Q is the poset whose underlying set is the Cartesian product {(p, q) | p ∈ P, q ∈ Q} and whose order relation is given by (p, q) ≤ P ×Q (p , q ) if p ≤ P p and q ≤ Q q .
2.1.Fibre theorems.An important tool to study the topology of posets is given by so called fibre lemmas comparing the connectivity properties of posets P and Q by analysing the fibres of a poset map between them.The first such fibre theorem appeared in [Qui73, Theorem A] and is know as Quillen's fibre lemma.For this text, we need the following refinement due to Björner.
Lemma 2.1 ([Bjö03, Theorem 2]).Let P and Q be connected posets and f : P → Q a poset map.Suppose that the fibre f −1 (Q ≤x ) is k-connected for all x ∈ Q.Then the induced map on homotopy groups f * j : π j (P ) → π j (Q) is an isomorphism for all j ≤ k.Using Whitehead's theorem, one gets in particular: If f −1 (Q ≤x ) is contractible for all x ∈ Q, then f induces a homotopy equivalence on geometric realisations.
For a poset P = (P, ≤), let P op = (P, ≤ op ) be the poset defined by x ≤ op y :⇔ y ≤ x.Using the fact that one has a natural identification ∆(P ) ∼ = ∆(P op ), one can draw the same conclusion as in the last lemma if one shows that Another result that we will frequently use is: Qui78]).If f, g : P → Q are poset maps such that f (x) ≤ g(x) for all x ∈ P , then they induce homotopic maps f , g on geometric realisations.In particular, if f : P → P is monotone, i.e. f (x) ≤ x for all x ∈ P or f (x) ≥ x for all x ∈ P , then f is homotopic to the identity.
Usually, the connectivity results one can obtain using fibre lemmas is bounded above by the degree of connectivity of the fibre.The following lemma gives a sufficient condition for obtaining a slightly better degree of connectivity.We will make use of it in Section 7.
Lemma 2.3.Let P and Q be connected posets and Proof.Applying Lemma 2.1, one gets that P is k-connected.
We now show that π k+1 (P ) also vanishes, which implies that P is in fact (k + 1)-connected.Consider a map i : S k+1 → P from the (k + 1)-sphere to P .Using simplicial approximation we can (after possibly precomposing with a homotopy) assume that i is simplicial with respect to a simplicial structure τ on S k+1 .We wish to show that i extends to a map î : B k+2 → P , where B k+2 is the (k + 2)-ball and î| ∂B k+2 = i.
Consider the simplicial map h )-connected, it extends to a map ĥ : B k+2 → Q such that ĥ| ∂B k+2 = h.Simplicial approximation applied to the pair (B k+2 , S k+1 ) (see e.g.[Spa66]) allows us to assume that ĥ is simplicial with respect to a simplicial structure τ on B k+2 such that τ agrees with τ on ∂B k+2 = S k+1 .For this, we might need to do barycentric subdivision and replace i by a homotopic map again.We now show that ĥ lifts to a map h : B k+2 → P such that h| ∂B k+2 = i by defining ĥ inductively on the simplices of τ .
To start, let v be a vertex of τ .If v ∈ τ , then h(v) := i(v); otherwise set h(v) to be any vertex in f −1 ( ĥ(v)).Now assume by induction that for m ≤ k+1, the map h has been defined on every (m−1)-simplex where q m−1 is the largest vertex in ĥ(σ m−1 ) and h restricts to i on τ .Let σ m be an m-simplex )) in π k+1 (P ) is trivial, the map h extends to σ.Thus we have shown that P is (k + 1)-connected.
2.2.The nerve of a covering.The nerve of a family of sets (X i ) i∈I is the simplicial complex N (X i ) i∈I that has vertex set I and where a finite subset σ ⊆ I forms a simplex if and only if i∈σ X i = ∅.The Nerve Theorem is another standard tool that exists in various versions.For simplicial complexes, it can be stated as follows: Lemma 2.4 ([Bjö95, Theorem 10.6]).Let X be a simplicial complex and (X i ) i∈I a family of subcomplexes such that X = i∈I X i .Suppose that every non-empty finite intersection X i 1 ∩. ..∩X i k is contractible.Then X is homotopy equivalent to the nerve N ((X i ) i∈I ).
2.3.Alexander duality and Whitehead's theorem.Alexander duality allows one to compute homology groups of compact subspaces of spheres by looking at the homology of their complement.We will need the following poset version of it which is due to Stanley.
Lemma 2.5 ([Sta82], [Wac07, Theorem 5.1.1]).Let P be a poset such that P is homeomorphic to an n-sphere and let Q ⊂ P be a subposet.Then for all i, one has Hi ( Q ; Z) ∼ = Hn−i−1 ( P \ Q ; Z).
In order to deduce information about the homotopy type of a space from its homology groups, we need a corollary of the theorems of Whitehead and Hurewicz.
Theorem 2.7 (Whitehead's theorem, [Hat02,Corollary 4 Corollary 2.8.Let X be a simply-connected CW-complex such that Then X is homotopy equivalent to a wedge of λ spheres of dimension n. Proof.By the Hurewicz theorem, X is in fact (n − 1)-connected and π n (X) ∼ = Hn (X) = Z λ .Now take a disjoint union µ≤λ S µ of n-spheres.For each µ ≤ λ, choose a generator S µ → X of the µ-th summand of π n (X).This gives rise to a map f : Y → X where Y is the space obtained by wedging together the S µ along their base points.This induces an isomorphism f * on all homology groups, so the claim follows from Whitehead's theorem.
Remark 2.9.A CW complex is n-spherical if it is homotopy equivalent to a wedge of n-spheres.By the preceding theorems, an n-dimensional complex X is n-spherical if and only if π i (X) is trivial for all i < n.

Outer space and its relatives
Throughout this section, let F be a free group of finite rank n ≥ 2.
3.1.Outer Space, its spine and the free splitting complex.Identify F with π 1 (R, * ) where R is a rose with n petals.A marked graph G is a graph of rank n equipped with a homotopy equivalence m : R → G called a marking.The marking determines an identification of F with π 1 (G, m( * )).
(Unreduced) Culler-Vogtmann Outer space CV n , defined in [CV86], is the space of equivalence classes of marked metric graphs G of volume one such that every vertex of G has valence at least three.Outer space can be decomposed into a disjoint union of open simplices, where the missing faces are thought of as "sitting at infinity".There is a natural simplicial completion obtained by adding the missing faces at infinity.The subspace of this completion consisting of all the open faces sitting at infinity is called the simplicial boundary ∂ s CV n of Outer space.
A free splitting S of F is a non-trivial, minimal, simplicial F-tree with trivial edge stabilisers.The vertex group system of a free splitting S is the (finite) set of conjugacy classes of its vertex stabilisers.Two free splittings S and S are equivalent if they are equivariantly isomorphic.We say that S collapses to S if there is a collapse map S → S which collapses an F-invariant set of edges.The poset of free splittings FS n is given by the set of all equivalence classes of free splittings of F where S ≤ S if S collapses to S .The free splitting complex is the order complex ∆(FS n ) of the poset of free splittings.Outer space naturally embeds as a subspace of FS n .
In fact, the free splitting complex is naturally identified with the barycentric subdivision of the simplicial completion of CV n .Each free splitting S can equivalently be seen as a graph of groups decomposition of F with trivial edge groups by taking the quotient S/F.We will often adopt this point of view in later sections without further notice.
The spine L of CV n is given by the subposet of FS n consisting of all free splittings that have trivial vertex stabilisers.We can interpret L as a subspace of CV n .It consists of all marked metric graphs G satisfying the following property: The subgraph spanned by the set of all edges of G not having maximal length forms a forest.In [CV86], Culler and Vogtmann showed that L is a contractible deformation retract of CV n .By the definitions above, we have a homeomorphism An edge e of a graph G is called a separating edge if removing it from G results in a disconnected graph.The subspace of CV n consisting of all marked graphs that do not contain separating edges is called reduced Outer space, denoted CV r n .It is an equivariant deformation retract of CV n .Similarly to the unreduced cased, there is a poset K such that CV r n retracts to K .It is the subposet of L consisting of all marked graphs having no separating edges and is called the spine (of reduced Outer space).
The barycentric subdivision of the simplicial closure of reduced Outer space is given by the order complex of the poset FS r n consisting of all those free splittings S ∈ FS n such that the quotient S/F does not have any separating edges.Just as in the unreduced case, we have 3.2.Relative Outer space and its spine.In [GL07], Guirardel and Levitt define relative Outer space for a countable group that splits as a free product They also prove contractibility of relative Outer space.We will later on consider the case where G = F splits as ]} be the associated free factor system of F. Subgroups of F that are conjugate into a free factor in A are called peripheral subgroups.An (F, A)-tree is an R-tree with an isometric action of F, in which every peripheral subgroup fixes a unique point.Two (F, A)-trees are equivalent if there exists an F-equivariant isometry between them.A Grushko (F, A)-graph is the quotient by F of a minimal, simplicial metric (F, A)-tree, whose set of point stabilisers is the free factor system A and edge stabilisers are trivial.Relative Outer space is the space of homotethy classes of equivalence classes of Grushko (F, A)-graphs.The spine of relative Outer space, denoted by L(F, A), is the subposet of FS n consisting of all free splittings whose system of vertex stabilisers is given by A. Its realisation can be seen as a subspace of relative Outer space.Since relative Outer space deformation retracts onto its spine, L(F, A) is contractible.

Posets of graphs
In this section, we study (finite) posets of subgraphs of a given graph G.For the combinatorial arguments we use, let us set up the following notation: In what follows, all graphs are allowed to have loops and multiple edges.For a graph G, we denote the set of its vertices by V (G) and the set of its edges by E(G).If e ∈ E(G) is an edge, then G − e is defined to be the graph obtained from G by removing e and G/e is obtained by collapsing e and identifying its two endpoints to a new vertex v e .A graph is called a tree if it is contractible.It is called a forest if it is a disjoint union of trees.
Throughout this section, we will only care about edge-induced subgraphs, i.e. when we talk about a "subgraph H of G", we will always assume that H is possibly disconnected but does not contain any isolated vertices.Hence, we can interpret any subgraph of G as a subset of E(G).Definition 4.1.A core subgraph H of a graph G is a proper subgraph such that the fundamental group of each connected component of H is non-trivial and no vertex of H has valence one in H. Every graph G contains a unique maximal core subgraph that we will refer to as the core of G, denoted by G.
Note that, in contrast to the convention introduced in [BF00], our core subgraphs are allowed to have separating edges.
4.1.The poset of all core subgraphs.Definition 4.2.Let G be a graph.We define the following posets of subgraphs of G; all of them are ordered by inclusion: (1) Sub(G) is the poset of all proper subgraphs of G that are non-empty.Equivalently, Sub(G) can be seen as the poset of all proper, non-empty subsets of E(G).
(2) For(G) denotes the poset of all proper, non-empty subgraphs of G which are forests.
(3) X(G) is defined to be the poset of proper subgraphs of G that are non-empty and where at least one connected component has non-trivial fundamental group.(4) C(G) is the poset of all proper core subgraphs of G.

Clearly one has
Examples of the realisation of X(G) can be found in the Appendix, see Figure 9.The proof of the following lemma is fairly standard and we will use the argument several times throughout this article.For the sake of completeness, here we will spell it out once.

Lemma 4.3. X(G) deformation retracts to C(G).
Proof.Every subgraph H ∈ X(G) contains a unique maximal core subgraph H and if H 1 ⊆ H 2 , one has H1 ⊆ H2 .Hence, sending each H to this core subgraph H defines a poset map f : X Proposition 4.4.Let G be a finite connected graph whose fundamental group has rank n ≥ 2 and assume that every vertex of G has valence at least 3. Then X(G) is contractible if and only if G has a separating edge.If G does not have a separating edge, then X(G) is homotopy equivalent to a wedge of spheres of dimension n − 2.
Proof.Note that Sub(G) can be seen as the poset of all proper faces of a simplex with vertex set E(G).Hence, its realisation Sub(G) is homeomorphic to a sphere of dimension |E(G)| − 2.
By [Vog90, Proposition 2.2], the poset For(G) is contractible if and only if G has a separating edge and is homotopy equivalent to a wedge of (|V (G)| − 2)-spheres if it does not contain a separating edge.We want to use Alexander duality as stated in Lemma 2.5 to describe the homology groups of X(G) = Sub(G) \ For(G).If G has a separating edge, it immediately follows from Alexander duality that all reduced homology groups of X(G) vanish.If on the other hand G does not have a separating edge, then the only non-trivial homology group of X(G) appears in dimension where it is given by a direct sum of Z's.
We next want to show that for n ≥ 4, the realisation of X(G) is simply-connected in order to apply Whitehead's theorem.
Denote by Sub(G) (k) the subposet of Sub(G) given by those subgraphs having precisely (|E(G)|− k) edges.As n ≥ 4, removing at most three edges from G results in a graph with non-trivial fundamental group.Hence, we have Sub(G) forms a subspace of X(G) that is homeomorphic to the 2-skeleton of an (|E(G)| − 2)-simplex.In particular, it is simply-connected.Now let ρ be a closed edge path in X(G) given by the sequence of vertices (H = H 1 , H 2 , ..., H k = H).We want to show that it can be homotoped to a path in Sub(G) (1) ∪ Sub(G) (2) .Whenever we have an edge (H i−1 ⊂ H i ) such that H i has at least two edges less than G, there is a subgraph and hence assume that every second vertex crossed by ρ lies in Sub(G) (1) (see the left hand side of Figure 1).Next take a segment ( This argument shows that every closed path can be homotoped to a path that lies in Sub(G) (≤3) .As this is a simply-connected subset of X(G) , it follows that X(G) itself is simply-connected for n ≥ 4. Applying Corollary 2.8 yields the result.
The only cases that remain are those where n = 2 or 3.However, as we assumed that every vertex of G has valence at least 3, there are only finitely many such graphs.Using Lemma 4.3, it is not hard to verify the claim using a case-by-case analysis.For completeness, the proof for n = 3 can be found in the Appendix 9.
Remark 4.5.Assuming that each vertex of G has valence at least 3 does not impose any restrictions for the considerations in this article as every graph in Outer space satisfies this condition.However, note that we only used this assumption in the case where n = 2 or 3 and there it only shortened the argument and could easily be dropped.4.2.The poset of connected core subgraphs.Definition 4.6.For a graph G, we define cX(G) to be the poset of all proper connected subgraphs of G which are not trees ordered by inclusion.Let cC(G) by the poset of all proper connected core subgraphs of G.
Later on, we will be interested in the homotopy type of cC(G) as it occurs as the fibre of a map we will use to study higher connectivity of F n .However, it is easier to describe the structure of cX(G), so we set up the following Lemma.
Proof.As we allow our core subgraphs to have separating edges, the unique maximal core subgraph H contained in a connected non-tree subgraph H is connected as well.Hence, sending H to H defines a monotone poset map cX(G) → cC(G).Using Lemma 2.2 as in the proof of Lemma 4.3, the claim follows.
Definition 4.8 (Valence-2-homotopy). Let G be a finite connected graph and v ∈ V (G) be a vertex of valence two with adjacent edges e 1 = e 2 .We define G v to be the graph obtained from G by replacing the segment e 1 ve 2 by a new edge e v ; i.e.
and e v connects the endpoints of e 1 and e 2 which are not equal to v.
The graphs we want to study have no vertices of valence smaller than 3.In order to preserve this property throughout the induction procedure used in the proof of Proposition 4.10, we need the following: Lemma 4.9.Let G be as in Definition 4.8.Then cX(G) cX(G v ) .
Proposition 4.10.Let G be a finite connected graph whose fundamental group has rank n ≥ 2 and assume that every vertex of G has valence at least 3. Then cX(G) is homotopy equivalent to a wedge of (n − 2)-spheres.
Proof.We do induction on n.If n = 2, there are exactly three graphs having only vertices of valence at least 3.It is easy to check that for all of them, the poset of proper connected core subgraphs is a disjoint union of two or three points, i.e. a wedge of 0-spheres.Now assume G is a graph whose fundamental group has rank n > 2. If every edge of G is a loop, G is a rose with n petals and every proper non-empty subset of E(G) is an element of cX(G).Hence, the order complex of cX(G) is given by the set of all proper faces of a simplex of dimension n − 1 whose vertices are in one-to-one correspondence with the edges of G. Now assume that G has an edge e which is not a loop.Whenever H ∈ cX(G), the set H \ {e} can be seen as a connected non-tree subgraph of G/e.If H is not equal to G − e, then H \ {e} is a proper subgraph of G/e.Consequently, we get a poset map On the other hand, if we take a subgraph K ∈ cX(G/e) that contains the vertex v e to which e was collapsed, it is easy to see that K ∪ {e} is an element of cX(G) \ {G − e}.This way, we can define a poset map One has ψ • φ(H) ⊇ H and φ • ψ(K) = K, so using Lemma 2.2, the realisations of these two posets are homotopy equivalent.When e is a separating edge, the graph G − e is not connected so in particular not an element of cX(G).It follows that cX(G) is homotopy equivalent to cX(G/e) .As G/e has one edge less than G and every vertex in G/e has valence at least 3, we can apply induction.
If on the other hand e is not a separating edge, G − e is a connected graph having the same number of vertices as G and one edge less.This implies that rk(π 1 (G − e)) = n − 1.After possibly applying Lemma 4.9, we may assume that each vertex in G − e has valence at least 3. cX(G) is obtained from cX(G) \ {G − e} by attaching the star of G − e along its link.The link of G − e in cX(G) is naturally isomorphic to cX(G − e) which is by induction homotopy equivalent to a wedge of (n − 3)-spheres.The star of a vertex is always contractible and gluing a contractible set to an (n − 2)-spherical complex along an (n − 3)-spherical subcomplex results in an (n − 2)-spherical complex, so the claim follows (see e.g.[BSV18, Lemma 6.2]).

Contractibility of relative free splitting complexes
For a free splitting S, let V(S) denote its vertex group system.Given a free factor system A in F, the poset of free splittings of F relative to A, denoted FS n (A), is the subposet of FS n consisting of all free splittings S ∈ FS n such that A V(S).Its realisation is the relative free splitting complex studied in [HM14], where the authors showed that it is non-empty, connected and hyperbolic.
The aim of this section is to show that for every free factor system A in F, the poset FS n (A) is contractible (Theorem 5.17).In order to prove this, we work with several relativisations of the complexes in question and need to introduce some notation in order to keep track of them.To make the proof more accessible, we will first give an informal outline before spelling out the technicalities in greater detail.5.1.Outline of the proof.Fix a free factor system A in F. The poset of all free splittings of F having vertex group system equal to A is nothing else but the spine L(F, A) of Outer space relative to A, so we know that it is contractible.This way, we can see FS n (A) as being assembled from the contractible pieces L(F, B) where B ranges over all free factor systems with A B. In order to understand the (order) relation between these different pieces, we need a way of organising them.The natural choice is to use the ordering of free factor systems in FF n .
Roughly speaking, the order " " of FF n is coarser than the one on FS n : It is true that if S is greater or equal to S in FS n , i.e. if there is a collapse map S → S , one has V(S) V(S ).Hence, the map FS n → FF n , sending S to V(S) is an order-inverting poset map.However, if one is given S ∈ FS n such that A V(S), it is not necessarily true that there is a collapse map S → S such that S has vertex group system V(S ) = A.
Recall that in the order complex of FS n , an edge between S and S is added if and only if one collapses to the other.So in order to understand how the spines of the different relative Outer spaces are glued together to form all of the relative free splitting complex, we need to understand the following situation: If B A, which elements S ∈ L(F, B) collapse to some free splitting S ∈ L(F, A)? Adopting the graph of groups point of view, one can intuitively see S as being obtained from S by collapsing a subgraph of groups.This is why in this case, we will say that S has a "subgraph with fundamental group A" and denote the poset of all such S ∈ L(F, B) by X(B : A) (see below for the precise definitions).Those are the posets whose connectivity properties we want to understand.
Eventually, our inductive argument requires us to consider intersections and unions of such X(B : A) as well, so we are lead to consider slightly more general versions of these posets and need to show that they all are contractible.

Definitions and Notation.
Here is some notation which is used in this section.
• We will drop the square brackets used to denote the conjugacy class of a free factor.
• For G ∈ L and H a subgraph of G, we will write G/H for the splitting obtained by collapsing H. • We will use the letter G to denote a free splitting in CV n or L (which hence has trivial vertex groups) and use S to denote any splitting in FS n .
Let G be a graph in L and A a conjugacy class of finitely generated subgroups of F.
Definition 5.1 (A|G).In [BF14], Bestvina and Feighn define A|G to be the core of the covering space of G corresponding to A. There is a canonical immersion from A|G into G which gives A|G a marking.We say G has a subgraph with fundamental group A if A|G → G is an embedding.
For example, take G to be a rose with 3 petals and labels a, b, c.Consider the subgroups A 1 = abc and A 2 = a, b of F 3 = a, b, c .Then A 2 |G → G is an embedding and we say G has a subgraph with fundamental group A 2 .But A 1 |G → G is an immersion which is not an embedding.See Figure 2.
We extend the above definition to any free splitting S ∈ FS n .
Definition 5.2 (A|S).Let A be the conjugacy class of a proper free factor such that for every V ∈ V(S), the intersection A ∩ V is either trivial or equal to V .Consider a blow-up Ŝ ∈ L of S obtained by blowing up all the vertex groups of S to roses.Then there is an immersion A| Ŝ → Ŝ.
We say S has a subgraph with fundamental group A or A|S is a subgraph of S if A| Ŝ → Ŝ is an embedding for (some such) Ŝ ∈ L. Define A|S to be the graph obtained by collapsing in A| Ŝ the roses corresponding to each In order to see that A|S is well-defined, consider two splittings Ŝ and Ŝ defined as in Definition 5.2.Such splitting can only differ in the roses corresponding to the vertex stabilisers of S. Thus collapsing the roses for V ∈ V(S) in A| Ŝ and A| Ŝ yields the same graph A|S.See Figure 3 for some examples of A|S.b, c, d, e, f , A Definition 5.3 (A|S).Let A be a free factor system of F such that V(S) A. Define A|S to be the collection {A|S} A∈A .We say S has a subgraph with fundamental group A or A|S is a subgraph of S, if for every A ∈ A, the splitting S has a subgraph with fundamental group A.
Definition 5.4.For a chain of free factor systems of F given by B 0 . . .B l A 0 . . .A m , • let X(1 : A 0 , . . ., A m ) be the poset of all free splittings S such that V(S) is trivial and A i |S is a subgraph of S for every 0 ≤ i ≤ m; • let X(B 0 , . . ., B l : A 0 , . . ., A m ) be the poset of all free splittings S such that one has V(S) ∈ {B 1 , . . ., B l } and A i |S is a subgraph of S for every 0 ≤ i ≤ m.See Figure 4 for examples of these posets.The main technical result we need to prove in order to show contractibility of the relative free splitting complex is the following: Proposition 5.5.For a chain of free factor systems of F given by B 0 . . .B l A 0 . . .A m , the posets X(1 : A 0 , . . ., A m ) and X(B 0 , . . ., B l : A 0 , . . ., A m ) are contractible.This is proved over the next three subsections.We will first prove contractibility in the special case X(1 : A 0 ) in Section 5.3.Then we generalise the arguments to X(1 : A 0 , . . ., A m ) in Section 5.4 and to X(B 0 , . . ., B l : A 0 , . . ., A m ) in Section 5.5 where it is used to show that FS n (A) is contractible.5.3.Contractibility of X(1 : A 0 ).Before we start to prove contractibility of X(1 : A 0 ), note that whenever one has a marked graph (G, m), every valence-2-homotopy of the combinatorial graph G (as defined in Definition 4.8) induces a valence-2-homotopy of marked graphs, changing (G, m) to (G v , m ) for a marking m , well-defined up to equivalence of marked graphs (see [BH92]).
Proof.We will use the technique of Stallings' folding paths to exhibit connectedness.The reader is referred to [BF14] for details on folding paths.We will also view X(1 : A 0 ) ⊂ L as a subset of CV n .
Let's first suppose that A 0 is a single free factor A. Let G, G ∈ X(1 : A).By definition, A|G (resp.A|G ) is a subgraph of G (resp.G ).By collapsing some forest we may assume A|G (resp.A|G ) is a rose and all edges of G \ A|G (resp.G \ A|G ) incident at A|G (resp.A|G ) are actually incident at the vertex of A|G (resp.A|G ).Both A|G and A|G can be viewed as points in the spine of Outer space of A, denoted L(A).Since L(A) is connected, there is a folding path between A|G and A|G guided by the change of marking f A : A|G → A|G .
Let G be the graph obtained after replacing A|G by A|G in G.The graph G is well-defined and unique because A|G and A|G are roses and all edges of G incident at A|G are in fact incident at the vertex of A|G.Extend f A by the identity on G \ A|G to a change of marking f 1 : G → G .Then the folding path in CV n guided by f 1 projects to a path in X(1 : A).Also the change of marking f 2 : G → G restricts to the identity on A|G .Thus the folding path guided by f 2 also projects to a path in X(1 : A).Then the concatenation of f 1 and f 2 projects to a path in X(1 : A) connecting G and G .
If A 0 has more than one free factor, then by doing the above process one at a time for each free factor in A 0 we get a path in X(1 : A 0 ) connecting G and G .
Let's now lay out the set-up for proving contractibility of X(1 : A 0 ).Let A 0 = {A 1 , . . ., A k }.Recall L(F, A 0 ) denotes the spine of the Outer space of F relative to A 0 and L(A i ) is the spine of the Outer space of the free group A i .Denote by ) be the product of these posets.Consider the poset map such that Ψ(G) := (Γ, H) where H = A 0 |G and Γ = G/H.After performing valence-2-homotopies, we may assume Given (Γ, H) ∈ L(F, A 0 ) × L(A 0 ), we can view G as being obtained from Γ by blowing up the vertex v i stabilised by A i into the graph H i .When trying to reconstruct G from the pair (Γ, H), one has to face two ambiguities: The first one occurs because for each i, one can choose where to attach the edges of Γ which are adjacent to v i to the graph H i .The second ambiguity arises because before blowing up v i to H i , one can change the marking of each of the adjacent edges of Γ by an element of π 1 (H i ) = A i .These two choices can be used to parametrise the fibre Ψ −1 (Γ, H).
This point of view is the basis for the proof of Lemma 5.7.We will now use it to show that the fibres of the map Ψ are contractible.Then because of the contractibility of L(F, A 0 ) and L(A 0 ), and using Lemma 2.1, it will follow that X(1 : A 0 ) is contractible.
Proof.For simplicity, let us first assume that A 0 is a single free factor A and that A has rank at least 2. Subdivide all loops incident at the vertex v of Γ that is stabilised by A into two edges.Let m be the number of edges of Γ incident at v and let E 1 , . . ., E m denote the outgoing edges incident at v. Let H be the universal cover of H.The space Ψ −1 (Γ, H) is the subcomplex of X(1 : A 0 ) spanned by marked graphs obtained by blowing up the vertex v of Γ to the subgraph H. Its geometric realisation Ψ −1 (Γ, H) can be naturally seen as a subspace of CV n and we claim that it is homeomorphic to Hm , the m-fold product of H.We will now construct a function f : Hm → Ψ −1 (Γ, H) and show that it is a homeomorphism.The idea for defining the map is that a gluing point on H for an edge E i of Γ is parametrised by a point in H.
We first set up some notation and choose a lift of H to its universal cover H. Let e 1 , e 2 , . . ., e q be the collection of edges of H. Choose a base point v 1 in H and a maximal forest F H . Using the marking of H, the edges of H not contained in F H are labelled and oriented.Also choose orientations for the edges in F H . Let o j and t j denote the initial and terminal end points of the edge e j , respectively.Denote the label of e j by α j ∈ A, where α j is trivial if e j ∈ F H . Consider a metric on H where each edge has length one.Choose a lift ṽ1 of v 1 in H. Let FH be the lift of F H that contains ṽ1 .Let v 1 , . . ., v l denote the vertices of H and ṽ1 , . . ., ṽl the respective lifts which are contained in FH .Let ẽj be the lift of e j such that õj ∈ FH .The tree H gets the lifted metric from H.
Consider a point P = (p 1 , . . ., p m ) ∈ Hm .If p i is a vertex of H, then there exists h i ∈ π 1 (H, v 1 ) and 1 ≤ j ≤ l such that p i = h i ṽj .If p i is in the interior of an edge of H, then there is a j such that p i is specified by the pair (h i ẽj , l j (i)) for h i ∈ π 1 (H, v 1 ) and l j (i) ∈ (0, 1).Given P , we will first construct a marked metric graph (G, m, ) in CV n .Then we will show that this marked metric graph is actually in L seen as a subspace of CV n .
• The graph G : For each 1 ≤ j ≤ q, order the numbers 0 < l j (i 1 ) ≤ • • • ≤ l j (i j ) < 1.Now subdivide the edge e j of H (which has length one) according to the numbers l j (i r ), i 1 ≤ i r ≤ i j and denote the vertices by u j (i r ).It is possible that a vertex has multiple labels.Let H be the graph obtained from H by this subdivision.The graph G is obtained as follows: if p i = (h i ẽj , l j (i)) or h i ṽj , then attach the initial end point of E i at the vertex u j (i) of H . • The marking m: For each 1 ≤ j ≤ q, if there exists an l j (i r ) ∈ (0, 1), then e j gets subdivided.Define a marking of H as follows: for every edge e j that gets subdivided, label the edge [u j (i j ), t j ] of H , where t j denotes the terminal end point of e j , by α j .For the edges that didn't get subdivided, keep the same label as in H.For p i = (h i ẽj , l j (i)) or h i ṽj , mark the edge E i by multiplying the label it inherits from Γ from the left by h i .The remaining edges of G retain the marking from Γ.To see that this is indeed a marking of G -collapse the edges of H that are unlabelled (α j being trivial is considered a labelling).Then for an edge E i which got a prefix h i , the last letter of h i coincides with a label of an edge incident at the initial vertex of E i and hence can be folded away (using Stalling's folds).Continue inductively.• The metric : We say an edge of H is an edgelet if at least one of its end points is a valence 2 vertex of H .Each edgelet e of H has a length l (e) induced by the metric on H.For each edge e j of H that got subdivided, let l j (max) be the length of a longest edgelet contained in e j .Define l j (max) to be 1 if e j didn't get subdivided.Let M = q j=1 l j (max).Now assign lengths to the edges of H as follows: For each edgelet e that was part of e j ∈ E(H), set the length of e to be M l j (max) • l (e).Edges of Γ are assigned the length M .Thus we get a metric on G. Now normalise the metric on G to volume one.Now set f (P ) = (G, m, ).We claim that f (P ) ∈ Ψ −1 A (Γ, H) .Indeed, the set C of edges of G of non-maximal length is precisely the set of non-maximal edgelets of H .These form a forest, so f (P ) is an element of L , seen as a subspace of CV n .Furthermore, f (P ) by construction contains a subgraph which differs from H only by valence-2-homotopies and collapsing this subgraph yields the marked graph Γ.
The image f (P ) depends continuously on P : The definition of f is set up in a way such that moving the point P ∈ H m inside the product of these universal covers corresponds to sliding the "feet" of the edges E i , i.e. the points to which these edges are attached, along the graph H. Assume one has P ∈ H m and slightly perturbs it such that: none of its coordinates crosses a vertex of H; and for each edge e j of H, the order l j (i 1 ) ≤ l j (i 2 ) ≤ • • • ≤ l j (i j ) given by the positions of the attaching points of the E i on e j does not change, i.e. no foot overtakes another one.Then neither the combinatorial graph nor the marking of f (P ) change.Hence, f (P ) moves inside an open simplex of CV n .This movement is specified by the metric of f (P ) and it is not hard to see that it depends continuously on P .If on the other hand one of the coordinates passes a vertex of H or the order of the attaching points on some edge of H changes, the image f (P ) continuously moves to an adjacent simplex of CV n passing through a face of smaller dimension.
Next we claim that f is a homeomorphism.Indeed, the map f is injective because for two different points in Hm their images will, by construction, differ either in the combinatorial graph, or the marking or the metric.To see that f is also surjective, take a point (G, m, ) ∈ Ψ −1 (Γ, H) .Then the data from G and m provides the information h i ẽj or h i ṽj and the metric allows to solve for the lengths l j (i r ) to give the precise gluing points.
If A is a rank 1 free factor, then a blow-up of Γ is invariant under conjugation by elements of A. Thus fixing the attaching point for (any) one edge incident at v, we get that Ψ −1 (Γ, H) is homeomorphic to H m−1 .Since H is contractible, we get the desired result.Now consider the general case when A 0 = {A 1 , . . ., A k } and (Γ, H) = (Γ, H 1 , . . ., H k ).Let m i be the number of edges of Γ (after subdividing the loops) incident at the vertex of Γ which has vertex group then the exponent changes to m i − 1).To define the homeomorphism, take P ∈ H1 and apply the blow-up construction described above at each vertex v i independently to get a (combinatorial) marked graph (G, m).The graph G now has a (disconnected) subgraph H whose components are subdivisions of H 1 , . . ., H k .The metrics on the H i determine a length function on the edges of H .This allows us just as above to define a metric on G giving the same length to all maximal edgelets of H and the edges coming from Γ.
be seen as a subgraph of G .We claim that this subgraph is a forest.Suppose not.Then there is a loop l ∈ F crossing each edge at most once.As F H := k i=1 F H i ⊂ H is a forest, we know that l cannot be completely contained in H ⊂ G .Hence, collapsing H maps l to a non-trivial loop in F Γ ⊂ Γ which is a contradiction.
Collapsing F ⊂ G , we get a graph G which lies in Ψ −1 (Γ, H).This defines a map The claim now follows from Lemma 2.2.
Proof.Using Lemma 5.7, Lemma 5.8 and applying Lemma 2.1 to the map Ψ, we conclude that X(1 : A 0 ) is contractible.
Proof.Given G, G ∈ X(1 : A 0 , . . ., A m ), we will construct a folding path between G and G by "folding one free factor system at a time".Using Lemma 5.6 and passing to appropriate graphs by collapse, we begin by performing, on G, the folds dictated by the change of marking between A 0 |G and A 0 |G .Let the new graph be denoted by G 1 ∈ X(1 : A 0 , . . ., A m ).Now perform on G 1 the folds guided by the change of marking from A 1 |G 1 to A 1 |G to get a graph G 2 .If G t denotes a graph in this fold path, then A 0 |G t is isomorphic to A 0 |G and hence is not affected by the folding.Continue this process for all A 2 , . . ., A m to get a path connecting G and G in X(1 : A 0 , . . ., A m ).
Notation 5.11.In this subsection, we add extra decoration to the notation X(1 : A) to emphasise that this poset contains free splittings of F. We do this by writing X(1 is the poset of free splittings S of A that have trivial vertex groups and such that B|S is a subgraph of S. We recall some terminology used to describe a rooted tree that will be used in the proof of the next lemma.In a rooted tree, the parent of a vertex is the vertex connected to it on the path to the root; every vertex except the root has a unique parent.A child of a vertex v is a vertex of which v is the parent.A descendant of a vertex v is any vertex which is either the child of v or is recursively the descendant of any of the children of v.The height of a vertex in a finite rooted tree is defined to be the length of the longest downward path to a leaf from that vertex.The depth of a vertex is defined to be the length of the path to its root.Definition 5.12 (Rooted tree T for {A 0 , . . ., A m }).For 0 ≤ i ≤ m, let A i = {A i,1 , . . ., A i,l i }.Let A be the collection of conjugacy classes of free factors in A i as i varies from 0 to m.Then A is a partially ordered set under inclusion.We define a labelled rooted tree of free factors, denoted T, which captures the partially ordered set A. The root is labelled by F. The children of F are labelled by the free factors in A m .The set A is the set of descendants of the root.As i varies from 1 to m, the children of a vertex labelled A i,j are given by A i−1,r for all r such that A i−1,r ⊆ A i,j up to conjugation.
Note that the collection of free factors in T at depth i > 0 is the free factor system A i .
Proof.Let T be the labelled rooted tree for the chain A 0 . . .A m .Let A 1 be a height one vertex of T. Then its children form a free factor system is in fact equal to L(A 1 ), which is contractible.Otherwise, the poset X(1 : B 1 )[A 1 ] is contractible by Lemma 5.9 applied to the pair (A 1 , B 1 ) instead of the pair (F, A 0 ).
Let A h be a height h vertex of T and let T(A h ) be the subtree of T rooted at A h .In T(A h ), the free factors at depth i > 0 form a free factor system B h i of A h .Thus we have a chain B h h . . .B h 1 of free factor systems of A h .Since we are interested in the poset X(1 we may assume that the B h i are all distinct.By induction on height, suppose X(1 ] is contractible for all h ≤ h 0 for some h 0 > 0. Then for a height h 0 + 1 vertex A h 0 +1 , we will now show that X(1 : . Now apply Lemma 5.7 and Lemma 5.8 to the map Ψ.Note that the only difference is that for every i the contractible poset L(A i ) is replaced by its subposet X(1 : which is also contractible.Thus by Lemma 2.1, we get contractibility of X(1 : ].By induction on height, we conclude that X(1 . Rooted tree for the proof of Lemma 5.13.5.5.Contractibility of X(B 0 , . . ., B l : A 0 , . . ., A m ).Recall that for a chain of free factor systems B 0 . . .B l A 0 . . .A m , we defined X(B 0 , . . ., B l : A 0 , . . ., A m ) to be the poset of all free splittings S such that V(S) ∈ {B 0 , . . ., B l } and for every 0 ≤ i ≤ m, the core A i |S is a subgraph of S.Here we allow A m to be the "non-proper" system A m = F and interpret X(B 0 , . . ., B l : F) as the poset of all free splittings S such that V(S) ∈ {B 0 , . . ., B l }.
Proof.We first show that X(B : A 0 , . . ., A m ) is connected for a free factor B. For this, let S, S ∈ X(B : A 0 , . . ., A m ) and let G ∈ L (resp.G ) be obtained by blowing up the vertex stabilised by B in S (resp.S ).Then G, G ∈ X(1 : B, A 0 , . . ., A m ) which is connected by Lemma 5.10.A path G t between G and G in X(1 : B, A 0 , . . ., A m ) can be projected to a path in X(B : A 0 , . . ., A m ) by collapsing B|G t in every G t .Now for connectedness of X(B 0 : A 0 , . . ., A m ) we can use connectedness of X(1 : B 0 , A 0 , . . ., A m ) together with the argument in the previous paragraph.Note that X(B 0 , . . ., B l : A 0 , . . ., A m ) is a union of X(B i : A 0 , . . ., A m ) for 0 ≤ i ≤ l, each of which is connected.For i < j, there exist splittings S i ∈ X(B i : A 0 , . . ., A m ) and S j ∈ X(B j : A 0 , . . ., A m ), such that S i collapses onto S j , namely S i is such that B j |S i is a subgraph of S i .Thus X(B i : A 0 , . . ., A m ) and X(B j : A 0 , . . ., A m ) are connected in X(B 0 , . . ., B l : A 0 , . . ., A m ).Hence we conclude that X(B 0 , . . ., B l : A 0 , . . ., A m ) is connected.
For simplicity of the exposition, we will first prove contractibility of X(B 0 , . . ., B l : A 0 , . . ., A m ) in special cases.The purpose of the next lemma is basically to remark on the blow-up construction of Lemma 5.7 in the case when the free splittings have non-trivial vertex groups and to provide the base case for Lemma 5.16 .
Lemma 5.15.Let A, B be conjugacy classes of free factors and B 0 , A 0 , . . ., A m free factor systems of F, such that B ⊂ A and B 0 A 0 . . .A m .Then (1) X(B : A) is contractible.
(2) X(B 0 : Proof.X(B 0 : F) is equal to the spine of the Outer space of F relative to B 0 which we know to be contractible.That is why we will assume that there are proper inclusions of free factor systems A 0 , A F.
(1) For each G ∈ X(B : A), the subgraph A|G can (after performing valence-2-homotopies) be seen as an element of L(A, B).This allows us to define a poset map as follows: for G ∈ X(B : A), set Ψ(G) := (Γ, H) where H = A|G and Γ = G/H.We claim that Ψ −1 (Γ, H) is homeomorphic to Hm , where m is the number of half-edges of Γ incident at the vertex v of Γ stabilised by A and H is the Bass-Serre tree of H.We associate a unique marked metric graph (G, m, ) to a point P ∈ Hm in exactly the same manner as in Lemma 5.7.It is not hard to see that G and are well-defined in this setting.But perhaps it is not immediately clear that m is well-defined because H has a non-trivial vertex group.Using notation from Lemma 5.7, suppose the attaching point of an edge E i of Γ is specified by p i = (h i ẽj , l j (i)) or h i ṽj in H, where h i ∈ A. Furthermore, suppose that the label of E i is obtained by multiplying the label it inherits from Γ from the left by h i .If p i lies in the interior of an edge of H, the element h i ∈ A is well-defined because the edge stabilisers of H are trivial.If p i = h i ṽj is a vertex of H with non-trivial stabiliser, the element h i is only well-defined up to right multiplication with an element of the vertex stabiliser Stab A (ṽ j ) which is contained in the conjugacy class B. In this case however, the terminal end point of E i gets attached to the vertex v j which has vertex group Stab(ṽ j ).It follows that any element from the coset h i • Stab A (ṽ j ) determines the same marking of G.
(2) For A 0,i ∈ A 0 , let B 0,i be the collection of free factors in B 0 that are contained in A 0,i .Thus B 0,i is a (possibly trivial) free factor system of A 0,i .If B 0,i = ∅, then L(A 0,i , B 0,i ) is equal to L(A 0,i ) and if B 0,i = A 0,i , then L(A 0,i , B 0,i ) is just a point.Consider the poset map Then Ψ can be shown to be homotopy equivalence by applying part (1) simultaneously to all free factors in A 0 .
(3) This follows by a very slight modification of the rooted tree construction of Lemma 5.13.
Form the rooted tree T for the chain A 0 . . .A m and for each vertex A h of height h, consider the poset X(B(h where B(h) is the collection of free factors in B 0 that are contained in A h .By induction on height and using part (2), we may assume that X(B(h) : B h h , . . ., B h 1 )[A h ] is contractible for all h ≤ m.Then to show that the poset associated to the height m + 1 vertex F is contractible, we use the same arguments as in Lemma 5.13.Lemma 5.16.X(B 0 , . . ., B l : A 0 , . . ., A m ) is contractible.In particular, X(B 0 , . . ., B l : F) is contractible.
Then in particular, the posets as well as X l := X(B l : A 0 , . . ., A m ) are contractible.By definition X l−1,l is the subposet of X l−1 consisting of all those S ∈ X l−1 that collapse to some free splitting in X l .For each such S ∈ X l−1 , there is a unique maximal splitting S ∈ X l , on which S collapses, namely S = S/(B l |S).Hence, the map It follows that X(B 0 , . . ., B l : A 0 , . . ., A m ) = X l−1 ∪ X l is obtained by gluing together X l−1 and X l along X l−1,l .Now X l−1 , X l−1,l and X l are contractible by assumption, whence the claim follows.
We are now ready to prove Theorem C which we restate as follows: Theorem 5.17.
(1) For all free factor systems A in F, the poset of free splittings FS n (A) of F relative to A is contractible.
(2) For all conjugacy classes A of free factors in F, the poset FS 1 n (A) consisting of all free splittings having exactly one non-trivial vertex group B ⊇ A is contractible.
Proof.We start by proving the first claim: Each simplex σ in the order complex ∆(FS n (A)) is of the form S 0 → . . .→ S k where each S i is a free splitting of F collapsing to S i+1 .Furthermore, the vertex group systems of these free splittings form a chain V(S 0 ) . . .V(S k ) of free factor systems such that A V(S i ) for all i.It follows that σ is contained in ∆(X(A, V(S 0 ), . . ., V(S k ) : F)).Hence the realisation FS n (A) can be written as a union By Lemma 5.16, each X(A, A 1 , . . ., A l : F) is contractible.Furthermore, one has Consequently, all intersections of these sets are contractible and Lemma 2.4 implies that FS n (A) is homotopy equivalent to the nerve of this covering.However, as all of these sets contain X(A : F) , they intersect non-trivially, so this nerve complex is contractible.
By the same arguments, FS 1 n (A) is homotopy equivalent to the nerve of its covering given by all the set X(A, A 1 , . . ., A l : F) where each A i is the conjugacy class of a free factor in F such that A A i .Again, the intersection of all of these sets is non-empty as it contains X(A : F) , so the second claim follows.

Factor complexes at infinity
In this section, we are interested in various subposets of FS n which sit at the boundary of the simplicial completion of CV n .Namely, we are interested in FS n \ L, FS 1 n and FS r n \ K where FS 1 n is defined to be the subposet of FS n given by free splittings that have exactly one non-trivial vertex group.For fixed n ≥ 2, these posets will be denoted by In order to use the various fibre lemmas for posets, we need to establish that all these complexes are connected.We will then show that FS * and FS 1 are homotopy equivalent to the better known complexes FF n and F n , respectively.
The poset of free splittings FS n is contractible by a result of [Hat95] and in particular connected.It is also well-known that the free factor complex, denoted F n by abuse of notation, is connected if n ≥ 3. Lemma 6.1.FF n is connected.
Proof.For n = 2, the complex FF n is the barycentric subdivision of the Farey graph and hence connected.For n ≥ 3, the free factor complex F n is connected.Moreover, F n is a subcomplex of FF n and every vertex in FF n is connected to a vertex in F n .Thus FF n is connected.Lemma 6.2.FS * is connected for n ≥ 2.
Proof.Let S, S ∈ FS * with vertex group systems A and A , respectively.Since FF n is connected by Lemma 6.1, there is a path A = A 0 , A 1 , . . ., A k = A in FF n .For each i, choose a free splitting S i ∈ FS * with vertex group system equal to A i such that S 0 = S and S k = S .We will exhibit paths in FS * connecting S i and S i+1 for all 0 ≤ i < k.Without loss of generality suppose A i A i+1 .Then there exists a free splitting S i,i+1 with vertex group system A i and a subgraph with fundamental group A i+1 .Since L(F, A i ) is connected, there is a path in L(F, A i ) ⊂ FS * connecting S i and S i,i+1 .Also, there is a collapse map from S i,i+1 to S i+1 .It follows that S i and S i+1 are connected by a sequence of edges in FS * .As this is true for all 0 ≤ i < k, we get a path in FS * connecting S and S .Lemma 6.3.FS 1 is connected for n ≥ 3.
Proof.After replacing FF n by F n , the proof is basically the same as the one of Lemma 6.2:For free splittings S, S ∈ FS 1 , each of their vertex group systems is given by a single conjugacy class of free factors [A] and [A ] respectively.As the free factor complex F n is connected for n ≥ 3, it contains a path connecting [A] and [A ].For any free factor B, the spine L(F, [B]) is contained in FS 1 .Now using connectedness of relative Outer space as in Lemma 6.2, one can construct a path from S to S which is entirely contained in FS 1 .Lemma 6.4.FS r, * is connected for n ≥ 3.
Proof.First note that FS r n is connected.Indeed, given two splittings S, S ∈ F S r n , let S ∈ K (resp.S ) be a blow-up of S (resp.S ).Since K is connected, in fact contractible, there is a path connecting S and S in K. Thus S and S are connected in FS r n .Without loss, such a path is obtained by alternately blowing up and collapsing subgraphs.
Let S, S ∈ FS r, * .Since FS r n is connected, there is a path joining S and S in FS r n .If there is a subpath S ← S 1 → S 2 ← S 3 → S where S 1 , S 2 , S 3 ∈ K, then we can replace S 2 by Ŝ2 / ∈ K obtained by collapsing a proper core subgraph of S 2 .Thus we get a path that alternates between splittings in FS r, * = FS r n \ K and K.It suffices to show that any path S = S 0 , S 1 , S 2 = S where S 1 ∈ K and S, S ∈ FS r, * can be homotoped to a path between S and S which is entirely contained in FS r, * .
Suppose we have such a path S = S 0 , S 1 , S 2 = S where S 1 is in K, i.e. S 1 has no separating edges.Then S and S are obtained from S 1 by collapsing subgraphs H and H , respectively.Enlarge H, H to H, H , respectively, to contain all but one edge.Then the intersection of H and H contains a circle c.For a subgraph Γ of S 1 let S 1 /Γ denote the splitting obtained by collapsing the subgraph Γ.Then S and S are connected by a path in FS r, * as follows: Indeed, S 1 ∈ K does not have separating edges and the subgraphs H, H , H, H , c have non-trivial fundamental group, therefore the corresponding splittings are in FS r n \ K.The claimed homotopy equivalences now follow almost immediately from the contractibility of the relative free splitting complexes: Proposition 6.5.
(1) FS * is homotopy equivalent to FF n for n ≥ 2.
(2) FS 1 is homotopy equivalent to F n for n ≥ 3.
Proof.Assigning to each splitting S ∈ FS * the free factor system V(S) given by its non-trivial vertex stabilisers defines a poset map f : FS * → FF op n .As there is a natural isomorphism of the order complexes ∆(FF op n ) ∼ = ∆(FF n ), we will interpret f as an order-inverting map f : FS * → FF n .For any free factor system A in F, the fibre f −1 ((FF n ) ≥A ) is equal to the poset FS n (A) of free splittings relative to A. This poset is contractible by the first point of Theorem 5.17.
The image f (FS 1 ) is equal to F n , so we can consider its restriction g : FS 1 → F n .Now for any conjugacy class [A] of free factors in F, the preimage ), so the second point of Theorem 5.17 finishes the proof.Remark 6.6.The map f : FS * → F F n defined in the proof of Proposition 6.5 has already been used to study the geometry of the complexes in question: In [HM14, Section 6.2], the authors define "projection maps" π : FS n → FF n and show that these maps are Lipschitz with respect to the metrics on the 1-skeleta of FS n and FF n assigning length 1 to each edge.The map f can be seen as the restriction of such a projection map to FS * and hence is Lipschitz as well.
Using the language of sphere systems (see Section 8.2), Hilion and Horbez in [HH17, Section 8] consider the poset FS c n ⊂ FS 1 of all free splittings whose corresponding graph of groups is a rose with non-trivial vertex group, i.e. those free splittings of FS 1 having only one orbit of vertices.

They show that the inclusion FS c
n ⊂ FS 1 defines a quasi-isometry of the 1-skeleta and that the restriction f : FS c n → F n has quasi-convex fibres.This is used to deduce hyperbolicity of F n .

Higher connectivity of factor complexes
In this section, we will combine the results obtained so far in order to establish higher connectivity properties of the various complexes defined in the introduction.Fix n ≥ 2 and define FS * , FS 1 and FS r, * as in Section 6.
Let L × F S * denote the product poset of the spine of (unreduced) Outer space and its simplicial boundary.We define Z to be the subposet of L × F S * given by all pairs (G, S) such that G ∈ L and S = G/H is obtained by collapsing a proper core subgraph H ⊂ G. Let p 1 : Z → L and p 2 : Z → FS * be the natural projection maps.
We want to use Z to study the connectivity properties of FS * .The methods we use for this can also be applied to understand the topology of the free factor complex and the boundary of jewel space.So we will in fact prove connectivity results for all these complexes at the same time.For this we need to introduce two subposets of Z: For the first one, we set Z 1 to be the subposet of L × FS 1 given by all pairs (G, S) such that S = G/H is obtained by collapsing a proper connected core subgraph H ⊂ G. Let q 1 : Z 1 → L and q 2 : Z 1 → F S 1 be the natural projection maps.The poset Z 1 is a subposet of Z and q 1 and q 2 are the restrictions of the projections p 1 and p 2 .
Secondly, we define Z r to be the subposet of K × F S r, * given by all pairs (G, S) such that S = G/H is obtained by collapsing a proper core subgraph H ⊂ G.Note that if a graph G does not contain a separating edge, neither does G/H for any subgraph H ⊂ G.It follows that Z r is the subposet of Z consisting of all (G, S) such that G ∈ K.The natural projection maps r 1 : Z r → K and r 2 : Z r → FS r, * are obtained by restricting the maps p 1 and p 2 .
We think of Z, Z 1 and Z r as thickened versions of FS * , FS 1 and FS r, * , respectively.In order to deduce connectivity results about these three complexes, we proceed in two steps: First we show that the projections p 2 , q 2 and r 2 to the second factors define homotopy equivalences; then we apply the results of Section 4 to understand the fibres of the projections p 1 , q 1 and r 1 .7.1.Projections to the second factor.We first deformation retract the fibres of p 2 , q 2 and r 2 to simpler subposets: Lemma 7.1.
Proof.We define a map f : p −1 2 (FS * ≥S ) → p −1 2 (S) as follows: If (G , S ) ∈ p −1 2 (FS * ≥S ), then there are collapse maps G → S and S → S. Concatenating these maps, we see that S is obtained from G by collapsing a subgraph H ⊂ G .As S ∈ FS * = FS n \ L, the graph H has non-trivial fundamental group.If H is a core subgraph, then we set f (G , S ) := (G , S = G /H ).If H is not a core subgraph, then the set of edges contained in H \ H forms a forest T in G .We set f (G , S ) := (G /T , S).As S = (G /T )/ H , this indeed is an element of p −1 2 (S).For (2), if (G , S ) ∈ q −1 2 (FS 1 ≥S ), then the splitting S is obtained from G by collapsing a connected core subgraph and S is obtained from S by collapsing a subgraph.Concatenating these two collapse maps, one sees that S = G /H for a subgraph H ⊂ G .This subgraph may be disconnected, but only one of its components has non-trivial fundamental group.It follows that its core H is connected.Using this observation, the map f : p −1 2 (FS * ≥S ) → p −1 2 (S) restricts to a monotone poset map q −1 2 (FS 1 ≥S ) → q −1 2 (S).So the second claim follows from Lemma 2.2 as well.For (3), recall that if a graph G does not contain a separating edge, then neither does G/H for any subgraph H ⊂ G.It follows that for all S ∈ F S r, * , the map f also restricts to a deformation retraction r −1 2 (FS r, * ≥S ) → r −1 2 (S).Hence, instead of studying arbitrary fibres, it suffices to consider the preimages of single vertices.We start by using the results from Section 5 to show: Proposition 7.2.For all S ∈ FS * , the preimage p −1 2 (S) is contractible.Proof.Fix a free splitting S ∈ F S * and let [A 1 ], . . ., [A k ] be the components of V(S).Every element in p −1 2 (S) is given by a pair (G, S) such that there is a unique core subgraph H ⊂ G having connected components H 1 , . . ., H k where π 1 (H i ) = [A i ] and S = G/H.After possibly performing a valence-2-homotopy, we may assume that H i is an element of L(A i ).Now define a poset map As in Section 5, we will use the same letter H to denote both a core subgraph of G and the tuple (H 1 , . . ., H k ).We claim that for all restricting to the identity on φ −1 (H).Hence, Lemma 2.2 shows that we have a retraction However, in the notation of Section 5, one has φ −1 (H) = Ψ −1 (S, H) and this is contractible by Lemma 5.7.Lemma 2.1 implies that φ is a homotopy equivalence which proves the claim.
The following shows that Proposition 7.2 also provides us with sufficient information about the fibres of q 2 and r 2 .Proposition 6.5 immediately implies the following corollary which completes the proof of Theorem B.
Corollary 7.13.The complex FF n of free factor systems is (n − 2)-connected.
Note that in contrast to the situation here, these arguments cannot be used to deduce (n − 2)connectivity of ∂J n as the fibres of the map r 1 a priori do not satisfy the conditions needed to apply Lemma 2.3.For more comments on the optimality of the result obtained here, see Section 8.3.8. Some remarks 8.1.Relative complexes.In [HM14], the authors do not only study the whole poset of free factor systems, but also relative versions of it.For a given free factor system A, the poset of free factor systems of F relative to A consists of all free factor systems B in F such that there are proper inclusions A B F. In other words, this poset is given by (FF n ) >A .
Replacing CV n by Outer space relative to A one can apply the arguments used in the previous sections in order to show higher connectivity of these relative complexes of free factor systems.As already in the "absolute" case, we make use of relative Outer space, most proofs can be taken literally for the relative setting as well.The fibres one obtains and needs to analyse here correspond to posets of graphs with a labelling of the vertices (as described e.g. in [BF00]).This can be done using similar arguments as in the proof of Proposition 4.10.

Sphere systems.
There is an equivalent description of the free splitting complex in terms of sphere systems.For this, let M n be the connected sum of n copies of S 1 × S 2 .This manifold has fundamental group isomorphic to F. A collection {S 1 , . . ., S k } of disjointly embedded 2-spheres in M n is called a sphere system if no S i bounds a ball in M n and no two spheres are isotopic.The set of isotopy classes of such sphere systems has a partial order given by inclusion of representatives.The order complex S(M n ) of this poset is called the complex of sphere systems.Considering the fundamental group of its complement, each sphere system induces a free splitting of F. In fact, the free splitting complex FS n is the barycentric subdivision of S(M n ).
Following this, our considerations in this article can be translated in the language of such sphere systems: The complex FS * corresponds to the complex S ∞ ⊂ S(M n ) consisting of all sphere systems σ whose complement M n \ σ has at least one connected component which is not simplyconnected.The complex FS 1 on the other hand corresponds to S 1 (M n ) ⊂ S(M n ), the subcomplex of S(M n ) consisting of sphere systems whose complement has exactly one component which is not simply-connected.
Using this description, (n − 3)-connectivity of FS * can be deduced very quickly as follows: Proof of (n − 3)-connectivity via sphere systems [Vog18].Whenever one takes a sphere system σ consisting of at most (n−1)-many spheres, it induces a free splitting of π 1 (M n ) ∼ = F with at most n−1 orbits of edges.It follows that at least one of the vertex groups of this splitting must be non-trivial, implying that the complement M n \ σ contains at least one connected component with non-trivial fundamental group.Hence, the entire (n−2)-skeleton of S(M n ) is contained in S ∞ ∼ = FS * .However, the complex S(M n ) is contractible (see [Hat95]), so we have {0} ∼ = π n−3 (S(M n )) ∼ = π n−3 (S ∞ ).
The same argument also shows (n − 3)-connectivity of FS r, * ∂J n , the second part of Theorem D. However, we would like to point out that this does a priori not give a proof for (n − 2)connectivity of FS * and also in particular does not show connectivity properties of FS 1 F n .8.3.The simplicial boundaries of CV 2 and CV r 2 .The difference in the degree of connectivity between the reduced and the unreduced setting might be surprising at first glance, but in fact it can easily be seen when one considers the case where n = 2.
Here, reduced Outer space CV r n can be identified with the tesselation of the hyperbolic plane by the Farey graph (an excellent picture of this tesselation can be found in [Vog08]).The triangles of this tessellation correspond to the three-edge "theta graph".Each side of such a triangle is given by graphs which are combinatorially roses with two petals and obtained by collapsing one of the edges of the theta graph; as the rose is a graph of rank 2, these edges are contained in the interior of CV r 2 .In contrast to that, the vertices of the triangles correspond to loops obtained by collapsing two edges of the theta graph and hence are points sitting at infinity.Hence, the simplicial boundary of CV r 2 is homeomorphic to Q, a countable join of 0-spheres.Starting from reduced Outer space, unreduced CV 2 is obtained by adding "fins" above each edge of the Farey graph.These fins are triangles corresponding to the "dumbbell graph" which consists of two loops connected by a separating edge.Collapsing this separating edge, one obtains the side of the triangle which corresponds to the rose.On the other hand, collapsing one of the two loops of the dumbbell yields a graph of rank one, forcing the other two sides of the triangle to sit at infinity.Inside the simplicial boundary ∂ s CV 2 , the concatenation of these sides now connects two vertices of the adjacent theta graph triangles as depicted in Figure 6.It follows that ∂ s CV 2 is isomorphic to the barycentric subdivision of the Farey graph which is in turn homotopy equivalent to a countable wedge of circles.
This argument answers Question 1.1 for n = 2: Here the lower bounds we get for the degree of connectivity of the simplicial boundaries ∂ s CV r n ∂J n and ∂ s CV n FF n are optimal and furthermore, the homology of these complexes is concentrated in dimension n − 2 and n − 1, respectively.For higher rank, this is however not clear at all as ∂J n and FF n have dimension 2n − 3.In the case of ∂J n , there are obvious (n − 2)-spheres one might expect to be non-trivial elements of π n−2 (∂J n ).If G is a rose, the realisation of X(G) = Sub(G) is the boundary of a triangulated 1-sphere.For the graphs b) -e) in Figure 8, the complex X(G) is depicted in Figure 9.
As the graphs f), g) and h) do not contain any disconnected core subgraphs, the claim here follows from Proposition 4.10.The only disconnected core subgraph of i) consists of the edges 1, 4 and 5. Hence, C(G) is derived from cC(G) by attaching the star of the vertex {1, 4, 5} along its link.It is an easy exercise to check that the result is homotopy equivalent to a circle.The same is true for j) whose only non-connected core subgraph is {1, 2, 4, 6}.
For the remaining graphs k) -p), the following tables define Morse functions φ : C(G) → R with contractible descending links: As an illustration, we explain why all the descending links for n) are contractible: st({1, 2, 3, 4}) is obviously contractible as this is true for any star in a simplicial complex.The vertices of C(G) not contained in this star are precisely the proper core subgraphs of G containing the (separating) edge 5.The descending link of {1, 2, 3, 5} contains a unique maximal element and hence is contractible; this cone point is given by {1, 2, 3} which is the unique maximal core subgraph of {1, 2, 3, 5} not containing 5. As {2, 3, 5} does not contain 4, it is contained in {1, 2, 3, 5} which hence forms a cone point of its descending link.Lastly, the link of {1, 3, 4, 5} is coned off by {1, 3}.
The interested reader may complete this argument to an alternative proof of Proposition 4.4 for arbitrary n ≥ 2 in the case where G contains at least one separating edge.

Figure 4 .
Figure 4. Let F 4 = a, b, c, d .The left figure shows some graphs in X(1 : a, b ) and the right figure shows some graphs in X( b : a, b, c ).

Figure 6 .
Figure 6.A part of CV 2 .The turquoise bottom part is reduced Outer space, together with the red fins on top it forms unreduced Outer space.The faces at infinity are coloured in plum, where the three round vertices are the only points contained in the reduced boundary ∂ s CV r n .

Figure 8 .
Figure 8.All combinatorial types of graphs in CV 3We want to show that for each such graph G, the poset X(G) is homotopy equivalent to a (nontrivial) wedge of circles if G does not contain a separating edge and is contractible otherwise.Using Lemma 4.3, it suffices to show the same statement for the poset C(G) of all core subgraphs.