Proceedings of the Edinburgh Mathematical Society (1990) 33, 367-379 ( LOCAL STRUCTURE OF SOME OUT(Fn)-COM?LEXES by KAREN VOGTMANN* (Received 19th July 1988, revised 25th October 1989) In previous work of the author and M. Culler, contractible simplicial complexes were constructed on which the group of outer automorphisms of a free group of finite rank acts with finite stabilizers and finite quotient. In this paper, it is shown that these complexes are Cohen-Macauley, a property they share with buildings. In particular, the link of a vertex in these complexes is homotopy equivalent to a wedge of spheres of codimension 1. 1980 Mathematics subject classification (1985 Revision): 2OJO5. 0. Introduction Let Out(Fn) be the group of outer automorphisms of a finitely generated free group. In [1] a contractible space X = X(n) was constructed on which Out(Fn) acts discretely with finite stabilizers. This space may be thought of as analogous to the homogeneous space of an algebraic group, with a discrete action by an arithmetic subgroup, or to the Teichmuller space of a surface with the action of the mapping class group of the surface. Two Out(Fn)-invariant deformation retracts K = K(n) and L = L{n) of X were also described in [1]; these are locally finite simplicial complexes with finite quotient. In [1] the complex K was used to prove cohomological finiteness properties of the group Out(Fn). In particular, it was shown that Out(Fn) is VFL and has virtual cohomological dimension 2M-3. In contrast with homogeneous spaces and Teichmuller spaces, the space X is not a manifold, and standard methods in manifold theory, such as Poincare duality, cannot be used to study the group action and quotient space. Borel and Serre encountered the same difficulty when studying S-arithmetic groups, where the role of the homogeneous space is played by a Euclidean building. Euclidean buildings are simplicial complexes, but are not triangulated manifolds; in particular, the link of a vertex is not homeo- morphic to a sphere of codimension 1. However, there is a uniform local structure to buildings which makes them homotopically similar to manifolds: the link of each vertex is homotopy equivalent to a wedge of spheres of codimension 1. The purpose of this paper is to show that the simplicial complexes K and L have similar local properties. I would like to thank the referee for helpful comments and for pointing out an error in the original version of this paper. •Partially supported by a grant from the National Science Foundation. 367 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 16:04:47, subject to the Cambridge Core terms of use. 368 K. VOGTMANN FIGURE 1. Forest collapse. 1. Background We briefly recall from [1] the definition of the complexes K and L and some basic properties. Let Ro be an n-leafed rose, i.e. a connected graph with one vertex and n edges. Vertices of L are equivalence classes of pairs (g, G), where G is a connected graph with vertices of valence at least 3, and g is a homotopy equivalence from Ro to G. Two pairs (g, G) and {g',G') are equivalent if there is a homeomorphism h:G-*G' such that hog~g'. Vertices vo,...,vk of L span a /c-simplex if representatives (g0, Go),...,(gk, Gk) can be chosen so that G, is obtained from Gf.j by collapsing each component of a forest in G,_! to a point, and g( is the composition of #,_! with the collapsing map. Here a forest in G is a subset of the edges of G which contains no cycle. This operation is called a forest collapse (see Fig. 1). The complex L can be thought of as the geometric realization of the poset (partially ordered set) of its vertices, where the partial ordering is (g, G)^(g', G') if (g',G') can be obtained from (g, G) by a forest collapse. An edge e of a graph G is called a bridge if G minus the interior of e is disconnected. There is a deformation retraction of L onto the subcomplex K spanned by points (g, G) such that G has no bridges. In [1] it is shown that K and L are contractible of dimensions In — 3, and Out(Fn) acts on K and L with finite stabilizers and finite quotient. This implies the cohomo- logical finiteness results mentioned in the introduction. 2. The Cohen-Macauley property for L We recall some standard facts about posets. We refer to [2] for a more complete discussion and proofs. Let P be a poset, and let p e P. The height of p, ht(p), is the length of the longest totally ordered chain of elements of P which are all less than p. The height of the poset P is the maximum of the heights of its elements. A poset is said to be k-spherical if its geometric realization is /c-dimensional and (k— l)-connected. Note that the geometric realization of a k-spherical poset is homotopy equivalent to a bouquet of /c-spheres. Definition A poset P of height h is Cohen-Macauley if P is /i-spherical and Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 16:04:47, subject to the Cambridge Core terms of use. LOCAL STRUCTURE OF SOME OC/T(Fn)-COMPLEXES 369 the link of every simplex of dimension k in the geometric realization of P is (h — k — l)-spherical. Define subposets P>p = {seP:s<p}, P<p={seP:s>p} and (p,q) = {seP:q<s<p}. The Cohen-Macauley property for P is equivalent to the following properties: (i) P is /i-spherical, (ii) P>p is (ht(p) — l)-spherical for each peP, and (iii) P<p is (h — ht(p) — l)-spherical for each peP. (iv) (p,q) is (ht(q) — ht(p) — l)-spherical for every p<q in P. The following standard lemma is useful in determining the homotopy type of a poset. A map f:P-*P is a poset map if p^q implies f(p)^f(q)- Lemma 2.1. (Poset Lemma) Let f:P-*P be a poset map such that f(p)^p for all peP or f(p)^p for all peP. Then the geometric realization of P is homotopy equivalent to the geometric realization of f(P). The rest of this section is devoted to showing that L is Cohen-Macauley. Since L is contractible and the dimension of L is equal to its height as a poset, L satisfies (i) above. Let (g, G) be a vertex of L. Then L<(gG) can be identified with the partially ordered set of non-empty forests in the graph G. The partial ordering is given by inclusion. Proposition 2.2. Let G be a finite connected graph, and F(G) the poset of non-empty forests in G. Then the geometric realization of F(G) is homotopy equivalent to a wedge of spheres of dimension v — 2, where v = v(G) is the number of vertices of G. The geometric realization of F(G) is contractible if and only if G has a bridge. Proof. If e is an edge of G, we denote by G — e the graph obtained by deleting the interior of the edge e, and by G/e the quotient graph obtained by collapsing e to a point. Let e denote the number of edges of G. We will proceed by induction on e + v. If e + v= 1, the theorem is trivial. If an edge e of G is a loop, then F(G) = F(G—e); thus we may assume that G has no loops. If G has a bridge E, then 0u{e} is a forest whenever <p is. Then <p-KpKj{e}-*{e} are poset maps giving a contraction of the geometric realization of F(G) to a point. We now assume that G has no bridges. Fix an edge e of G. Let Fx = F{G) — {e} be the set of all forests except the forest consisting of the single edge e, and let Fo be the set of all forests which do not contain e. Then $->$ — {e} is a poset map from F, onto Fo giving a homotopy equivalence of their realizations. Fo is naturally isomorphic to F(G-E). Since e is not a bridge, G-e is connected. By induction, Fo is homotopy equivalent to a wedge of (v — 2)-spheres. Now F(G) = Fl^> (star (e)), and F1n(star(6)) = link(e), where star and link are defined as for simplicial complexes, i.e. the star is the set of forests <j> which contain e. The map #-></>/e is a poset isomorphism from link(e) to F(G/e). G/e is connected and has no bridges since G is connected and has no bridges. Furthermore, G/e has one less vertex than G since e is not a loop. Therefore the geometric realization of F(G/e) is homotopy Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 16:04:47, subject to the Cambridge Core terms of use. 370 K. VOGTMANN equivalent to a wedge of (v — 3)-spheres by induction. It follows by the Van Kampen and Mayer-Vietoris theorems that ^ v S"'2 v susp( v Sv~3)^ vS""2. Corollary 2.3. L>(g<G) is (ht(#, G) - l)-spherical. Proof. We have already remarked L>igG) can be identified with F(G).