The cohomology of Bestvina-Brady groups

Ian J. Leary^{∗}  Müge Saadetoğlu^{†}

July 25, 2006

Abstract

For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus, our computation gives the homology of the groups introduced by M. Bestvina and N. Brady in [3]. We compute the cohomological dimension of each of these groups.

1 Introduction

Let 𝕋 be the circle, or 1-dimensional unitary group, given a CW-structure with one 0-cell and one 1-cell. Suppose also that the identity element of the group is chosen to be the 0-cell. For a set V, let T (V) denote the direct sum T (V) = ⨁_{v∈V} 𝕋. There is a natural CW-structure on T (V) in which the i-cells are in bijective correspondence with i-element subsets of V.

For the purposes of this paper, a simplicial complex will be defined abstractly as a non-empty set of finite sets which is closed under inclusion. The one element members of the set of sets are the vertices of the simplicial complex. Every simplicial complex (including the empty simplicial complex) contains a unique −1-simplex corresponding to the empty set.

If σ is an finite subset of V, the closure in T (V) of the cell corresponding to σ is equal to T (σ), and consists of all the cells corresponding to subsets of σ. It follows that there is a bijective correspondence between simplicial complexes whose vertex set is contained in V and non-empty subcomplexes of T (V) (see [8, 3.23] for this statement in the case when V is finite). The empty simplicial complex corresponds to the subcomplex T_{∅} consisting of just the single 0-cell of T (V), and a non-empty simplicial complex L corresponds to the complex T_{L} defined by

∗Partially supported by NSF grant DMS-0505471

†Supported by the British Council and by the Ohio State Mathematical Research Institute

T_{L} = ⋃_{σ∈L} T (σ).

The fundamental group of T_{L} and the cohomology ring of T_{L} are easily described in terms of L, and there is a characterisation of those L for which T_{L} is aspherical. (We shall describe all of these results below.)

A point in T (V) is a vector (t_{v}) of elements of 𝕋 indexed by V, such that only finitely many t_{v} are not the identity element. The group multiplication induces a map μ : T (V) → 𝕋 which takes the point (t_{v}) to the product of all of the non-identity t_{v}’s. For each L this induces a cellular map μ_{L} : T_{L} → 𝕋, and μ_{L} is surjective when L is non-empty. Our aim is to study the homology and cohomology of the space T̃_{L}, the infinite cyclic cover of T_{L} obtained by pulling back the universal cover of 𝕋 via μ_{L}.

\begin{matrix}T̃_{L} & ⟶ & ℝ \\ ↓ & & ↓ \\ T_{L} & ⟶^{μ_{L}} & 𝕋 \end{matrix}

For each L, we describe the homology of T̃_{L}, together with information about the ℤ-action induced by the action of ℤ by deck transformations on T̃_{L}. We deduce that for any non-trivial ring R, the R-homology of T̃_{L} is finitely generated as an R-module if and only if L is finite and R-acyclic. In all cases when L is R-acyclic, we give a complete description of the cohomology ring H^{∗}(T̃_{L}; R).

Let G_{L} denote the fundamental group of T_{L}. Since the fundamental group of a CW-complex depends only on its 2-skeleton, the group G_{L} depends only on the 1-skeleton of the simplicial complex L. The presentation for G_{L} coming from the cell structure on T_{L} has one generator for each vertex of L, subject only to the relation that the generators v and w commute whenever {v, w} is an edge in L. These groups are known as right-angled Artin groups. It can be shown that T_{L} is aspherical if and only if L is a flag complex. Every simplicial complex may be completed to a flag complex with the same 1-skeleton (just add in a simplex for each finite complete subgraph of the 1-skeleton) and so one sees that the spaces of the form T_{L} include models for the classifying spaces of all right-angled Artin groups.

When L is non-empty, T̃_{L} is connected and the fundamental group of T̃_{L} is the kernel of the induced map μ_{∗} : G_{L} → ℤ, which sends each of the generators for G_{L} to 1 ∈ ℤ. Call this group H_{L}. The groups H_{L} are known as Bestvina-Brady groups. In the case when L is a finite flag complex, M. Bestvina and N. Brady showed that the homological finiteness properties of H_{L} are determined by the homotopy type of L. For example, they show that H_{L} is finitely presented if and only if L is 1-connected [3]. For an explicit presentation for H_{L} for any L, see [6].

Let L be an n-dimensional flag complex. It is easy to show that in this case, the cohomological dimension of the group G_{L} is equal to n + 1. The cohomological dimension of G_{L} over any non-trivial ring R is also equal to n+1. It also follows easily that the cohomological dimension of H_{L} is equal to either n or n+1. Our computations together with some of the results from [3] allow us to determine the cohomological dimension of H_{L}, at least in the case when R is either a field or a subring of the rationals. If n = 0 and L is a single point, then H_{L} is the trivial group. Otherwise, if there exists an R-module A such that H^{n}(L; A) ≠ 0, then H_{L} has cohomological dimension n + 1 over R. If there exists no such A, then H_{L} has cohomological dimension n over R. Note that in contrast to the case of G_{L}, the cohomological dimension of H_{L} may vary with the choice of ring R. As a corollary we deduce that the trivial cohomological dimension and cohomological dimension of H_{L} are equal.

Some of these results appeared, with slightly different proofs, in the Southampton PhD thesis of the second named author. For some results, we give a brief sketch of a second proof. Some computations of low-dimensional ordinary cohomology (and many other algebraic invariants) for a special class of the Bestvina-Brady groups also appear in a recent preprint of S. Papadima and A. Suciu [12].

2 Homology and cohomology of T_{L}

The differential in the cellular chain complex for T (V) is trivial, and hence so is the differential in the cellular chain complex for T_{L}, for any L. It follows that for any ring R, H_{i}(T_{L}; R) is a free R-module with basis the i-cells of T_{L}, or equivalently the (i − 1)-simplices of L. The differential in the cellular cochain complex is also trivial. The group H^{i}(T_{L}; R) is isomorphic to a direct product of copies of R indexed by the (i − 1)-simplices of L. To describe the ring structure on the cohomology, we first consider the case of the torus T (V).

The cohomology ring H^{∗}(T (V); R) can be described as the exterior algebra Λ_{R,V}^{∗}. A homogeneous element f ∈ Λ_{R,V}^{n} is an alternating function f : V^{n} → R, where we say that a function is alternating if the following two conditions are satisfied:

1. f(v_{1}, …, v_{n}) = 0 whenever there exists 1 ≤ i < j ≤ n with v_{i} = v_{j};

2. f(v_{1}, …, v_{i}, v_{i+1}, …, v_{n}) = −f(v_{1}, …, v_{i+1}, v_{i}, …, v_{n}) for any i with 1 ≤ i < n.

If f ∈ Λ^{i} and g ∈ Λ^{n−i}, the product f.g is the so-called ‘shuffle product’. This is defined in terms of the pointwise product by the equation

f.g(v_{1}, …, v_{n}) = ∑_{π} ϵ(π)f(v_{π(1)}, …, v_{π(i)})g(v_{π(i+1)}, …, v_{π(n)}),

where ϵ(π) ∈ {±1} denotes the sign of the permutation π, and the summation ranges over all permutations π such that

π(1) < π(2) < ⋯ < π(i) and π(i + 1) < π(i + 2) < ⋯ < π(n).

(The ‘shuffles’ or permutations of the above type are chosen because they are a set of coset representatives in S_{n} for the subgroup S_{i} × S_{n−i}, so that each i-element subset of {1, …, n} is equal to {π(1), …, π(i)} for exactly one such π. Any other set of coset representatives could be used instead.)

There is a similar description of the ring structure on H^{∗}(T_{L}; R) for any simplicial complex L, as the exterior face ring Λ_{R}^{∗}(L) of L. If V is the vertex set of L, Λ_{R}^{∗}(L) is the quotient of Λ_{R,V}^{∗} by the homogeneous ideal I_{L}, with generators the functions that vanish on every n-tuple (v_{1}, …, v_{n}) which does not span a simplex of L. The inclusion of T_{L} in T (V) induces a homomorphism of cohomology rings

Λ_{R,V}^{∗} ≅ H^{∗}(T (V); R) → H^{∗}(T_{L}; R),

and it is easy to check (from the additive description of H^{∗}(T_{L}; R) given above) that this homomorphism is surjective and that its kernel is I_{L}. Hence one obtains a theorem which was first stated in [10] in the case when L is finite:

Theorem 1 For any simplicial complex L and ring R, the cohomology ring H^{∗}(T_{L}; R) is isomorphic to the exterior face ring Λ_{R}^{∗}(L).

For any path-connected space X, there is a natural isomorphism between H^{1}(X; ℤ) and Hom(π_{1}(X), ℤ). The element of H^{1}(T_{L}; ℤ) = Λ_{ℤ}^{1}(L) that corresponds to the homomorphism μ_{∗} : G_{L} → ℤ is the element β_{L}, the constant function which takes each vertex of L to 1 ∈ ℤ. By a slight abuse of notation, write β_{L} also for the element of Λ_{R}^{1}(L) that takes each vertex of L to 1 ∈ R.

In any anticommutative ring, multiplication by an element of odd degree gives rise to a differential. The cochain complex structure on Λ_{R}^{∗}(L) given by multiplication by β_{L} is easily described.

Theorem 2 For any ring R, there is a natural isomorphism of cochain complexes

(Λ_{R}^{∗}(L), β×) ≅ C_{+}^{∗−1}(L; R)

between the exterior face ring of L with differential given by left multiplication by β_{L}, and the augmented simplicial cochain complex of L shifted in degree by one.

Proof. In degree i, each of the two graded R-modules is isomorphic to a direct product of copies of R indexed by the (i − 1)-simplices of L, or equivalently the R-valued functions on the oriented (i−1)-simplices of L, where f(−σ) = −f(σ) if −σ is the same simplex as σ with the opposite orientation. It remains to show that this isomorphism is compatible with the differentials on the two cochain complexes.

Let f be an R-valued function on the (i − 1)-simplices of L, and compare the functions β.f and δf, the image of f under the differential on C_{+}^{∗−1}(L; R). If (v_{0}, …, v_{i}) is the vertex set of an oriented i-simplex of L, then

β.f(v_{0}, …, v_{i}) = ∑_{j=0}^{i}(−1)^{j}β(v_{j})f(v_{0}, …, v_{j−1}, v_{j+1}, …, v_{i})

= ∑_{j=0}^{i}(−1)^{j}f(v_{0}, …, v_{j−1}, v_{j+1}, …, v_{i})
= δf(v_{0}, …, v_{i}).

This completes the proof.

3 Higher homotopy of T_{L}

Recall that a full subcomplex M of a simplicial complex L is a subcomplex such that if σ is any simplex of L and each vertex of σ is in M, then σ is in M.

Proposition 3 If M is a full subcomplex of L, then T_{M} is a retract of T_{L}.

Proof. Let W be a subset of V. There is an isomorphism of topological groups T (V) ≅ T (W) ⊕ T (V − W). The inclusion

i : T (W) ≅ T (W) ⊕ {1} → T (V)

and projection

π : T (V) → T (V)/{1} ⊕ T (V − W) ≅ T (W)

satisfy π ∘ i = 1_{T (W)}, and show that T (W) is a retract of T (V).

Now suppose that L is a simplicial complex with vertex set V and that M is the full subcomplex with vertex set W ⊆ V. Then T_{L} is a subcomplex of T (V), and T_{M} is a subcomplex of T (W). The maps i and π, when restricted to T_{M} and T_{L}, show that T_{M} is a retract of T_{L} as claimed.

Recall that a simplicial complex L is said to be flag if every finite complete subgraph of the 1-skeleton of L is the 1-skeleton of a simplex of L. Any full subcomplex of a flag complex is flag.

Proposition 4 T_{L} is aspherical if and only if L is a flag complex.

Proof. A subset of a CW-complex that meets the interior of infinitely many cells contains an infinite discrete set, so cannot be compact. Hence any map from a sphere to a CW-complex has image inside a finite subcomplex and any homotopy between maps of a sphere into a CW-complex has image contained in a finite subcomplex. Thus it suffices to consider the case when L is finite.

Suppose that L is a finite flag complex with vertex set V. If L is an n-simplex, then T_{L} is an (n + 1)-torus, and so T_{L} is aspherical. If L is not a simplex, then there exist v_{1}, v_{2} ∈ V so that there is no edge in L from v_{1} to v_{2}. For i = 1, 2, let L_{i} be the full subcomplex of L with vertex set V − {v_{i}}, and define L_{3} by L_{3} = L_{1} ∩L_{2}. Then L = L_{1} ∪L_{2}, and each of L_{1}, L_{2} and L_{3} is flag. By induction, T_{L_{i}} is aspherical for i = 1, 2, 3. Also T_{L_{3}} = T_{L_{1}} ∩ T_{L_{2}} is a subcomplex of both T_{L_{1}} and T_{L_{2}}. The fundamental group of T_{L_{3}} maps injectively to the fundamental group of each of T_{L_{1}} and T_{L_{2}}, since T_{L_{3}} is a retract of each of T_{L_{1}} and T_{L_{2}}. A theorem of Whitehead [8, 1.B.11] implies that T_{L} = T_{L_{1}} ∪ T_{L_{2}} is aspherical.

Conversely, suppose that L is not flag. Then L contains a full subcomplex M which is equal to the boundary of an n-simplex for some n > 1. Then T_{M} is the n-skeleton of an (n + 1)-torus, and so π_{n}(T_{M}) is non-zero. Since T_{M} is a retract of T_{L} it follows that π_{n}(T_{L}) is non-zero too.

Remark 5 There is also a metric proof that T_{L} is aspherical whenever L is a finite flag complex: in this case the geodesic metric induced by the standard product metric on T (V) is locally CAT(0). A version of the Cartan-Hadamard theorem shows that any locally CAT(0) metric space is aspherical [3]. We give the above proof instead to emphasize that the metric technology is not needed.

4 Homology of T̃_{L}

Let Z denote the fundamental group of 𝕋, an infinite cyclic group. Since T̃_{L} is defined in terms of μ_{L} : T_{L} → 𝕋 by pulling back the universal covering space of 𝕋, Z acts via deck transformations on T̃_{L}. When L is non-empty, the map μ_{L} induces an isomorphism G_{L}/H_{L} ≅ Z. In this section we describe the cellular chain complex and homology of T̃_{L} as a Z-module, for every L. Let C_{∗}^{+}(L) denote the augmented cellular chain complex of L, and let d = d_{L} be its differential.

Proposition 6 The cellular chain complex C_{∗}(T̃_{L}) is isomorphic to ℤ[Z] ⊗ C_{∗−1}^{+}(L) with differential (1 − z) ⊗ d_{L}.

Proof. Each (n − 1)-simplex σ of L corresponds to a cubical n-cell in T_{L}, whose opposite faces are identified. In T̃_{L} this lifts to a free Z-orbit of ncells. The ith opposite pair of faces are no longer identified, but differ by the translation action of x. By picking an orbit representative in each orbit of cells, we establish a Z-equivariant bijection between the set of n-cells of T̃_{L} and the direct product of Z with the set of (n − 1)-simplices of L. The free abelian group with this basis is naturally isomorphic to ℤ[Z] ⊗ C_{n−1}^{+}(L). Let v be any fixed 0-cell of T̃_{L}. In each orbit of higher-dimensional cells, pick the orbit representative that has v as a vertex but does not have z^{−1}v as a vertex. With respect to this choice of orbit representatives, the boundary map is as claimed.

Corollary 7 For any L, for any abelian group A, and for any n ≥ 0, there are short exact sequences of ℤ[Z]-modules:

0 → B_{n−1}^{+}(L; A) → H_{n}(T̃_{L}; A) → ℤ[Z] ⊗ \overline{H}_{n−1}(L; A) → 0,
0 → ℤ[Z] ⊗ \overline{H}_{n−1}(L; A) → H_{n}(T̃_{L}; A) → Z_{n−1}^{+}(L; A) → 0,

where Z acts trivially on Z_{∗}^{+}(L; A) and on B_{∗}^{+}(L; A), the cycles and boundaries in C_{∗}^{+}(L; A). The inclusion of the Z-fixed points in H_{n}(T̃_{L}; A) gives rise to the first sequence, and the map of H_{n}(T̃_{L}; A) onto its largest Z-invariant quotient gives rise to the second sequence.

In the case when A = R, a ring, each sequence admits an R[Z]-module structure. In this case, the first sequence is split if \overline{H}_{n−1}(L; R) is R-projective, and the second sequence always admits an R-module splitting.

Proof. Take elements p(z) ∈ ℤ[Z], and c ∈ C_{n−1}^{+}(L; A). The chain p(z) ⊗ c is a cycle for (1 − z) ⊗ d if and only if d(c) = 0. The boundary of p(z) ⊗ c is (1 − z)p(z) ⊗ dc. Thus the cycles Z_{n} in C_{∗}(T̃_{L}; A) may be identified with ℤ[Z]⊗Z_{n−1}^{+}(L; A), and the boundaries B_{n} may be identified with (1−z)ℤ[Z]⊗ B_{n−1}^{+}(L; A). Between these lies B′_{n} = ℤ[Z] ⊗ B_{n−1}^{+}(L; A), and B′_{n} is a ℤ[Z]- submodule of Z_{n}. This gives a short exact sequence of ℤ[Z]-modules

0 → B′_{n}/B_{n} → H_{n}(T̃_{L}; A) → Z_{n}/B′_{n} → 0,

and one sees that B′_{n}/B_{n} ≅ B_{n−1}^{+}(L; A) with the trivial Z-action and that Z_{n}/B′_{n} ≅ ℤ[Z] ⊗ \overline{H}_{n−1}(L; A).

Now define Z′_{n} to be (1 − z)ℤ[Z] ⊗ Z_{n−1}^{+}(L; A), a ℤ[Z]-submodule of Z_{n}. As abelian groups, Z_{n} = Z′_{n} ⊕ (1 ⊗ Z_{n−1}^{+}(L; A)). It follows that the short exact sequence

0 → Z′_{n}/B_{n} → H_{n}(T̃_{L}; A) → Z_{n}/Z′_{n} → 0

is always ℤ-split. One sees that Z′_{n}/B_{n} ≅ ℤ[Z] ⊗ \overline{H}_{n−1}(L; A) and that Z_{n}/Z′_{n} ≅ Z_{n−1}^{+}(L; A).

To compute the Z-fixed points in H_{n}(T̃; A), apply the Z-fixed point functor to the first sequence. Since this functor is left-exact, one obtains an exact sequence:

0 → B_{n−1}^{+}(L; A) → H_{n}(T̃_{L}; A)^{Z} → 0.

To compute the maximal Z-fixed quotient of H_{n}(T̃; A), start with the short exact sequence

0 → B_{n} → Z_{n} → H_{n} → 0,

and apply the invariant quotient functor H_{0}(Z; −). This functor is rightexact, and it is easy to see that H_{0}(Z; Z_{n}) = Z_{n}/Z′_{n}. Hence one obtains an exact sequence:

Z_{n}/Z′_{n} → H_{n}(T̃_{L}; A)_{Z} → 0.

Since we have already shown that Z_{n}/Z′_{n} is a Z-invariant quotient of the homology group H_{n}(T̃_{L}; A), we see that the maximal Z-invariant quotient is isomorphic to Z_{n}/Z′_{n} as claimed.

In the case when A = R, a ring, the ℤ[Z]-modules and maps that appear in the short exact sequences also admit an R-module structure which commutes with the Z-action. If \overline{H}_{n−1}(T̃; R) is R-projective, then ℤ[Z] ⊗ \overline{H}_{n−1}(T̃; R) is R[Z]-projective, and so the first short exact sequence of R[Z]- modules splits. In any case, Z_{n−1}^{+}(L; R) is free as an R-module, and so the second short exact sequence admits an R-splitting.

Corollary 8 Suppose that L is a finite complex and R is a ring such that \overline{H}_{i}(L; R) = 0 for i < n. Suppose also that L has f_{i} i-dimensional simplices for i ≥ 0, and define f_{−1} = 1. For each i with 0 ≤ i ≤ n, H_{i}(T̃_{L}; R) is a free R-module of rank

∑_{j=0}^{i}(−1)^{i+j}f_{j−1}.

Proof. For each j, C_{j}^{+}(L; R) is a free R-module of rank f_{j}. For each i ≤ n, we know that

H_{i}(T̃_{L}; R) ≅ Z_{i−1}^{+}(L; R) = B_{i−1}^{+}(L; R),
C_{i}^{+}(L; R) ≅ Z_{i−1}^{+}(L; R) ⊕ Z_{i}^{+}(L; R),

and that Z_{i}^{+}(L; R) is a free R-module. Solving for the rank of Z_{i}^{+}(L; R) gives the claimed result.

Corollary 9 If L is finite and ℚ-acyclic, then the Euler characteristic of T̃_{L} is defined and is given by the formula:

χ(T̃_{L}) = ∑_{i≥0}(−1)^{i}(i + 1)f_{i}.

Proof. Since L is ℚ-acyclic, the reduced Euler characteristic ∑_{i≥−1}(−1)^{i}f_{i} of L is equal to zero. Let L be n-dimensional. Then the expression for the rank of H_{i}(T̃_{L}; ℚ) given by the previous corollary gives

χ(T̃_{L}) = ∑_{i=0}^{n}(−1)^{i} ∑_{j=0}^{i}(−1)^{i+j}f_{j−1} = ∑_{i=0}^{n}(−1)^{i}(n + 1 − i)f_{i−1}.

Hence we see that

χ(T̃_{L}) = ∑_{i=0}^{n}(−1)^{i}(n + 1 − i)f_{i−1} − (n + 1) ∑_{i=0}^{n+1}(−1)^{i}f_{i−1} = ∑_{i=0}^{n+1}(−1)^{i−1}if_{i−1},

as claimed.

Remark 10 There is a way to deduce Corollary 9 from results of Bestvina and Brady from [3]. This alternative proof was given in the second named author’s PhD thesis [13]. Let μ̃_{L} be the map defined by the pullback square

\begin{matrix}T̃_{L} & ⟶^{μ̃_{L}} & ℝ \\ ↓ & & ↓ \\ T_{L} & ⟶^{μ_{L}} & 𝕋. \end{matrix}

In the case when L is finite and ℚ-acyclic, Bestvina and Brady show that the inclusion μ̃_{L}^{−1}(x) in T̃_{L} is a rational homology isomorphism for any real number x [3]. In the case when x is not an integer, there is a cellular structure on μ̃_{L}^{−1}(x) with (i + 1)f_{i} i-cells for each i ≥ 0.

On the other hand, we know of no other proof of Corollary 7 or Corollary 8 than the proofs given above.

5 Cohomology of T̃_{L}

Proposition 11 Let C^{∗}(T̃; A) denote the cellular cochain complex for T̃_{L} with coefficients in the abelian group A. Each C^{n}(T̃_{L}; A) is isomorphic to a coinduced Z-module:

C^{n}(T̃_{L}; A) ≅ Hom(ℤ[Z], C_{+}^{n−1}(L; A)) ≅ ∏_{i∈ℤ}C_{+}^{n−1}(L; A).

If A is an R-module for some ring R, then this is an isomorphism of R[Z]- modules. The coboundary map is given by δ((f_{i})_{i∈ℤ}) = (δf_{i} − δf_{i+1})_{i∈ℤ}. The action of Z is the ‘shift action’: z(f_{i})_{i∈ℤ} = (f_{i+1}). The image of C^{∗}(T_{L}) in C^{∗}(T̃_{L}) is identified with the ‘constant sequences’, i.e., those with f_{i} = f_{j} for all i, j.

Proof. Most of the assertions follow immediately from the description of C_{∗}(T̃_{L}) given in Proposition 6, since

C^{∗}(T̃_{L}; A) ≅ Hom(C_{∗}(T̃_{L}), A) ≅ Hom(ℤ[Z] ⊗ C_{∗−1}^{+}(L), A)
≅ Hom(ℤ[Z], C_{+}^{∗−1}(L; A)).

The claim concerning the image of C^{∗}(T_{L}) is clear, since cochains that factor through the projection T̃_{L} → T̃_{L}/Z = T_{L} may be identified with cochains that are fixed by the Z-action.

Corollary 12 For any L, any abelian group A, and any n ≥ 0 there are short exact sequences of ℤ[Z]-modules:

0 → C_{+}^{n−1}(L; A)/B_{+}^{n−1}(L; A) → H^{n}(T̃_{L}; A) → M → 0,

0 → ∏_{Z} \overline{H}^{n−1}(L; A) → H^{n}(T̃_{L}; A) → C_{+}^{n−1}(L; A)/Z_{+}^{n−1}(L; A) → 0,

where Z acts trivially on C_{+}^{∗}(L; A), and Z_{+}^{∗}(L; A) (resp. B_{+}^{∗}(L; A)) denotes the cocycles (resp. coboundaries) in C_{+}^{∗}(L; A). The module M fits in to a short exact sequence:

0 → \overline{H}^{n−1}(L; A)⟶^{Δ} ∏_{Z} \overline{H}^{n−1}(L; A) → M → 0,

where Z acts by the ‘shift action’ on the product and where Δ is the inclusion of the constant sequences. The first short exact sequence is the inclusion of the Z-fixed points in H^{n}(T̃_{L}; A). If A is an R-module, then both short exact sequences admit an R[Z]-action.

Proof. Using the description of C^{∗} = C^{∗}(T̃_{L}; A) given in Proposition 11, we obtain descriptions of the coboundaries B^{n} and cocycles Z^{n} in C^{n}:

B^{n} ≅ ∏_{i∈ℤ}B_{+}^{n−1}(L; A),

Z^{n} ≅ {(f_{i}) ∈ ∏_{i∈ℤ}C_{+}^{n−1}(L; A) : δf_{i} = δf_{i+1}∀i ∈ ℤ}.

Let B^{′n} be the submodule of Z^{n} generated by B^{n} and the constant sequences {(f_{i})_{i∈ℤ} : f_{i} = f_{j}}, and let Z^{′n} be the submodule of Z^{n} consisting of sequences of cocycles, i.e., Z^{′n} = ∏_{i∈ℤ} Z_{+}^{n−1}(L; A). The first short exact sequence in the statement is equal to

0 → B^{′n}/B^{n} → H^{n}(T̃_{L}; A) → Z^{n}/B^{′n} → 0,

and the second one to

0 → Z^{′n}/B^{n} → H^{n}(T̃_{L}; A) → Z^{n}/Z^{′n} → 0.

The computation of the Z-fixed points in H^{n}(T̃_{L}; A) follows by applying the Z-fixed point functor to the first exact sequence.

Theorem 13 Let R be a ring. The image of the map

H^{∗}(T_{L}; R) → H^{∗}(T̃_{L}; R)

is equal to the Z-fixed point subring of H^{∗}(T̃_{L}; R) and is isomorphic to the quotient H^{∗}(T_{L}; R)/(β_{L}). In degree n, the cokernel of this map is isomorphic to an infinite product of copies of \overline{H}^{n−1}(L; R). In particular, the map is a ring isomorphism if and only if L is R-acyclic.

Proof. This follows from Corollary 12 and Theorem 2.

Corollary 14 Suppose that L is R-acyclic. There is an R-algebra isomorphism

H^{∗}(T̃_{L}; R) ≅ Λ_{R}^{∗}(L)/(β_{L}).

For each n, H^{n}(T̃_{L}; R) is isomorphic to a direct product of copies of R.

Proof. This follows from Theorem 13 and Theorem 1 Remark 15 The second named author’s PhD thesis contained a different proof of Corollary 14 in the case when L is finite, flag and R-acyclic [13]. Recall that we denote by G_{L} the fundamental group of T_{L}, and by H_{L} the fundamental group of T̃_{L}. Also recall from Proposition 4 that when L is flag, each of T_{L} and T̃_{L} is an Eilenberg-Mac Lane space for its fundamental group. In [13] an explicit chain homotopy was used to show that the action of Z on H^{∗}(T̃_{L}; R) is trivial when L is finite, flag and R-acyclic. The long exact sequence in group cohomology coming from the isomorphism G_{L} = H_{L} : Z was then used to establish the isomorphism of Corollary 14.

6 Cohomological dimension

The trivial cohomological dimension of a space X, tcd(X), is defined to be the supremum of those integers n for which there exists an abelian group A for which the singular cohomology group H^{n}(X; A) is non-zero. For any nontrivial ring R, tcd_{R}(X) is defined similarly except that only abelian groups A admitting an R-module structure are considered.

Now suppose that X is path-connected, and that G is the fundamental group of X. For M a G-module, we write H^{∗}(X; M) for the singular cohomology of X with twisted coefficients in M. If X admits a universal covering space X̃, then this is just the cohomology of the cochain complex Hom_{G}(C_{∗}(X̃), M) of G-equivariant singular cochains on X̃. The cohomological dimension cd(X) of X is the supremum of those integers n such that there is a G-module M for which H^{n}(X; M) ≠ {0}. For a non-trivial ring R, cd_{R}(X) is defined similarly except that only RG-modules M are considered.

Each of the invariants depends only on the homotopy type of X. In the case when X is a classifying space or Eilenberg-Mac Lane space for G, we write cd(G), cd_{R}(G), tcd(G) and tcd_{R}(G) for the corresponding invariants of X.

We summarize the properties of these invariants below.

Proposition 16 In the following statements, X is any path-connected space, and R is any non-trivial ring. Rings are assumed to have units, and ring homomorphisms are assumed to be unital.

1. tcd(X) = tcd_{ℤ}(X), cd(X) = cd_{ℤ}(X).

2. tcd_{R}(X) ≤ cd_{R}(X).

3. If there is a ring homomorphism ϕ : R → S, then tcd_{S}(X) ≤ tcd_{R}(X) and cd_{S}(X) ≤ cd_{R}(X).

4. If R is a subring of S in such a way that R is a direct summand of S as an R-R-bimodule, then tcd_{R}(X) = tcd_{S}(X) and cd_{R}(X) = cd_{S}(X).

5. If Y is a covering space of X, then cd_{R}(Y) ≤ cd_{R}(X).

6. If H ≤ G, then cd_{R}(H) ≤ cd_{R}(G).

7. For any group G, cd_{R}(G) = proj.dim._{RG}(R).

Proof. A ℤG-module is the same thing as a G-module, establishing (1). Similarly, any R-module A may be viewed as an RG-module by letting each element of G act via the identity, which establishes (2). A ring homomorphism ϕ as above allows one to define an R-module structure on any S-module and an RG-module structure on any SG-module, which proves (3). Under the hypotheses of (4), any R-module A is isomorphic to a direct summand of the S-module S ⊗_{R} A and any RG-module M is isomorphic to a direct summand of the SG-module SG ⊗_{RG} M. Since cohomology commutes with finite direct sums, this shows that tcd_{R}(X) ≤ tcd_{S}(X) and cd_{R}(X) ≤ cd_{S}(X). The opposite inequalities follow from (3).

If the fundamental group of Y is the subgroup H ≤ G, and M is any RH-module, define the coinduced RG-module by

Coind(M) = Hom_{ℤH}(ℤG, M).

Then there is an isomorphism H^{∗}(X; Coind(M)) ≅ H^{∗}(Y; M), which establishes (5). Now (6) is just the special case of (5) in which X is a classifying space for G, since then the covering space of X with fundamental group H is a classifying space for H.

If X is a classifying space for G, then the universal covering space X̃ is contractible, and so C_{∗}(X̃) is a free resolution of ℤ over ℤG. Similarly, C_{∗}(X̃; R) is a free resolution of R over RG. Hence H^{∗}(X; M) is isomorphic to Ext_{ℤG}(ℤ, M), and if M is an RG-module H^{∗}(X; M) is also isomorphic to Ext_{RG}(R, M), which establishes (7). (See for example [4] for more details.)

Remark 17 It is easy to find groups G for which tcd(G) < cd(G). There are many acyclic groups, or groups for which the group homology H_{i}(G; ℤ) = 0 for all i > 0 (see for example [1]). For any such G, tcd(G) = 0, while cd(G) ≠ 0 unless G is the trivial group.

Before stating the next proposition, we remind the reader that the dimension of a simplicial complex is the supremum of the dimensions of its simplices, so that the empty simplicial complex has dimension −1.

Proposition 18 For any simplicial complex L and any non-trivial ring R,

tcd_{R}(T_{L}) = cd_{R}(T_{L}) = dim(T_{L}) = dim(L) + 1.

Proof. Immediate from Theorem 1.

Proposition 19 For any simplicial complex L with at least two vertices and for any R,

tcd_{R}(T̃_{L}) = max{dim(L), 1 + tcd_{R}(L)}.

Proof. Immediate from Corollary 12.

The following theorem is the first result in this paper for which we rely on techniques from Bestvina-Brady [3].

Theorem 20 Let L be a simplicial complex with at least two vertices and let R be a non-trivial ring.

1. dim(L) ≤ tcd_{R}(T̃_{L}) ≤ cd_{R}(T̃_{L}) ≤ dim(L) + 1.

2. If L is R-acyclic, then tcd_{R}(L) = cd_{R}(T̃_{L}) = dim(L).

Proof. The claims for arbitrary L follow easily from earlier results. Proposition 19 implies that dim(L) ≤ tcd_{R}(T̃_{L}). By Proposition 16, one has that tcd_{R}(T̃_{L}) ≤ cd_{R}(T̃_{L}) ≤ cd_{R}(T_{L}), and of course cd_{R}(T_{L}) ≤ dim(T_{L}) = dim(L) + 1.

It remains to show that when L is R-acyclic, cd_{R}(T̃_{L}) ≤ dim(L). Following Bestvina and Brady [3], let X_{L} be the universal covering space of T_{L}, or equivalently of T̃_{L}. Now let f_{L} : X_{L} → ℝ be the composite

X_{L} → X_{L}/H_{L} = T̃_{L}⟶^{μ̃}ℝ,

where μ̃ is the lift of the map μ_{L} : T_{L} → 𝕋. Now define Y_{L} = f^{−1}(0) ⊆ X_{L}. There is a natural cubical CW-structure on X_{L}, whose cells are the lifts to X_{L} of the cells of T_{L}. One can also put a CW-structure on Y_{L}, such that each cell of Y_{L} is the intersection of Y_{L} with a cell of X_{L}. For this CW-structure, the dimension of Y_{L} is equal to dim(L). It can be shown that X_{L} is homotopy equivalent to Y_{L} with infinitely many subspaces homotopy equivalent to L coned off (this is from [3], but see also [11] which explicitly checks this in the case when L is infinite).

Since L is R-acyclic, it follows that the inclusion of Y_{L} in X_{L} induces an isomorphism of R-homology. The cellular chain complexes C_{∗}(Y_{L}; R) and C_{∗}(X; R) consist of free RH_{L}-modules, and so it follows that for any RH_{L}- module M,

H^{∗}(T̃_{L}; M) ≅ H^{∗}(Y_{L}/H_{L}; M).

This shows that cd_{R}(T̃_{L}) ≤ dim(L) as required.

Remark 21 In the case when L is empty, T̃_{L} is 0-dimensional, and consists of a single free Z-orbit of points. In the case when L is a single point, T̃_{L} is homeomorphic to ℝ, with Z acting via the translation action of ℤ. It is clear that the case when L is empty is exceptional for T̃_{L}. The reason why the case when L is a single point needs to be excluded from Proposition 19 and from Theorem 20 is that the formulae for H ∗ (T̃_{L}; R) given in Section 5 involve the reduced cohomology of L, whereas the definition of tcd(L) involves the unreduced cohomology of L. This only makes a difference when L is both 0-dimensional and R-acyclic, i.e., the case when L is a single point.

7 Bestvina-Brady groups

In this section, we give a complete calculation of the cohomological dimension of Bestvina-Brady groups, or equivalently the cohomological dimension of the spaces T̃_{L} for L a flag complex. We impose some conditions on the coefficient ring R that were not previously required. The reason why we work only with flag complexes in this section is that we need to know that when L ≤ K is a full subcomplex of K, then T̃_{L} is homotopy equivalent to a covering space of T̃_{K}.

Theorem 22 Let R be either a field or a subring of ℚ. Let L be a flag complex with at least two vertices. (This implies that H_{L} is infinite, and that T̃_{L} is a classifying space for H_{L}.) The following equations hold:

cd_{R}(H_{L}) = tcd_{R}(H_{L}) = max{dim(L), 1 + tcd_{R}(L)}.

The proof of the theorem will require two lemmas, the second of which is a strengthening of a lemma from [2].

Lemma 23 Suppose that L is a flag complex and is a subcomplex of a simplicial complex K. The relative barycentric subdivision (K, L)′ is a flag complex containing L as a full subcomplex.

Proof. Before recalling the definition of the relative barycentric subdivision (K, L)′, we remind the reader that each simplicial complex contains a unique −1-simplex corresponding to the empty subset of its vertices. The vertex set of (K, L)′ is the disjoint union of the vertex set of L and the set of simplices of K not contained in L. An n-simplex of (K, L)′ has the form (σ_{0} < σ_{1} < ⋯ < σ_{r}), for some r satisfying 0 ≤ r ≤ n + 1. Here σ_{0} denotes an (n − r)-simplex of L, each σ_{i} is a simplex of M, and σ_{i} is not contained in L if i > 0.

Suppose that a finite subset S of the vertices of (K, L)′ has the property that any two of its members are joined by an edge. Since L is flag, this implies that the set S ∩ L is the vertex set of a simplex σ_{0} of L. Each element of S − L is a simplex σ_{i} of M not contained in L. The existence of an edge between each element of S ∩ L and each element of S − L implies that each σ_{i} contains σ_{0}, and the existence of an edge between each pair of elements of S − L implies that the σ_{i} are totally ordered by inclusion. This shows that (K, L)′ is a flag complex.

From the description of the simplices of (K, L)′, it is easy to see that any simplex of (K, L)′ whose vertex set lies in L is in fact a simplex in L, which verifies that L is a full subcomplex of (K, L)′.

Lemma 24 Let L be a flag complex, let R be either a subring of ℚ or a field of prime order, and suppose that tcd_{R}(L) < dim(L). Then there is an R-acyclic flag complex K containing L as a full subcomplex such that dim(L) = dim(K).

Proof. We may assume that dim(L) = n is finite, and we may assume that n ≥ 2. Let C′L denote the cone on the (n − 2)-skeleton of L, and let L_{1} be the union L_{1} = L ∪ C′L. Now L_{1} is (n − 2)-connected, and the inclusion map L → L_{1} induces an isomorphism H^{n}(L_{1}; A) → H^{n}(L; A) for any abelian group A.

In each case, R is a principal ideal domain, and so by the universal coefficient theorem for cohomology [9, V.3.3], for any R-module A we have that

H^{n}(L_{1}; A) ≅ Hom_{R}(H_{n}(L_{1}, R), A) ⊕ Ext_{R}^{1}(H_{n−1}(L_{1}, R), A).

The hypotheses therefore imply that H_{n}(L_{1}; R) = 0, and that H_{n−1}(L_{1}; R) is a projective R-module. (Since L_{1} is (n − 2)-connected, one also has that \overline{H}_{i}(L_{1}; R) = 0 for each i < n − 1.) Every projective module for a principal ideal domain is free [9, I.5.1], and so H_{n−1}(L_{1}; R) is a free R-module. Since \overline{H}_{n−2}(L_{1}; ℤ) = 0 (because L_{1} is (n − 2)-connected), the universal coefficient theorem for homology [9, V.2.5] tells us that H_{n−1}(L_{1}; R) ≅ H_{n−1}(L_{1}; ℤ)⊗R.

Hence there exist integral cycles z_{i} ∈ C_{n−1}(L_{1}; ℤ) whose images in the group C_{n−1}(L_{1}; R) map to an R-basis for H_{n−1}(L_{1}; R). By the Hurewicz theorem, each z_{i} is realized by some map f_{i} : S^{n−1} → L_{1}. Replace f_{i} by a simplicial approximation f′_{i} : S_{i} → L_{1}, where S_{i} is some triangulation of the (n − 1)-sphere. Using a simplicial mapping cylinder construction as in [8, 2C.5], use each f′_{i} to attach a triangulated n-cell to L_{1} to produce L_{2}, an R-acyclic simplicial complex with dim(L_{2}) = n and such that L ≤ L_{1} ≤ L_{2}. By Lemma 23, we may take K to be the relative barycentric subdivision K = (L_{2}, L)′.

Proof of Theorem 22. If S is any field, and R is the smallest subfield of S, then R and S satisfy the conditions of statement (4) of Proposition 16. Hence there are equalities of functions cd_{R} = cd_{S} and tcd_{R} = tcd_{S}. Thus it suffices to prove Theorem 22 in the case when R is either the field of p elements or a subring of ℚ.

We may assume that dim(L) = n < ∞. In the case when tcd_{R}(L) = n, Proposition 19 and part (1) of Theorem 20 imply that

n + 1 = tcd_{R}(T̃_{L}) ≤ cd_{R}(T̃_{L}) ≤ dim(T̃_{L}) = n + 1,

as claimed. If tcd_{R}(L) < n, then by Lemma 24, there is an n-dimensional R-acyclic flag complex K containing L as a full subcomplex. Part (2) of Theorem 20 tells us that tcd_{R}(K) = cd_{R}(K) = n. Now H_{L} is a retract of H_{K}, and so by part (6) of Proposition 16

cd_{R}(H_{L}) = cd_{R}(T̃_{L}) ≤ cd_{R}(H_{K}) = cd_{R}(T̃_{K}) = n.

This gives

n = tcd_{R}(T̃_{L}) ≤ cd_{R}(T̃_{L}) ≤ cd_{R}(T̃_{K}) = n,

as claimed.

8 Examples

Example 25 Let L = L(m) be a flag triangulation of the space constructed by attaching a disc to a circle via a map of degree m. This L has the property that tcd_{R}(L) = 0 if m is a unit in R and tcd_{R}(L) = 2 if m is not a unit in R. Let H = H_{L} be the corresponding Bestvina-Brady group. From Theorem 20 and Proposition 19, it follows that for any ring R, cd_{R}(H) = 2 if m is a unit in R and cd_{R}(H) = 3 if m is not a unit in R.

Example 26 Let L be a flag triangulation of a 2-dimensional Eilenberg-Mac Lane space for the additive group of ℚ. Then tcd_{F} (L) = 1 for any field F, while tcd_{ℤ}(L) = 2. From Theorem 22 it follows that the Bestvina-Brady group H = H_{L} has the property that cd_{F} (H) = 2 for any field F, while cd_{ℤ}(H) = 3.

Groups having similar properties to those given in Example 25 have been constructed previously by many authors, using finite-index subgroups of right-angled Coxeter groups [2, 5, 7]. The examples coming from Coxeter groups have the advantage that they are of type F P, whereas they have the disadvantage that the trivial cohomological dimension of these groups appears to be unknown.

Finite-index subgroups of (non-finitely generated) Coxeter groups were used in [5] to construct groups having similar properties to those given in Example 26. Again, the trivial cohomological dimension of those examples appears to be unknown. It can be shown that for any group G of type F P, there is a field F such that cd_{F} (G) = cd_{ℤ}(G). (See Proposition 9 of [5].)

References

[1] G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980) 1–47.

[2] M. Bestvina, The virtual cohomological dimension of Coxeter groups, in Geometric Group Theory Volume I (edited by G. A. Niblo and M. Roller), London Math. Soc. Lecture Notes 181 Cambridge Univ. Press (1993) 19–23.

[3] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997) 445–470.

[4] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87 Springer Verlag (1982).

[5] W. Dicks and I. J. Leary, On subgroups of Coxeter groups, in Geometry and Cohomology in Group Theory (edited by P. H. Kropholler, G. A. Niblo and R. Stöhr), London Math. Soc. Lecture Notes 252, Cambridge Univ. Press (1998) 124–160.

[6] W. Dicks and I. J. Leary, Presentations for subgroups of Artin groups, Proc. Amer. Math. Soc. 127 (1999) 343–348.

[7] A. N. Dranishnikov, The virtual cohomological dimension of Coxeter groups, Proc. Amer. Math. Soc. 125 (1997) 1885–1891.

[8] A. Hatcher, Algebraic Topology, Cambridge University Press (2001).

[9] P. J. Hilton and U. Stammbach, A Course in Homological Algebar, Graduate Texts in Mathematics 4 Springer Verlag (1971).

[10] K. H. Kim and F. W. Roush, Homology of certain algebras defined by graphs, J. Pure Appl. Algebra 17, (1980), 179–186.

[11] I. J. Leary, On finite subgroups of groups of type VF, Geometry and Topology 9 (2005), 1953–1976.

[12] S. Papadima and A. Suciu, Algebraic invariants for Bestvina-Brady groups, preprint (2006).

[13] M. Saadetoğlu, Finiteness conditions and Bestvina-Brady groups, PhD Thesis, School of Mathematics, University of Southampton (2005).

Authors’ addresses: Ian Leary: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174, United States.

and School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom.

leary@math.ohio-state.edu

Müge Saadetoğlu: School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom.

and AS277, Department of Mathematics, Eastern Mediterranean University, Gazimağusa, Mersin 10, Turkey

muge.saadetoglu@emu.edu.tr