Compactly supported cohomology of buildings

发布于:2021-11-29 04:42:18

Compactly supported cohomology of buildings

arXiv:0806.2412v1 [math.GR] 15 Jun 2008

M.W. Davis?

J. Dymara? T. Januszkiewicz? B. Okun

J. Meier§

Abstract We compute the compactly supported cohomology of the standard realization of any locally ?nite building. AMS classi?cation numbers. Primary: 20F65 Secondary: 20E42, 20F55, 20J06, 57M07. Keywords: Building, cohomology of groups, Coxeter group.

Introduction
A building consists of a set Φ (the elements of which are called “chambers”) together with a family of equivalence relations (“adjacency relations”) on Φ indexed by a set S and a “W -valued distance function,” Φ × Φ → W , where W is a Coxeter group with fundamental set of generators S. So, associated to any building there is a Coxeter system (W, S), its type. There is a construction which associates a topological space to Φ. This construction admits some freedom of choice. The idea is to choose a space X as a “model chamber” and then glue together copies of it, one for each element of Φ. To do this, it is ?rst necessary to choose a family of closed subspaces {Xs }s∈S so that copies of X corresponding to s-adjacent chambers are glued together along Xs . (We call such a family, {Xs }s∈S , a “mirror structure” on X.) Let U(Φ, X) denote the topological realization of Φ where each chamber is realized by a copy of the model chamber X. (Details are given in Section 2.) Classically, interest has centered on buildings of spherical or a?ne type, meaning that W is a spherical or Euclidean re?ection group, respectively. For example, each algebraic group over a local ?eld has a corresponding a?ne building. However, we are mainly interested in buildings which are not classical in that their associated Coxeter systems are neither spherical nor a?ne. This is a large class of spaces, many of which have a great deal of symmetry. For example, such buildings arise in the theory of Kac-Moody groups (e.g., see [6, 21, 22]). Also, nonclassical buildings associated to arbitrary right-angled
? The ? The

?rst author was partially supported by NSF grant DMS 0706259. second author was partially supported by KBN grant N201 012 32/0718 ? The third author also was partially supported by NSF grant DMS 0706259. § The fourth author was partially supported by a Richard King Mellon research grant.

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Coxeter groups have been a subject of recent interest in geometric group theory (e.g., see [8, 19, 26]). Two choices for a model chamber X stand out. The ?rst is X = ?, a simplex of dimension Card(S) ? 1, with its codimension one faces indexed by S. This was Tits’ original choice (cf. [1]). We call U(Φ, ?) the “classical realization” of Φ. The other choice for X is the “Davis chamber” K, de?ned as the geometric realization of the poset S of spherical subsets of S (see [9, Chapters 7, 18]). U(Φ, K) is the “standard realization” of Φ. Both realizations are contractible. The standard realization is important in geometric group theory, the reason being that in this ?eld one is interested in discrete group actions which are both proper and cocompact and these conditions are more likely to hold for the action of a group of automorphisms on the standard realization than on the classical realization. (If Φ has ?nite thickness, U(Φ, K) is locally ?nite, while U(Φ, ?) need not be.) If a discrete group Γ acts properly and cocompactly on a locally ?nite, contractible CW complex Y , then the compactly supported cohomology of Y is isomorphic to the cohomology of Γ with ZΓ coe?cients. In particular, it determines the virtual cohomological dimension of Γ, as well as, the number of ends of Γ, and it determines if Γ is a duality group. (For more information, see [17, Part IV].) However, as we will explain, even if one is only interested in cohomological computations in the case of the standard realization of Φ, it is necessary to carry out similar computations for various other realizations, in particular, for the classical realization. In the classical case of an (irreducible) a?ne building, the two notions of model chamber agree: ? = K. So, in the a?ne case the study of the cohomology of cocompact lattices in Aut(Φ) is closely tied to the study of the cohomological properties of U(Φ, ?). For example, in [2] Borel and Serre calculated ? the compactly supported cohomology, Hc (U(Φ, ?)), for any (irreducible) a?ne building and then used this calculation to derive information about the cohomol? ogy of “S-arithmetic” subgroups. The calculation of [2] was that Hc (U(Φ, ?)) is concentrated in the top degree (= dim ?) and is free abelian in that degree. ? Our main result, Corollary 8.2, is a calculation of Hc (U(Φ, K)), generalizing the theorem of Borel-Serre. In the case where Φ = W , this was done in [7, 10, 12]. For a general (thick) building, in the case where (W, S) is right-angled, it was done in [10, Theorem 6.6]. It was claimed in full generality in [12]; however, there is a mistake in the proof (see [13]). In order to write the formula, we need more notation. Let A be the free abelian group of ?nitely supported, Z-valued functions on Φ. For each subset T ? S, let AT denote the subgroup of all functions f ∈ A which are constant on each residue of type T . (A “residue” of type T is a certain kind of subset of Φ; in the case of the building W , a residue of type T is a left coset of WT , the subgroup of W generated by T .) N.B. The empty set, ?, is a spherical subset and a residue of type ? is just a single chamber; hence, A? = A. If U ? T , then AU ? AT . Let A>T ? AT denote the Z-submodule, U?T AU . (Throughout this paper we will use the convention that ? denotes containment and ? will be reserved for strict containment. Also, we will use the symbol for an internal 2

sum of modules and to mean either an external direct sum or an internal sum which we have proved is direct.) ? We shall show in Section 7 that A>T is a direct summand of AT . Let AT be a complementary summand. As in [10] the main computation is a consequence the following Decomposition Theorem (proved as Theorem 7.4 in Section 7). The Decomposition Theorem. A=
U∈S

? AU

and, in fact, for any T ∈ S,

AT =
U?T

? AU .

The point is that this theorem provides a decomposition of a coe?cient system which can be used to calculate the compactly supported cohomology of any of the various realizations of Φ. The calculation in which we are most interested is the following (proved as Corollary 8.2 in Section 8). The Main Theorem. Suppose Φ is a building of ?nite thickness and type (W, S). Let K be the geometric realization of the poset S of spherical subsets of S. Then ? ? Hc (U(Φ, K)) ? H ? (K, K S?T ) ? AT . =
T ∈S

(For any subset U of S, K U denotes the union of the Ks , with s ∈ U .) The above theorem applies to all buildings. A general building Φ will not be highly symmetric, in that its automorphism group, Aut(Φ), can have in?nitely many orbits of chambers. However, if Φ/ Aut(Φ) is ?nite and if Γ is a torsionfree cocompact lattice in Aut(Φ), then the Main Theorem implies that the cohomological dimension of Γ is equal to the virtual cohomological dimension of the corresponding Coxeter group. Moreover, this dimension is cd(Γ) = vcd(W ) = max{k | H k (K, K S?T ) = 0, for some T ∈ S} (cf. Corollary 9.4). As another example, such a torsion-free cocompact lattice is an n-dimensional duality group if and only if for each T ∈ S, H ? (K, K S?T ) is free abelian and concentrated in degree n (cf. Corollary 9.5). ? The central objective of [10] was to calculate Hc (U(W, K)) as a W -module. ? In that paper we showed there is a ?ltration of Hc (U(W, K)) by W -submodules so that the associated graded terms look like the terms on the right hand side of the formula in the Main Theorem. Similarly, one can ask about the G-module ? structure of Hc (U(Φ, K)) for any subgroup G ? Aut(Φ). The methods of [10] are well adapted to the present paper. In particular, for each T ∈ S, the free abelian group AT is a G-module, as is its quotient DT := AT /A>T . So, as ? in [10], there is a ?ltration of Hc (U(Φ, K)) by G-submodules and we get the following (proved as Theorem 9.1). Theorem. Suppose G is a group of automorphisms of Φ. There is a ?ltration ? of Hc (U(Φ, K)) by right G-submodules with associated graded term in ?ltration degree p: H ? (K, K S?T ) ? DT .
T ∈S |T |=p

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As we mentioned earlier, although the Main Theorem is the result of importance in geometric group theory, it is no harder to do similar calculations when the chambers are modeled on an arbitrary X (or at least on X f , the complement of the faces of X which have in?nite stabilizers in W ). In fact, as we explain below, the proof of the Decomposition Theorem depends on ?rst establishing a version of the Main Theorem for U(Φ, ?f ). So, the Main Theorem ultimately depends on ?rst proving a version of it in the case of the classical realization. This version (proved as Theorem 4.3) is the following.
? Theorem. When W in?nite, Hc (U(Φ, ?f )) is free abelian and is concentrated in the top degree n (= dim ?).

This result is obvious when Φ = W , for then U(W, ?f ) is homeomorphic to Euclidean space Rn (see Section 2). We prove it for a general Φ by showing the following (Theorem 3.1). Theorem. U(Φ, ?f ) admits a CAT(0) metric (extending Moussong’s CAT(0) metric on the standard realization). The existence of this CAT(0) metric on U(Φ, ?f ) is of independent interest. (See [20, 8] or [9] for a description of Moussong’s metric on U(Φ, K).) ? To ?nish the calculation of Hc (U(Φ, ?f )) we invoke a result of [4] which asserts that the compactly supported cohomology of such a CAT(0) space is concentrated in the top degree provided the cohomology of each “punctured link” vanishes except in the top degree. These links are spherical buildings and the vanishing of the cohomology groups, in degrees below the top, of their punctured versions is a result of [16] (and independently, [24]). As we have said, special cases of the Decomposition Theorem were proved in [10]. The method of [10] was simply to ?nd a basis for A adapted to its ? decomposition into the AT . In the general case, ?nding an explicit description of such a basis seems problematic. We use instead an idea coming from an analogy with the argument of [11]. In that paper we proved L2 versions of the Decomposition Theorem and of the Main Theorem. The proof of the L2 version of the Decomposition Theorem was homological: the key step was to show that certain “weighted L2 -homology” groups of certain auxiliary spaces associated to W vanished except in the bottom degree. We then applied a certain duality (not applicable here) to deduce the Decomposition Theorem in the cases of actual interest. Analogously, in this paper we prove the Decomposition Theorem by establishing the vanishing, except in the top degree, of the cohomology of certain auxiliary spaces. The most important of these auxiliary spaces is U(Φ, ?f ) and we indicated in the previous paragraphs how we prove the result in that case.

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Coxeter groups and buildings

A chamber system over a set S is a set Φ of chambers together with a family of equivalence relations on Φ indexed by S. Two chambers are s-equivalent if they are related via the equivalence relation with index s; they are s-adjacent 4

if they are s-equivalent and not equal. A gallery in Φ is a ?nite sequence of chambers (?0 , . . . , ?k ) such that ?j?1 is adjacent to ?j , 1 ≤ j ≤ k. The type of this gallery is the word s = (s1 , . . . , sk ) where ?j?1 is sj -adjacent to ?j . If each sj belongs to a given subset T of S, then the gallery is a T -gallery. A chamber system is connected (resp., T -connected ) if any two chambers can be joined by a gallery (resp., a T -gallery). The T -connected components of a chamber system Φ are its residues of type T . A Coxeter matrix over a set S is an S × S symmetric matrix M = (mst ) with each diagonal entry = 1 and each o?-diagonal entry an integer ≥ 2 or the symbol ∞. The matrix M de?nes a presentation of a group W as follows: the set of generators is S and the relations have the form (st)mst where (s, t) ranges over all pairs in S × S such that mst = ∞. The pair (W, S) is a Coxeter system (cf. [3, 9]). Given T ? S, WT denotes the subgroup generated by T ; it is called a special subgroup. (WT , T ) is itself a Coxeter system. The subset T is spherical if WT is ?nite. De?nition 1.1. The poset of spherical subsets of S (partially ordered by inclusion) is denoted S. Also, P (= P(S)) will denote the poset of all proper subsets of S. (In Section 3 the poset P ? S plays a role.) Suppose (W, S) is a Coxeter system and M = (mst ) is its Coxeter matrix. Following [23] (or [10]), a building of type (W, S) (or of type M ) is a chamber system Φ over S such that (i) for all s ∈ S, each s-equivalence class contains at least two chambers, and (ii) there exists a W -valued distance function δ : Φ × Φ → W . (This means that given a reduced word s for an element w ∈ W , chambers ? and ?′ can be joined by a gallery of type s from ? to ?′ if and only if δ(?, ?′ ) = w.) Example 1.2. The group W itself has the structure of a building: the sequivalence classes are the left cosets of W{s} and the W -valued distance, δ : W × W → W , is de?ned by δ(v, w) = v ?1 w. A residue of type T is a building; its type is (WT , T ). A building of type (W, S) is spherical if W is ?nite. A building has ?nite thickness if each sequivalence class is ?nite, for each s ∈ S. (This implies all spherical residues are ?nite.) Henceforth, all buildings will be assumed to have ?nite thickness.

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Geometric realizations of Coxeter groups and buildings

A mirror structure over a set S on a space X is a family of subspaces (Xs )s∈S indexed by S. Given a mirror structure on X, a subspace Y ? X inherits a mirror structure by Ys := Y ∩ Xs . If X is a CW complex and each Xs is a

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subcomplex, then X is a mirrored CW complex. For each nonempty subset T ? S, de?ne subspaces XT and X T by XT :=
s∈T

Xs

and X T :=
s∈T

Xs .

(2.1)

Put X? := X and X ? := ?. Given a cell c of (a CW complex) X or a point x ∈ X, put S(c) := {s ∈ S | c ? Xs }, S(x) := {s ∈ S | x ∈ Xs }. Suppose now that S is the set of generators for a Coxeter system (W, S). Let Υ(X) denote the union of the nonspherical faces of X and X f its complement in X, i.e., XT and X f := X ? Υ(X). (2.2) Υ(X) :=
T ∈S /

The mirror structure is W -?nite if Υ(X) = ?. Given a building Φ of type (W, S) and a mirrored space X over S, de?ne an equivalence relation ? on Φ × X by (?, x) ? (?′ , x′ ) if and only if x = x′ and δ(?, ?′ ) ∈ WS(x) (i.e., ? and ?′ belong to the same S(x)-residue). The X-realization of Φ, denoted U(Φ, X), is de?ned by U(Φ, X) := (Φ × X)/ ? . (2.3)

(Φ has the discrete topology.) Suppose X is a mirrored CW complex and that we are given a cell c of X and a chamber ? ∈ Φ. Then ? · c denotes the corresponding cell in U(Φ, X). Let U (i) denote the set of i-cells in U(Φ, X). Each such cell has the form ? · c for some ? ∈ Φ and i-cell c of X. The classical realization. ? denotes the simplex of dimension |S| ? 1, with its codimension one faces indexed by S. In other words, the mirror ?s is a codimension one face and ?T (the intersection of the ?s over all s ∈ T ) is a face of codimension |T |. The simplicial complex U(W, ?) is the Coxeter complex of (W, S) while U(Φ, ?) is the classical realization of the building Φ. Tits constructed a representation of W on RS called “the contragredient of the canonical representation” in [3] and the “geometric representation” in [9]. The elements of S are represented by re?ections across the codimension one faces of a simplicial cone C. The union of translates of C is denoted by W C. It is a convex cone and W acts properly on its interior I. If W is in?nite, W C is a proper cone. If C f denotes the complement of the nonspherical faces of C, then I is equivariantly homeomorphic to U(W, C f ). Assume W is in?nite. Then the image of C ? 0 in projective space can be identi?ed with the simplex ? obtained by intersecting C with some a?ne hyperplane; moreover, ?f is identi?ed with the intersection of C f and this hyperplane. The image of I in projective space is then identi?ed with U(W, ?f ). Since this image is the interior of a topological 6

disk (of dimension |S| ? 1), it follows that U(W, ?f ) is homeomorphic to a Euclidean space of that dimension. In particular, it is contractible. Geometric realizations of posets. Given a poset T , Flag(T ) denotes the set of ?nite chains in T , partially ordered by inclusion, i.e., an element of Flag(T ) is a ?nite, nonempty, totally ordered subset of T . If α = {t0 , . . . , tk } ∈ Flag(T ) where t0 < · · · < tk , then we will write α := {t0 < · · · < tk } and min α := t0 . Flag(T ) is an abstract simplicial complex with vertex set T and with k-simplices the elements of Flag(T ) of cardinality k + 1. The corresponding topological simplicial complex is the geometric realization of the poset T and is denoted by |T |. For example, if P is the poset of proper subsets of S, partially ordered by inclusion, then its opposite poset, P op , is the poset of nonempty faces of the simplex ?. Flag(P) is the poset of simplices in its barycentric subdivision, b?, and |P| = b?. The standard realization, U(Φ, K). As before, S denotes the poset of spherical subsets of S. Put K := |S|. It is a subcomplex of b? (provided W is in?nite). The mirror structure on ? induces one on K. More speci?cally, for each s ∈ S, put Ks := |S≥{s} | and for each T ∈ S, KT = |S≥T |. (K is the “compact core” of ?f .) K is sometimes called the Davis chamber of (W, S) and U(W, K), the Davis complex. Alternatively, U(Φ, K) is the geometric realization of the poset of spherical residues of Φ (see [8]). By construction U(Φ, K) is locally ?nite (since Φ is assumed to have ?nite thickness). It is proved in [8] that U(Φ, K) is contractible. The realization U(Φ, ?f ). ?f and K have the same poset of faces (indexed by S) and there is a face-preserving deformation retraction ?f → K. That is to say, ?f is a “thickened version” of K. Similarly, U(Φ, ?f ) is a thickened version of U(Φ, K). Like U(Φ, K), the space U(Φ, ?f ) has the advantage of being locally ?nite; however, the chamber ?f is not compact whenever ?f = ?.

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A CAT(0) metric on U(Φ, ?f )

Our goal in this section is to prove the following. Theorem 3.1. Let Φ be a building of type (W, S) with W in?nite. Then there is a piecewise Euclidean, CAT(0) metric on U(Φ, ?f ). Review of the Moussong metric. Suppose T is a spherical subset of S. WT acts on RT via the canonical representation. The Coxeter cell of type T , denoted PT , is de?ned to be the convex hull of the WT -orbit of a point x0 in the interior of the fundamental simplicial cone. As examples, if WT is a product of n copies of the cyclic group of order 2, then PT is an n-cube; if WT is the symmetric group on n + 1 letters, then PT is an n-dimensional permutohedron. Its boundary complex, ?PT , is the dual of the Coxeter complex of WT (the Coxeter complex is a triangulation of the unit sphere in RT ). The fact that ?PT is dual to a simplicial complex means that PT is a “simple polytope”. 7

The isometry type of PT is determined once we choose the distance from x0 to each of the bounding hyperplanes of the simplicial cone. (We assume, without further comment, that such a choice of distance has been made for each s ∈ S.) The intersection of PT with the fundamental simplicial cone is denoted BT and called the Coxeter block of type T . It is a convex cell combinatorially isomorphic to a cube of dimension |T | (because PT is simple). One can identify BT with the subcomplex |S≤T | of K in such a way that x0 is identi?ed with the vertex corresponding to ?. To be more precise, BT is the union of simplices of Flag(S) whose maximum vertex is ≤ T , i.e., |S≤T | is a subdivision of BT . The convex polytope BT has two types of faces. First, there are the faces which contain the vertex x0 . Each such face is a Coxeter block of the form BV for some V ? T . The other type of face is the intersection of BT with the face of the fundamental simplicial cone ?xed by WV for some nonempty V ? T . We denote such a face by BT,V and call it a re?ecting face of BT . For the purpose of unifying di?erent cases, we shall sometimes write BT,? instead of BT . The Moussong metric on K is the piecewise Euclidean metric on K in which each Coxeter block BT is given its Euclidean metric as a convex cell in RT (cf. [20] or [9, Section 12.1]). This induces a piecewise Euclidean metric on U(W, K) as well as one on U(Φ, K). The link of the central vertex corresponding to ? can be identi?ed with a certain simplicial complex L (= L(W, S)) called the nerve of the Coxeter system. The vertex set of L is S and a subset T ? S spans a simplex if and only if it is spherical. Thus, the poset of simplices in L (including the empty simplex) is S. The piecewise Euclidean metric on K induces a piecewise spherical metric on L such that whenever mst < ∞, the length of the edge corresponding to {s, t} is π ? π/mst . Moussong proved that this piecewise spherical metric on L is CAT(1) and from this he deduced that the piecewise Euclidean metric on U(W, K) is CAT(0) (cf. [20] and [9, Section 12.3]). Using this, it is proved in [8] that for any building Φ, the Moussong metric on the standard realization, U(Φ, K), is CAT(0). A piecewise Euclidean metric on ?f . We will de?ne a cell structure on ?f so that each cell will have the form BT,V × [0, ∞)m for some T ∈ S, V ? T and nonnegative integer m. When m > 0 such a cell will be noncompact. There are two types of such cells. First there are the compact cells BT,V where T ∈ S and V ? T . The remaining cells are in bijective correspondence with triples (T, V, α) where T ∈ S, V ? T and α ∈ Flag(P ? S) is such that T < min α. Let F be the set consisting of pairs (T, V ) and triples (T, V, α), where T ∈ S, V ? T and α ∈ Flag(P ? S) is such that T < min α. F is partially ordered as follows: ? (T ′ , V ′ ) ≤ (T, V ) if and only if T ′ ? T and V ′ ? V , ? (T ′ , V ′ , α′ ) ≤ (T, V, α) if and only if (T ′ , V ′ ) ≤ (T, V ) and α′ ? α, ? (T ′ , V ′ ) ≤ (T, V, α) if and only if (T ′ , V ′ ) ≤ (T, V ). The cell c(T, V, α) which corresponds to (T, V, α) is de?ned to be BT,V ×[0, ∞)α , where [0, ∞)α means the set of all functions from the ?nite set α to [0, ∞). For 8

the most part, it will su?ce to deal with the case V = ? since the cells of the form BT × [0, ∞)α cover ?f . The piecewise Euclidean structure on ?f will be de?ned by declaring each BT × [0, ∞)α to have the product metric. Thus, the piecewise Euclidean metric on ?f will extend the one on K. Lemma 3.2. ?f has a decomposition into the cells, {BT,V } ∪ {c(T, V, α)}, de?ned above, where (T, V ) and (T, V, α) range over F . To prove this, we need to set up a standard identi?cation of the open cone of radius 1 on a k-simplex σ with the standard simplicial cone [0, ∞)k+1 ? Rk+1 . Let {vb }b∈B be the vertex set of σ for some ?nite index set B and let (xb )b∈B be barycentric coordinates on σ. Let RB be the Euclidean space of all functions B → R. Let S+ (RB ) denote the intersection of the standard simplicial cone [0, ∞)B with the unit sphere. Thus, S+ (RB ) is an “all right” spherical simplex (i.e., all edge lengths and all dihedral angles are π/2). Let {eb }b∈B be the standard basis for RB . De?ne a homeomorphism θB : σ → S+ (RB ), taking vb to eb , by xb vb → xb eb ( x2 )1/2 . b (3.1)

Let r : [0, 1) → [0, ∞) be some ?xed homeomorphism. The open cone on σ can be regarded as the points [t, x] in the join, v ? σ, of σ with a point v such that the join coordinate t is = 1. The homeomorphism from the open cone to [0, ∞)B is de?ned by [t, x] → r(t)θB (x). Proof of Lemma 3.2. Suppose α ∈ Flag(P). Then α = {T0 < · · · < Tk+l }, where Tk ∈ S and Tk+1 ∈ S. Put α′ := {T0 < · · · < Tk } and α′′ := {Tk+1 < / · · · < Tk+l }. The simplex σα′ lies in BTk while the simplex σα′′ is in the nonspherical face ?Tk+1 . We have σα = σα′ ? σα′′ and a point in σα has coordinates [t, x, y] where t ∈ [0, 1], x ∈ σα′ and y ∈ σα′′ . The points in σα ? σα′′ are those ′′ where the join coordinate t is = 1. The identi?cation σα ? σα′′ → σα′ × [0, ∞)α is given by [t, x, y] → (x, r(t)θα′′ (y)). Next we want to consider the link of the central vertex v? (corresponding to ?) in this cell structure. Let ?op denote the simplex on S. The nerve L is a subcomplex of ?op . If W is spherical, then L = ?op , while if W is in?nite, then L is a subcomplex of ??op . Moreover, ??op is a triangulation of the (n ? 1)sphere, for n = |S|. Assume W is in?nite. The link of v? in ?f is a certain subdivision L′ of ??op , which we shall now describe. The vertex set of L′ is the disjoint union S ∪ (P ? S). There are three types of simplices in L′ : 1) simplices σT in L corresponding to spherical subsets T ∈ S>? , 2) simplices σα in b?? (= b??op ) corresponding to ?ags α ∈ Flag(P ? S), 3) joins σT ? σα , with T ∈ S>? , α ∈ Flag(P ? S) and T < min α. Lemma 3.3. L′ is a subdivision of ??op and L ? L′ is a full subcomplex.

9

Proof. Suppose U is a minimal element of P ?S. Let σU denote the corresponding simplex in ??op . Introduce a “barycenter” vU ∈ ??op and then subdivide σU to a new simplicial complex (σU )′ by coning o? the simplices in ?σU . Each new simplex will have the form vU ? σT for some T ? U . Next, let U be an arbitrary element of P ? S and suppose by induction that we have de?ned the subdivision of (σU ′ ) for each U ′ ? U and hence, a subdivision (?σU )′ of ?σU . Introduce a barycenter vU of σU and subdivide by coning o? (?σU )′ . Each new simplex will have the form vU ? σU ′ for some U ′ ? U . In other words, each new simplex will be either of type 2 or type 3 above. It is clear that the subcomplex L is full. The following lemma is also clear. Lemma 3.4. L′ is the link of the central vertex in the piecewise Euclidean cell structure on ?f . The piecewise Euclidean metric induces a piecewise spherical metric on L′ extending the given metric on L. Since the link of the origin in [0, ∞)k+1 is the all right spherical k-simplex we get the following description of the metric on L′ . Lemma 3.5. The simplices in L′ have spherical metrics of the following types. 1) For T ∈ S>? , the simplex σT in L is the dual to the fundamental simplex for WT on the unit sphere in RT . In other words, for distinct elements s, t in T , the edge corresponding to {s, t} has length π ? π/mst . 2) The simplex σα corresponding to α ∈ Flag(P ? S) has its all right structure. In other words, each edge of σα has length π/2. 3) The simplex σT ? σα has the structure of a spherical join. In other words, the length of an edge connecting a vertex in σT to one in σα is π/2. Remark 3.6. The simplicial complex L′ is the nerve of a Coxeter group (W ′ , S ′ ) which contains W as a special subgroup. Namely, S ′ is the disjoint union, S ∪ (P ? S). Two generators of S are related as before. If U , V ∈ P ? S, then put m(U, V ) := 2 whenever U ? V or V ? U and m(U, V ) := ∞, otherwise. Similarly, if s ∈ S and U ∈ P ? S, then m(s, U ) = m(U, s) = 2 when s ∈ U and it is = ∞, otherwise. Since L′ is a triangulation of S n?1 , with n = |S|, this shows that any n generator Coxeter group is a special subgroup of a Coxeter group which acts cocompactly on a contractible n-manifold (such a Coxeter group is said to be type HM n in [9]). CAT(0) and CAT(1) metrics. Gromov [18] proved that a piecewise Euclidean metric on a polyhedron Y is locally CAT(0) if and only if the link in Y of each cell is CAT(1). In his proof that L(W, S) was CAT(1), Moussong [20] gave a criteria for certain piecewise spherical structures on simplicial complexes to be CAT(1). We recall his criteria below. A spherical simplex has size ≥ π/2 if each of its edges has length ≥ π/2. A spherical simplex σ ? Sk ? Rk+1 with 10

vertex set {v0 , . . . , vk+1 } and edge lengths lij := cos?1 (vi · vj ) is determined up to isometry by the (lij ). Conversely, a symmetric (k + 1) × (k + 1) matrix (lij ) of real numbers in [0, π) can be realized as the set of edge lengths of a spherical simplex if and only if the matrix (cos lij ) is positive de?nite, cf. [9, Lemma I.5.1, p. 513]. Suppose N is a simplicial complex with a piecewise spherical structure (i.e., each simplex has the structure of a spherical simplex). N has size ≥ π/2 if each of its simplices does. N is a metric ?ag complex if it satis?es the following condition: given any collection of vertices {v0 , . . . , vk } which are pairwise connected by edges, then {v0 , . . . , vk } is the vertex set of a simplex in N if and only if the matrix of edge lengths (lij ) can be realized as the matrix of edge lengths of an actual spherical simplex. (In other words, if and only if (cos lij ) is positive de?nite.) Moussong’s Lemma is the following. Lemma 3.7. (Moussong’s Lemma, [20] or [9, Appendix I.7]). Suppose a piecewise spherical simplicial complex N has size ≥ π/2. Then N is CAT(1) if and only if it is a metric ?ag complex. The next result follows immediately from our previous description of the piecewise spherical complex L′ . Lemma 3.8. L′ has size ≥ π/2 and is a metric ?ag complex. Corollary 3.9. L′ is CAT(1). Moreover, L is a totally geodesic subcomplex. Proof. The last sentence follows from the fact that L is a full subcomplex. Since the link of any simplex in a metric ?ag complex of size ≥ π/2 has the same properties (cf. [20, Lemma 8.3] or [9, Lemma I.5.11]), the link of any simplex in L′ is also CAT(1). Since the link in ?f ∩ ?V of any cell of the form BT,V × [0, ∞)α can be identi?ed with the link of the corresponding simplex σT ? σα in L′ , it follows that the link of each cell in ?f is CAT(1). Corollary 3.10. The piecewise Euclidean metric on U(W, ?f ) is CAT(0). Proof. The union of WT -translates of a Coxeter block BT in U(W, ?f ) is a Coxeter cell PT and the complete inverse image of BT in U(W, ?f ) is a disjoint union of copies of PT . Hence, U(W, ?f ) has a cell structure in which the cells are either translates of Coxeter cells of the form PT or translates of cells of the form PT × [0, ∞)α for some α ∈ Flag((P ? S)>T ). In either case the link of such a cell in U(W, ?f ) is identi?ed with the link of the corresponding cell in ?f . By Corollary 3.9, U(W, ?f ) is locally CAT(0). A space is CAT(0) if and only if it is locally CAT(0) and simply connected (cf. [18, p. 119] or [5, Ch. II.4]). U(W, ?f ) is contractible (hence, simply connected), since it is homotopy equivalent to U(W, K). Therefore, it is CAT(0). We also want to consider links of cells of the form BT,V or BT,V × [0, ∞)α in various cell complexes. (Here BT,V is a re?ecting face of BT .)

11

Lemma 3.11. Suppose c is a cell in ?f of the form c = BT,V or c = BT,V × [0, ∞)α for some V ? T ∈ S>? and α ∈ Flag((P ? S)>T ). Let d = BT or BT × [0, ∞)α be the corresponding larger cell in ?f and let σd be the corresponding simplex in L′ . Then Lk(c, ?f ) = Lk(σd , L′ ) ? τ (V ) Lk(c, U(W, ? ) = Lk(σd , L ) ? S
f ′ V

(3.2) (3.3)

Here SV is the unit sphere in the canonical representation of WV on RV and τ (V ) ? SV is the fundamental simplex. Moreover, suppose that Φ is a building of type (W, S), that R is the spherical residue of type V containing the base chamber and that S(R) := U(R, τ (V )) is the spherical realization of R. Then Lk(c, U(Φ, ?f ) = Lk(σd , L′ ) ? S(R). Proof. Let ?f denote the face of ?f ?xed by WV (cf. (2.1). Then V Lk(c, ?f ) = Lk(c, ?f ) ? Lk(?f , ?f ) V V = Lk(d, ?f ) ? τ (V ) = Lk(σd , L′ ) ? τ (V ) and similarly for formulas (3.3) and (3.4). We can now prove the main result of this section. Proof of Theorem 3.1. Any spherical building, such as S(R), is CAT(1) (e.g., see [8]) and the spherical join of two CAT(1)-spaces is CAT(1) (e.g., see [5]). So, the theorem follows from (3.4). Alternatively, it can be proved from Corollary 3.10 by using the argument in [8, §11]. A variation. In Section 4 we will need the following modi?cation of the previous construction. Given a subset U ? S, we will de?ne a new piecewise Euclidean metric on ?f ? ?U and then show that it induces CAT(0) metrics on U(WS?U , ?f ? ?U ) and U(R, ?f ? ?U ) for any (S ? U )-residue R of Φ. ? For each spherical subset T , let CT ? RT be the simplicial cone determined by the bounding hyperplanes of BT passing through the vertex x0 . (In other ? words, CT is the dual cone to the fundamental simplicial cone.) Let SS?U := S(WS?U , S ? U ) be the poset of spherical subsets of S ? U and let LS?U := L(WS?U , S ? U ) be the nerve of WS?U . ?f ? ?U has a cell structure with cells of the following two types: (a) BT,V , where T ∈ SS?U and V ? T , (b) BT × [0, ∞)α , where α ∈ P ? SS?U and T, V are as above. (3.4)

12

As before, each such cell is given the natural product metric. In e?ect we are putting Υ(?) ∪ ?U at in?nity. The piecewise Euclidean metric on ?f ??U induces one U(R, ?f ??U ). Let f U us describe the link, L′ S?U , of the central vertex in the new metric on ? ? ? . ′ The vertex set of LS?U is (S ? U ) ∪ (P ? SS?U ). As before, there are three types of simplices: 1) simplices σT in LS?U corresponding to spherical subsets T ∈ SS?U , 2) simplices σα in b?? (= b??op ) corresponding to ?ags α ∈ Flag(P ? SS?U ), 3) joins σT ? σα , with T ∈ SS?U , α ∈ Flag(P ? SS?U ) and T < min α.
op and L′ S?U can be identi?ed with a subdivision of ?? . Moreover, just as before, L′ has size ≥ π/2 and is a metric ?ag complex; hence, it is CAT(1). This S?U proves the following.

Theorem 3.12. Suppose U ? S and R is any (S ? U )-residue in Φ. Then the piecewise Euclidean metric on U(R, ?f ? ?U ), de?ned above, is CAT(0).

4

Metric spheres in U(Φ, ?f )

An n-dimensional cell complex X is CM (for “Cohen-Macaulay”) if H ? (X) is concentrated in degree n and is a free abelian group in that degree. Similarly, an n-dimensional, noncompact, contractible space X is SI (for “spherical at ? in?nity”) if Hc (X) is concentrated in degree n and is a free abelian group in that degree. We want to prove that U(Φ, ?f ) is SI. This is a consequence of Theorems 4.1 and 4.2 below. Suppose N is a CAT(1), piecewise spherical polyhedron and that p ∈ N . Let B(p, π/2) ? N denote the open ball of radius π/2 centered at p. De?ne a space P Np , called N punctured at p, by P Np := N ? B(p, π/2). We are interested in this concept when N = Lk(c), the link of a cell c in some CAT(0) complex X. In this case we will write PLkp (c) for P Np and call it the punctured link of c at p. Theorem 4.1. (Brady, McCammond and Meier [4]). Let X be a CAT(0), piecewise Euclidean cell complex (with ?nitely many shapes of cells). If for each cell c in X and for each p ∈ Lk(c), the spaces Lk(c) and PLkp (c) are (n ? dim c)-acyclic, then X is n-acyclic at in?nity. In particular, if X satis?es this condition and is n-dimensional, then it is SI. Remarks. In [4] the hypothesis of the above theorem is that X is the universal cover of a ?nite, nonpositively curved complex; however, the proof clearly works with a weaker hypothesis such as ?nitely many shapes of cells (which holds in our case). The proof of the theorem uses Morse theory for polyhedral complexes. Roughly, it goes as follows. Let ρ : X → R be the distance from some base point x0 , i.e., ρ(x) := d(x, x0 ). The spheres S(r) of radius r centered at x0 ∈ X are the level sets of ρ. Call a point x a critical point (of ρ) if it is the closest point to x0 in 13

some closed cell c. It follows from the CAT(0) hypothesis that the critical points are isolated. If, for su?ciently small ε, there is no critical point in the annular region between S(r + ε) and S(r), then S(r + ε) and S(r) are homeomorphic. On the other hand, the e?ect of crossing a critical point x ∈ S(r) is to remove a contractible neighborhood of x in S(r) and replace it by the punctured link PLk(x)p where p is the direction at x of the geodesic from x to x0 . (If x lies in the relative interior of a k-dimensional cell c, then Lk(x) ? Sk?1 ? Lk(c).) = So, the e?ect on the homotopy type of the level sets is to replace a contractible neighborhood by a copy of a suspension of a punctured link. It follows that, under the hypotheses of Theorem 4.1, each metric sphere is CM . Since X is ? CAT(0), metric balls are contractible and since Hc (X) = lim H ? (B(r), S(r)) is ? → concentrated in the top degree, X is SI. Theorem 4.2. (Dymara–Osajda [16] and Schulz [24]). Suppose R is a spherical building of type (WT , T ) and that S(R) (:= U(R, τ (T ))) is its spherical realization. Then for any p ∈ S(R), the space P S(R)p is CM . An immediate corollary to Theorems 4.1 and 4.2 is the following. Theorem 4.3. (cf. [2]). With notation as in Section 3 and above, given any building Φ, its realization U(Φ, ?f ) is SI. As a corollary to Theorem 3.12 we get the following relative version of Theorem 4.1. Theorem 4.4. Suppose U ? S and R is any (S ? U )-residue in Φ. Then U(R, ?f ? ?U ) is SI. Corollary 4.5. Suppose U ? S. Then U(Φ, ?f ? ?U ) is SI. Proof. U(Φ, ?f ? ?U ) is the disjoint union of the spaces U(R, ?f ? ?U ) where R ranges over the (S ? U )-residues. By the previous theorem, the compactly supported cohomology of each such component is free abelian and concentrated in the top degree.

5

Cohomology with ?nite support and compact support

? Given a CW complex Y , C?n (Y ) denotes the complex of ?nitely supported cellu? lar cochains on Y and H?n (Y ) its cohomology. When Y is only required to be a ? topological space, Hc (Y ) denotes its compactly supported singular cohomology, i.e., ? Hc (Y ) := lim H ? (Y, Y ? C), ? → where the direct limit is over all compact subsets C ? Y . If Y is a locally ?nite ? CW complex, H?n (Y ) ? Hc (Y ). = ? ′ Suppose Z, Z are mirrored spaces over S. A map F : Z → Z ′ is mirrored ′ if F (ZT ) ? ZT for all T ? S; it is a mirrored homotopy equivalence if F |ZT : ′ ZT → ZT is a homotopy equivalence for all T (including T = ?).

14

Given a mirrored space X, let us say that Υ(X) is collared in X if there is an increasing family N = {Nε }ε∈(0,a] of open neighborhoods of Υ(X) (“increasing means that Nε ? Nε′ whenever ε < ε′ ) such that the following two properties hold: (i)
ε∈(0,a]

Nε = Υ(X) and

(ii) For each ε > 0, the inclusion Υ(X) ?→ N ε is a mirrored homotopy equivalence. For example, if X is the simplex ?, then Υ(?) is collared in ?. (Proof: Υ(?) is a union of closed faces and we can take Nε to be its ε-neighborhood in ?.) More generally, if X is any ?nite CW complex, then Υ(X) is collared in X. Property (ii) implies that U(Φ, Υ(X)) ?→ U(Φ, Nε ) is a homotopy equivalence. To simplify notation, in what follows we often write UX instead of U(Φ, X). Lemma 5.1. Suppose X is a ?nite, mirrored CW complex . Then
? ? Hc (UX f ) = H?n (UX , UΥ(X) ). ? ? Proof. Since X is a ?nite complex, H?n (UX , UΥ(X) ) = Hc (UX , UΥ(X) ). Since Υ(X) is collared in X, we have a family N = {Nε } of open neighborhoods. For a given N = Nε ∈ N , let ?N denote the boundary of N . Since N is mirrored homotopy equivalent to Υ(X), ? H?n (UX , UΥ(X) ) ? Hc (UX , UN ) ? Hc (U(X?N ) , U?N ), = ? = ?

where the second isomorphism is an excision. As ε → 0, X ? Nε → X f , so any compact subset C of UX f lies within some UX?Nε . Hence,
? ? Hc (UX f ) = lim H ? (UX f , UX f ? C) = Hc (UX?N , U?N ). ? →

Combining these equations, we get the result. Example 5.2. ?f is a thickened version of K and U(Φ, ?f ) is a thickened version of U(Φ, K). Hence, U(Φ, ?f ) and U(Φ, K) are homotopy equivalent. On the other hand, their compactly supported cohomology groups can be completely di?erent. For example, suppose W is the free product of three copies of Z/2. Then ?f is a triangle with its vertices deleted while K is a tripod. U(W, ?f ) can be identi?ed with the hyperbolic plane while U(W, K) is the regular trivalent tree. (This is the familiar picture of the congruence 2 subgroup of P SL(2, Z) acting on the hyperbolic plane.) The compactly supported cohomology of U(W, ?f ) is that of the plane (i.e., it is concentrated in degree 2 and is isomorphic to Z in that degree), while the compactly supported cohomology of U(W, K) is that of a tree (i.e., it is concentrated in degree 1 and is a countably generated, free abelian group in that degree).

15

6

Cohomology with coe?cients in I(A)

Let A (= A(Φ)) be the free abelian group of ?nitely supported, Z-valued functions on Φ. For each subset T of S, de?ne a Z-submodule of A: AT := {f ∈ A | f is constant on each residue of type T }. Note that A? = A. Also note that AT = 0 whenever T is not spherical. We have AU ? AT when T ? U . Let A>T be the Z-submodule of AT spanned by the AU with T a proper subset of U . Put DT := AT /A>T . Remark 6.1. When Φ = W , A is the group ring ZW and AT consists of elements in ZW which are constant on each left coset wWT . Let us assume T is spherical (otherwise AT = 0). Let In(w) := {s ∈ S | l(ws) < l(w)} be the set of letters with which a reduced expression for w can end. Since T is spherical, each left coset of WT has a unique representative w which has a reduced expression ending in the longest element of WT ; hence, this representative has T ? In(w). As a basis for DT , we can take images of the elements in AT corresponding to cosets which have longest representatives with T = In(w). De?nition 6.2. The abelian group A and the family of subgroups {AT }T ?S de?ne a “coe?cient system” I(A) on X so that the i-cochains with coe?cients in I(A) are given by C i (X; I(A)) := AS(c)
c∈X (i)

where X (i) denotes the set of i-cells in X. We continue to write UX for U(Φ, X). Let U (i) denote the set of i-cells in UX . Given a chamber ? ∈ Φ and an i-cell c in X, let ? · c denote the corresponding i-cell in UX . Given a ?nitely supported function α : U (i) → Z, and an i-cell c ∈ X (i) with S(c) spherical, we get an element f ∈ AS(c) de?ned by f (?) := α(? · c). Of course, when S(c) is not spherical, AS(c) = 0. So, for any ?nite, mirrored CW complex X, this establishes an isomorphism C i (X; I(A)) ? C?n (UX , UΥ(X) ). = i (6.1)

In other words, the isomorphism (6.1) is given by identifying a ?nitely supported function on the inverse image in U (i) of a cell c ∈ X (i) with a function on Φ (i.e., with an element of A) which is constant on S(c)-residues containing the cells ? · c. The coboundary maps in C ? (X; I(A)) are de?ned by using these isomorphisms to transport the coboundary maps on ?nitely supported cochains to C ? (X; I(A)). This means that the coboundary maps in C ? (X; I(A)) are de?ned by combining the usual coboundary maps in C ? (X) with the inclusions AU ?→ AT for U ? T . So, we have the following. 16

Lemma 6.3. For X a ?nite, mirrored CW complex,
? ? H? (X; I(A)) = H?n (UX , UΥ(X) ) = Hc (UX f ).

Proof. The second equation is from Lemma 5.1. Remark 6.4. If Φ = W , then A = ZW . C ? (X; I(ZW )) can be interpreted as the equivariant cochains on U(W, X) with coe?cients in ZW . When X is a ?nite complex, Lemma 6.3 asserts
? C ? (X; I(ZW ) = C?n (U(W, X), U(W, Υ(X))).

The corresponding cohomology groups were studied in [10, 7]. In particular, when X = K, these cohomology groups are isomorphic to H ? (W ; ZW ). In what follows the coe?cient system is usually I(A) and we shall generally omit it from our notation, writing H( ) and C( ) instead of H( ; I(A)) and C( ; I(A)). (In other words, the coe?cients I(A) are implicit when we use the calligraphic C or H notation.) As usual, ? is the simplex of dimension n = |S|?1 with its codimension one faces indexed by S. Theorem 6.5. (cf. [15, Theorem B] as well as [11, 14]). H? (?) is concentrated in degree n and is free abelian; moreover, Hn (?) = D? . More generally, for any subset U of S, H? (?, ?U ) is concentrated in degree n and is free abelian and Hn (?, ?U ) = A As
s∈S?U ? Proof. By Lemma 6.3, H? (?) = Hc (U?f ) and by Theorem 4.3, the right hand side is concentrated in degree n and is free abelian. The cochain complex looks like · · · → C n?1 (?) → C n (?) → 0,

where C n (?) = A and C n?1 (?) = n is the quotient A

As . It follows that cohomology in degree As = D? .
s∈S

? Similarly, H? (?, ?U ) = Hc (U?f ??U ), and by Corollary 4.5 the right hand side is concentrated in degree n and is free abelian. Since C n (?, ?U ) = A and C n?1 (?, ?U ) = s∈U As , we get the ?nal formula in the theorem. /

Corollary 6.6. D? is free abelian. Remark 6.7. We saw in Section 2 that whenever W is in?nite, U(W, ?f ) is homeomorphic to Euclidean space Rn . Hence, the compactly supported coho? mology of U(W, ?f ) is that of Hc (Rn ). Similarly, U(R, ?f ? (?f )U ) is homen omorphic to R , for each (S ? U )-residue R.

17

Remark 6.8. U(W, ?f ) is a thickened version of U(W, K). In the proof of Corollary 3.10 we explained the cellulation of U(W, K) by “Coxeter cells.” The corresponding cellular chain complex, C? (U(W, K)), has the form ZW ←?
s∈S

H s ←?
T ∈S (2)

H T ←? · · ·

where H T is the representation induced from the sign representation of WT and where S (k) is the set of spherical subsets with k elements (cf. [11, §8]). The cochain complex C ? (?) for n ? 2 ≤ ? ≤ n, looks like · · · ?→
T ∈S (2)

AT ?→
s∈S

As ?→ ZW.

n?? So, C? (U(W, K)) and Cc (U(W, ?f )) are Poincar? dual. e

7

The Decomposition Theorem

We want to prove a version of Theorem 6.5 for lower dimensional spherical faces of ?. Suppose T is a spherical subset of S. Put σ = ?T and m = n ? |T |. Let U be an arbitrary subset of S ? T . We continue the policy of omitting the coe?cient system I(A) from our notation. Proposition 7.1. Each of the following cohomology groups is concentrated in the top degree and is a free abelian group in that degree: H? (σ, σ U ), H? (σ U , ?(σ U )), and H? (σ U ) (The top degrees are m, m ? 1, and m ? 1, respectively.) Moreover, Hm (σ, σ U ) = AT
s∈(S?T )?U

AT ∪{s} , AT ∪{s} ,

(7.1) (7.2)

Hm?1 (σ U , ?(σ U )) =
s∈U

Hm?1 (σ U ) =
s∈U

AT ∪{s}
s∈U t∈(S?T )?U

AT ∪{s,t} ,

(7.3)

Proof. We ?rst prove H? (σ, σ U ) is concentrated in degree m and is free abelian. The proof is by induction on the number of elements in T . It holds for |T | = 0 by Theorem 6.5. Suppose T = T ′ ∪ {s}, U ′ = U ∪ {s} and τ = ?T ′ . The exact ′ sequence of the triple (τ, τ U , τ U ) gives · · · → H??1 (σ, σ U ) → H? (τ, τ U ) → H? (τ, τ U ) → · · ·
′ (This uses the excision, H??1 (τ U , τ U ) ? H??1 (σ, σ U ).) By inductive hypothe= sis, the last two terms are free abelian and concentrated in degree m + 1. Hence, ′

18

H? (σ, σ U ) is concentrated in degree m. It is free abelian since it injects into a free abelian group. That the other cohomology groups are free abelian and are concentrated in the top degree follows from various exact sequences. For example, for H? (σ U , ?(σ U )), consider the sequence of the triple (σ, ?σ, σ S?U ): → H??1 (σ U , ?(σ U )) → H? (σ, ?σ) → H? (σ, σ S?U ) → (7.4)

where we have used the excision H??1 (?σ, σ S?U ) ? H??1 (σ U , ?(σ U )) to identify = the ?rst term. The second and third terms are free abelian and concentrated in degree m; hence, the ?rst term is free abelian and concentrated in degree m ? 1. For H? (σ U ), we have the exact sequence of the pair (σ, σ U ): → H??1 (σ U ) → H? (σ, σ U ) → H? (σ) → (7.5)

where the second and third terms are free abelian and concentrated in degree m. Hence, H? (σ U ) is free abelian and is concentrated in degree m ? 1. It remains to verify formulas (7.1), (7.2) and (7.3). We have C m (σ, σ U ) = AT and C m?1 (σ, σ U ) = s∈U AT ∪{s} , so / Hm (σ, σ U ) = AT
s∈U /

AT ∪{s} ,

proving (7.1). (In particular, Hm (σ) = DT and Hn (?) = D? .) In the exact sequence (7.4), we have Hm (σ, ?σ) = AT and, by (7.1), Hm (σ, σ S?U ) = AT / s∈U AT ∪{s} ; hence, (7.2). Using (7.1) to calculate the second and third terms of (7.5), we get Hm?1 (σ U ) =
s∈S?T

AT ∪{s}
s∈(S?T )?U

AT ∪{s}

and this can be rewritten as (7.3). By Proposition 7.1, DT is the free abelian group Hm (?T ). So, for each T ∈ S, we can choose a splitting ιT : DT → AT of the projection map AT → DT . ? De?nition 7.2. Let AT := ιT (DT ). (It is a direct summand of the free abelian T group A .) Proposition 7.3. Hm (σ, σ U ) =
V ?T V ?T ?U

? AV ? AV
V ?T (V ?T )∩U=?

(7.6)

Hm?1 (σ U , ?(σ U )) = Hm?1 (σ U ) =
V ?T V ?T ?U

(7.7)

? AV

(7.8)

19

Proof. Assume by induction that (7.6) through (7.8) hold for dim σ = m?1 and assume as well that they hold when dim σ = m and U is replaced by U ′ with |U ′ | < |U |. Write U = {s} ∪ U ′ , for some s ∈ U . Consider the exact sequence of the triple (σ U , σs ∪ ?(σ U ), ?(σ U )): 0 → Hm?1 (σ U , ?(σ U )) → Hm?1 (σ U , ?(σ U )) → Hm?1 (σs , (σs )S?U ) → 0, (7.9) ′ ′ where we have used the excisions H? (σ U , σs ∪ ?(σ U )) = H? (σ U , ?(σ U )) and ′ H? (σs ∪ ?(σ U ), ?(σ U )) = H? (σs , (σs )S?U ) to rewrite the ?rst and third terms. By induction, Hm?1 (σ U , ?(σ U )) =
V ?T V ∩U ′ =?
′ ′ ′ ′ ′

? AV , ? AV
V ?T ∪{s} V ?T ∪{s}∪U ′

Hm?1 (σs , (σs )S?U ) =



Substituting these into the last two terms of (7.9), we get Hm?1 (σ U , ?(σ U )) =
V ?T V ∩U=?

? AV ,

which is (7.7). Next consider the Mayer-Vietoris sequence of σ U = σ U ∪ σs : 0 → Hm?2 ((σs )U ) → Hm?1 (σ U ) → Hm?1 (σ U ) ⊕ Hm?1 (σs ) → 0. By induction, Hm?2 ((σs )U ) =
V ?T ∪{s} V ?(T ∪{s})?U ′
′ ′ ′



(7.10)

? AV

and Hm?1 (σ U ) =
V ?T V ?T ?U ′



? AV

? and Hm?1 (σs )) = AT ∪{s} . Substituting these into (7.10) we get Hm?1 (σ U ) =
V ?T V ?T ?U

? AV ,

which is (7.8). Sequence (7.5) is 0 → Hm?1 (σ U ) → Hm (σ, σ U ) → Hm (σ) → 0. ? Substituting (7.8) for the ?rst term and AT for the third, we get formula (7.6) for the middle term. We have Hm (σ, ?σ) = AT . Hence, in the special case U = S ? T formulas (7.6) and (7.8) give the following theorem (cf. [25], [11, Thm. 9.11], [10, Cor. 3.3]). 20

Theorem 7.4. (The Decomposition Theorem). For each subset T of S AT =
V ?T

? AV .

For any Z-submodule B ? A, put B T := AT ∩ B. Suppose we have a direct sum decomposition of Z-modules, A = B ⊕ C, so that for each T ? S, AT = B T ⊕ C T . As explained in [10, §2] this leads to a decomposition of coe?cient systems I(A) = I(B)⊕I(C) so that for any mirrored CW complex X we have a decomposition of cochain complexes C ? (X; I(A)) = C ? (X; I(B))⊕ C ? (X; I(C)). Since ? AV if V ? T , ? (AV )T = 0 otherwise, ? the formula in the Decomposition Theorem satis?es (A? )T = V ?T (AV )T , for ? all T ? S. So, we get a decomposition of coe?cient systems I(A) = AV and a corresponding decomposition of cochain complexes: C ? (X; I(A)) =
V

? C ? (X; I(AV ))

(7.11)

8

Cohomology of buildings

Just as in [10, Thm. 3.5], the Decomposition Theorem (Theorem 7.4) implies the following. Theorem 8.1. (cf. [15, Cor. 8.2], [11, Thm. 10.3], [10, Thm. 3.5]). Suppose X is a ?nite, mirrored CW complex. Then
? Hc (U(Φ, X f )) ? = T ∈S ? Hc (U(Φ, X f ))

H ? (X, X S?T ) ? AT .

is the cohomology of C ? (X; I(A)). ForProof. By Lemma 6.3, mula (7.11) gives a decomposition of cochain complexes: C ? (X; I(A)) =
T ∈S

? C ? (X; I(AT ))

We have

? C k (X; I(AT )) =
c∈X (k)

? AT ∩ AS(c) =
c∈X (k) c?X S?T

? AT .

? ? So, an element of C k (X; I(AT )) is just an ordinary AT -valued cochain on X S?T which vanishes on X , i.e., ? ? C ? (X; I(AT )) = C ? (X, X S?T ) ? AT ; C ? (X; I(A)) =
T ∈S

so,

? C ? (X, X S?T ) ? AT .

Taking cohomology, we get the result. 21

The most important special case of the previous theorem is the following. Corollary 8.2.
? Hc (U(Φ, K)) ? = T ∈S

? H ? (K, K S?T ) ? AT .

9

The G-module structure on cohomology

Assume X has a W -?nite mirror structure (i.e., X = X f ). Suppose G is a group of automorphisms of Φ. Then A, AT , A>T and DT are naturally right ? G-modules and so is the cochain complex Cc (U(Φ, X)) as is its cohomology. The discussion in Section 8 is well adapted to studying the G-module structure ? of Hc (U(Φ, X)). As in [10], we should not expect a direct sum splitting of Gmodules analogous nonequivariant splitting of Theorem 8.1; rather there should be a ?ltration of cohomology by G-submodules with associated graded terms similar to those in the direct sum. We show below that this is the case. For each nonnegative integer p de?ne a G-submodule Fp of A by Fp :=
|T |≤p

AT .

This gives a decreasing ?ltration: A = F0 ? · · · ? Fp ? Fp+1 · · · . (9.1)

As in [10], it follows from the Decomposition Theorem that the associated graded terms are DT . Fp /Fp+1 =
|T |=p

As in Section 6, we get a coe?cient system I(Fp ), a cochain complex C ? (X; I(Fp )) and corresponding cohomology groups H? (X; I(Fp )). The ?ltration (9.1) leads ? to a ?ltration of H? (X; I(A)) (= Hc (U(Φ, X))) by right G-modules, H? (X; I(A)) ? · · · H? (X; I(Fp )) ? H? (X; I(Fp+1 )) · · · . The Decomposition Theorem implies that 0?→ H? (X; I(Fp+1 ))?→ H? (X; I(Fp ))?→ H ? is short exact. From this we deduce the following. Theorem 9.1. Suppose G is a group of automorphisms of Φ. Then the ?ltration ? (9.2) of Hc (U(Φ, X)) by right G-modules has associated graded term, H? (X; Fp )/H? (X; Fp+1 ) ? =
T ∈S (p)

(9.2)

C ? (X; I(Fp )) C ? (X; I(Fp+1 ))

?→ 0

H ? (X, X S?T ) ? DT .

22

Cocompact G-action. In this ?nal paragraph we assume that there are only ?nitely many G-orbits on Φ and that X is a ?nite complex. These hypotheses imply that the quotient space U(Φ, X)/G is compact and since X is a ?nite complex, each of the cohomology groups H ? (X) is ?nitely generated. As a corollary to Theorem 9.1, we have the following.
? Corollary 9.2. With the above hypotheses, Hc (U(Φ, X)) is a ?nitely generated G-module.

Proof. The assumption that G has only ?nitely many orbits of chambers implies that each AT is ?nitely generated G-module; hence, so is each DT . Example 9.3. Suppose G is chamber transitive, i.e., suppose Φ consists of a single G-orbit. Choose a base chamber ?0 , let B be its stabilizer and for each T ∈ S, let GT be the stabilzer of the T -residue containing ?0 . Then A is the G-module Z(G/B) of ?nitely supported functions on G/B and AT can be identi?ed with Z(G/GT ). For each U ? T , we have a natural inclusion Z(G/GU ) → Z(G/GT ) induced by the projection G/GT → G/GU and DT is the G-module formed by dividing out the sum of the images of Z(G/GU ) in Z(G/GT ) for all U ? T . Here are some further corollaries to Corollary 8.2 and Theorem 9.1. Corollary 9.4. Suppose Γ ? G is a cocompact lattice in G. Then H ? (Γ; ZΓ) has, in ?ltration degree p, associated graded Γ-module H ? (K, K S?T ) ? DT .
T ∈S (p)

In particular, when Γ is torsion-free, its cohomological dimension is given by cd(Γ) = max{k | H k (K, K S?T ) = 0, for some T ∈ S}. A torsion-free group Γ is an n-dimensional duality group if H ? (Γ; ZΓ) is free abelian and concentrated in dimension n. Following [12, De?nition 6.1], we say that the nerve of a Coxeter system has punctured homology concentrated in dimension n if for all T ∈ S, H ? (K S?T ) is free abelian and concentrated in dimension n. Corollary 8.2 gives us a (correct) proof of the following result, stated in [12, Theorem 6.3]. Corollary 9.5. Suppose Γ ? G is a torsion-free, cocompact lattice in G. Then the following are equivalent. 1) Γ is an n-dimensional duality group. 2) W is an n-dimensional virtual duality group. 3) The nerve of (W, S) has punctured homology concentrated in dimension n?1.

23

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