ON DIMENSIONS IN BREDON HOMOLOGY

BRITA E. A. NUCINKIS

Abstract. We shall define a homological and cohomological dimension of groups in the context of Bredon homology and compare the two quantities. We shall apply this to describe the Bredonhomological dimension of nilpotent groups in terms of the Hirschrank. In particular this implies that for virtually torsion-free nilpotent groups the Bredon cohomological dimension is equal to the virtual cohomological dimension.

1. Introduction

The definition of Bredon cohomology groups for G-CW -complexes for finite groups G goes back to Bredon [3] and was further investigated by Slominska, see e.g. [22]. The definitions can easily be extended to infinite groups G and have recently regained attention through their use in equivariant obstruction theory [17] and through their connection to group actions on proper G-spaces [13, 18, 2, 12, 19]. Bredon homology also features in the Baum-Connes conjecture; a nice exposition of how the two relate can be found in [18].

We say a G-CW -complex X is an \underline{E}G, or universal proper G-space, if X^{H} contractible if H ≤ G is finite and X^{H} is empty otherwise. The minimal dimension of such an \underline{E}G is denoted by \underline{gd}G. Analogously to ordinary cohomology one defines the Bredon-cohomological dimension \underline{cd}G and the Bredon-homological dimension \underline{hd}G of a group. The augmented cellular chain complex of an \underline{E}G gives a Bredon-projective resolution for the group and hence \underline{cd}G ≤ \underline{gd}G. A detailed introduction to Bredon homology follows in Section 2. Analogously to the torsionfree case [8] it was shown by Lück [13, Theorem 13.19], that, provided \underline{cd}G ≥ 3, the two quantities are equal. If either dimension is one, it follows from a result of Dunwoody [7], that they are equal. In fact, it turns out [2], that there are examples of groups where \underline{cd}G = 2 yet \underline{gd}G = 3.

A question, which has been receiving some attention is how \underline{cd}G and \underline{gd}G behave under group-extensions. Originally these questions arose for virtually torsion-free groups, without mentioning the spaces \underline{E}G or the dimensions \underline{cd}G and \underline{gd}G by the names used here. Let H ≤ G be a torsion-free subgroup of finite index, k, say. Then by Serre’s theorem, see [5, Theorems VIII.3.1, VIII.11.1], \underline{gd}G ≤ k(gdH). It was Brown who first asked whether it can be shown that \underline{gd}G = gdH [5, 4], a question which has become known as the Brown conjecture. In [12] Ian Leary and the author gave examples where there is a strict inequality \underline{gd}G > gdH. Lück [14] extented Serre’s result to arbitrary group extension H ↪ G ↠ Q, where the group Q has a bound d on the orders of its finite subgroups: \underline{gd}G ≤ \underline{gd}H + d(\underline{gd}Q). Martinez [16] developed spectral sequences in Bredon homology and cohomology for various families 𝔉 of subgroups. Applying these to some special families of finite subgroups she obtained similar bounds for \underline{cd}G in terms of \underline{cd}H and \underline{cd}Q.

Date: 26 September 2003.

The motivation for this work was to find classes of groups for which the Brown conjecture holds. It is known to hold for polycyclic groups, see [15, Example 5.25] We shall show it for nilpotent groups using a purely algebraic approach following ideas of Stammbach [23] and Bieri [1], who were relating the (co)homological dimension of soluble groups to the Hirsch length hG. For torsion-free soluble groups a result of Stammbach [23] implies, that hdG = hG, and, if in addition G is countable, then hG ≤ cdG ≤ hG + 1. For a detailed account of these facts see [1]. After proving some preliminary results in Section 3, including an analogue of Lazard’s result [11] for arbitrary rings, that flat modules are direct colimits of finitely generated free modules, we shall move on to looking at cohomological and homological dimensions of groups in the Bredon setting. The main result of Section 4 will be, that for arbitrary countable groups G,

\underline{hd}G ≤ \underline{cd}G ≤ \underline{hd}G + 1.

Furthermore, for soluble groups, hG ≤ \underline{hd}G. In Section 5 shall be concerned with group extensions. We shall give a short account of some of Martinez’ results applied to families of finite subgroups. In Section 6 we will apply these results to nilpotent groups. In particular, we will show that for nilpotent groups G (Theorem 6.1),

\underline{hd}G = hG.

and that for virtually torsion-free nilpotent groups (Theorem 6.3)

\underline{cd}G = vcdG,

which also implies that the Brown conjecture holds for this class of groups.

2. Preliminaries on Bredon modules

We shall begin with a few basic definitions concerning Bredon-mo-dules most of which can be found in Lück’s [13] or tom Dieck’s [6] books. Another good reference for background material is Mislin’s notes [18].

Let G be an arbitrary group and 𝔉 a family of subgroups closed under subgroups and conjugation. Even though our main applications will be for families 𝔉 of finite subgroups of G, the results of the next two sections hold in greater generality. We shall consider the orbit category O_{𝔉}G, which is a category with objects the cosets G/H, where H ∈ 𝔉 and the morphisms are G-maps G/H → G/K, where G/H and G/K are objects in O_{𝔉}G. We now consider functor categories from O_{𝔉}G to 𝔄𝔟, the category of abelian groups. We denote by Mod_{𝔉}-G, the category of contravariant functors M(−) : O_{𝔉}G → 𝔄𝔟 and by G-Mod_{𝔉} the category of covariant functors N(−) : O_{𝔉}G → 𝔄𝔟. Since for 𝔉 = {e}, the category Mod_{𝔉}-G is just the category of right G-modules and G-Mod_{𝔉} is the category of left G-modules we shall usually call the objects of these categories right O_{𝔉}G-modules or left O_{𝔉}G-modules respectively. In case we are just mentioning O_{𝔉}G-modules we imply that the statement holds for both categories.

Many of the properties of G-modules can be, more or less directly, taken over to O_{𝔉}G-modules. For example, by taking things objectwise we have kernels and cokernels as well as arbitrary coproducts. A sequence M′ → M → M′′ of O_{𝔉}G-modules is exact if and only if M′(G/H) → M(G/H) → M′′(G/H) is exact for all H ∈ 𝔉. Also submodules, intersections and sums of modules are defined objectwise. We can also define chain complexes, exact sequences, resolutions and chain homotopies in the usual manner.

We shall now describe how to construct projectives in Mod_{𝔉}-G . A nice thorough account can be found in [18]. We say a module P in Mod_{𝔉}-G is projective if the functor:

mor(P, −) : Mod_{𝔉}G → 𝔄𝔟

is exact. Let K ∈ 𝔉 and consider the right O_{𝔉}G-module P_{K}, which is defined as follows:

P_{K}(G/H) = ℤmor(G/H, G/K).

ℤmor(G/H, G/K) denotes the free abelian group on the set mor(G/H, G/K). An application of a Yoneda-type argument, see [18] yields

mor(P_{K}, M) ≅ M(G/K),

which in turn gives the desired exactness of mor(P_{K}, −). Obviously, a coproduct of projectives is again projective and therefore we have, for an arbitray M ∈ Mod_{𝔉}-G the following epimorphism of a projective onto M:

∐_{H∈𝔉}( ∐_{M(G/H)} P_{H}) ↠ M.

In particular ever right O_{𝔉}G-module M admits a projective resolution, denoted P_{∗} ↠ M which is used to define the derived functors Ext_{𝔉}^{∗}(M, N), which are contravariant in the first and covariant in the second variable:

Ext_{𝔉}^{∗}(M, N) = H^{∗}(mor(P_{∗}, N))

There also is the obvious definition of the projective dimension, denoted pd_{𝔉}M, of a right O_{𝔉}G-module M. It is the minimal length of a projective resolution of M.

We also use projective resolutions to define a T or_{∗}^{𝔉}(M, N). To do this we shall introduce tensor products in our category. There are two such products, which are of interest to us, firstly, the tensor product over ℤ and secondly the tensor product over the Orbit-category O_{𝔉}G. Let M(−) ∈ Mod_{𝔉}-G and N(−) ∈ G-Mod_{𝔉}. We define the tensor product over ℤ as the composite of the following two functors:

M(−) ⊗ N(−) : O_{𝔉}G × O_{𝔉}G → 𝔄𝔟 × 𝔄𝔟⟶^{⊗ℤ}𝔄𝔟.

The definition of a tensor product over O_{𝔉}G needs a little bit more work. On can define it explicitly or use a universal property as done by Yoneda. For a nice account concerning the universal property as well as other results on tensor products of functors see [9]. For definitions in our sense and some of the facts now to follow see [13] and [18]. Let φ : G/K → G/L be a G-map. We now define the categorical tensor product to be the abelian group

M(−) ⊗_{𝔉} N(−) = ⨁_{K∈𝔉}M(G/K) ⊗ N(G/K)/ ∼,

where ∼ is the equivalence relation generated by φ^{∗}(m)⊗n = m⊗φ_{∗}(n) with m ∈ M(G/L) and n ∈ N(G/K).

Let Tor_{n}^{𝔉}(−, N) denote the n-th left derived functor of the tensor product − ⊗_{𝔉} N. We say a right O_{𝔉}G-module M is flat if the functor M ⊗_{𝔉} − is exact. Since, by the Yoneda isomorphism,

P_{K}(−) ⊗_{𝔉} N(−) ≅ N(G/K)

for all N ∈ G−Mod_{𝔉} and the tensor product commutes with coproducts one can conclude that projective right O_{𝔉}G-modules are flat. Define the flat dimension (fld_{𝔉}) of a right O_{𝔉}G-module as the minimal length of a flat resolution. Hence, for all M ∈ Mod_{𝔉}-G:

fld_{𝔉}M ≤ pd_{𝔉}M.

To define free right O_{𝔉}G-modules we shall follow [13, 9.16, 9.17]. Firstly we have to say what we mean by an Obj{O_{𝔉}G}-set Δ, or 𝔉-set Δ for short. This is a family {Δ_{K} | K ∈ 𝔉} of sets Δ_{K}. An 𝔉-map between two 𝔉-sets Δ and Ω is a family of maps {Δ_{K} → Ω_{K} | K ∈ 𝔉}. There is an alternative way of thinking of 𝔉-sets, namely as pairs (Δ, φ) with φ : Δ → Obj(O_{𝔉}G), where Δ_{K} = φ^{−1}(G/K). By forgetting the structure we obtain an 𝔉-set from any O_{𝔉}G-module. We can now define the free O_{𝔉}G-module on an 𝔉-set as the left adjoint to the just mentioned forgetful functor Mod_{𝔉}-G → 𝔉-set and we have the usual universal property: A right O_{𝔉}G-module F is called free over the 𝔉-set Δ if Δ is a 𝔉-subset of F, i.e. Δ_{K} ⊂ F (G/K) for each K ∈ 𝔉 and if each 𝔉-map f : Δ → N with N an arbitrary right O_{𝔉}G-module has a unique extension by an O_{𝔉}G-module map f̃: F → N.

As an example, the right modules P_{K} defined above are free on the 𝔉set Δ with Δ_{K} = {G/K}, the set with one element G/K, and Δ_{H} = ∅ for H ≠ K. Also, the module ∐_{K∈𝔉} P_{K} is free on the 𝔉-set Δ with Δ_{K} = {G/K} as above for all K ∈ 𝔉. In general, given an 𝔉-set (Δ, φ), we can construct the free module F_{Δ} as

F_{Δ} = ∐_{δ∈Δ} P_{φ(δ)}.  (1)

A free right O_{𝔉}G-module is finitely generated, if it is free on a finite 𝔉-set Δ or, equivalently if it is a finite coproduct of free modules of the form P_{K} for K ∈ 𝔉. Now let M be a right O_{𝔉}G-module and S ⊂ M an 𝔉-subset. Then the submodule generated by S is the smallest O_{𝔉}G-submodule containing S. Hence, a right O_{𝔉}G-module is finitely generated if and only if it is a quotient of a finitely generated free module. A right O_{𝔉}G-module M is said to be finitely presented if there is a short exact sequence N ↪ F ↠ M of O_{𝔉}G-modules with F finitely generated free and N finitely generated.

Later on we shall require O_{𝔉}G-modules, which have similar properties to duals of finitely generated projective ℤG-modules. It turns out, that in the Bredon setting we can’t make as general a definition as for ordinary G-modules.

Let P_{K} be the free right O_{𝔉}G-module of rank 1 as explained above. We define its dual

P^{K} = ℤmor(G/K, −),

which is now a left O_{𝔉}G-module. For arbitrary finitely generated free right O_{𝔉}G-modules F = ∐_{K∈𝔉′} P_{K} (𝔉′ a finite subfamily of 𝔉) we set

F^{∗} = ∐_{K∈𝔉′} P^{K}.

Analogously to the contravariant case ([18]) one can prove that

mor(P^{K}, N) = N(G/K)

for every N ∈ G-Mod_{𝔉} and every K ∈ 𝔉. Hence, the P^{K} and also F^{∗} are projective left O_{𝔉}G-modules and one can also show that every left O_{𝔉}G-module has a free mapping onto it.

An application of the above also yields a one-to-one correspondence between mor(P_{K}, P_{L}) and mor(P^{L}, P^{K}), since

\begin{aligned}mor(P_{K}, P_{L}) & = P_{L}(G/K) = ℤmor(G/K, G/L) \\ & = P^{K}(G/L) = mor(P^{L}, P^{K}) \end{aligned}

by the above. Hence, for abitrary finitely generated free O_{𝔉}G-modules P and Q we have a one-to-one correspondence

mor(P, Q) = mor(Q^{∗}, P^{∗}).  (2)

We should maybe mention that the whole construction works the other way around, too. For arbitrary finitely generated free left O_{𝔉}Gmodules P we have a dual P^{∗}, which is a finitely generated free right O_{𝔉}G-module and

P^{∗∗} = P.

in either setting. Now let M ∈ Mod_{𝔉}-G and P^{K} = ℤmor(G/K, −) ∈ G-Mod_{𝔉}. We define a map

Ψ : M(G/K) → M(−) ⊗_{𝔉} P^{K}(−)

by m ↦ m ⊗ id ∈ M(G/K) ⊗ P^{K}(G/K). This map is obviously injective. Take an arbitrary basis-element m ⊗ φ ∈ M(G/H) ⊗ P^{K}(G/H), where φ : G/K → G/H. This φ can be viewed as the composite φ∘id, hence m ⊗ φ(id) = φ^{∗}(m) ⊗ id ∈ M(G/K) ⊗ P^{K}(G/K) and hence Ψ is onto.

Applying this and the fact that the tensor-product and mor commute with finite coproducts, we obtain the following isomorphism

M ⊗_{𝔉} P^{∗} ≅ mor(P, M)  (3)

for finitely generated free P ∈ Mod_{𝔉}-G and arbitrary M ∈ Mod_{𝔉}-G.

3. Flat Bredon modules

To begin this section we shall collect a few facts about colimits in general and also in the category Mod_{𝔉}-G. A general definition of a colimit can be found in Weibel’s book [24], pages 54–56. Let I be an indexing category and (M_{i})_{i∈I} a family of objects in Mod_{𝔉}-G. The colimit colimM_{i} is a functor from Mod_{𝔉}-G^{I} to Mod_{𝔉}-G, which is left adjoint to the diagonal functor Δ : Mod_{𝔉}-G → Mod_{𝔉}-G^{I}. Hence colimits are right exact in Mod_{𝔉}-G if they exist. Recall, that the coproduct for a family (M_{i})_{i∈I}, of objects in Mod_{𝔉}-G is defined objectwise by (∐ M_{i})(G/H) = ⨁(M_{i}(G/H)). By a standard result, see e.g. [24], Proposition 2.6.8, it follows that the colimit colimM_{i} can be viewed as the cokernel of the following map:

∐_{φ:i→j} M_{i} → ∐_{i∈I} M_{i}  (4)

and therefore exists and is defined objectwise. Let I be a filtered category we the denote the filtered colimit of a family M_{i} by \underrightarrow{colim}M_{i}. Since filtered colimits are exact for modules over arbitrary rings, we have shown the following.

Lemma 3.1. Filtered colimits are exact in Mod_{𝔉}-G. □

The main result of this section will be the following Bredon analogue to Lazard’s [11] theorem.

Theorem 3.2. The following conditions on a right O_{𝔉}G-module M are equivalent.

(i) M is flat.

(ii) For every finitely presented O_{𝔉}G-module P and O_{𝔉}G-module homomorphisms φ : P → M there exists a finitely generated free O_{𝔉}G-module F such that φ factors through F.

(iii) M is the filtered colimit of finitely generated free O_{𝔉}G-modules.

Proof. (i) ⇒ (ii): Since P is finitely presented there exists an exact sequence

F_{1}⟶^{f}F_{0}⟶^{g}P → 0

with F_{1} and F_{0} both finitely generated free right O_{𝔉}G-modules. Taking duals we obtain an exact sequence of left O_{𝔉}G-modules

F′⟶^{h}F_{0}^{∗}⟶^{f̂}F_{1}^{∗}  (5)

where F′ ∈ G-Mod_{𝔉} is free. Since M is flat this yields an exact sequence

M ⊗_{𝔉} F′⟶^{id⊗h}M ⊗_{𝔉} F_{0}^{∗}⟶^{id⊗f̂}M ⊗_{𝔉} F_{1}^{∗}  (6)

By (3) we have the following commutative diagram:

mor(F_{0}, M) ≅ M ⊗_{𝔉} F_{0}^{∗}
f^{∗}↓ ↓_{id⊗f̂}
mor(F_{1}, M) ≅ M ⊗_{𝔉} F_{1}^{∗}

Since φ∘g∘f = f^{∗}(φ∘g) = 0 and (6) is exact there exists an element x ∈ M ⊗_{𝔉} F′ such that (id ⊗ h)(x) = φ ∘ g. Now there exists a finitely generated free left O_{𝔉}G-submodule F^{∗} of F′ such that

x ∈ M ⊗_{𝔉} F^{∗} ↪ M ⊗_{𝔉} F′

Denote by k̂ the composite F^{∗} ↪ F′⟶^{h}F_{0}^{∗} and by k it’s transpose F_{0} → F, where F = F^{∗∗}.

Since (5) is exact it follows from (2) that 0 = k ∘ f ∈ mor(F_{1}, F). Passing onto quotients we obtain a well defined map α ∈ mor(P, F) such that α ∘ g = k. Since F^{∗} is a finitely generated free left O_{𝔉}Gmodule the isomorphism (3) yields that x ∈ M ⊗_{𝔉} F^{∗} defines a map β ∈ mor(F, M). Another look at a diagram similar to the above gives that

φ ∘ g = (id ⊗ k̂)(x) = k^{∗}(β) = β ∘ k.

Passing onto quotients now yields the desired equality βα = φ.

(ii) ⇒ (iii): Let F_{X} denote the free right O_{𝔉}G-module on the 𝔉-set X. We can view the right O_{𝔉}G-module M as an 𝔉-set (M, φ) by forgetting the structure (φ : M → Obj(O_{𝔉}G) and φ^{−1}(G/K) = M(G/K) with the module structure forgotten).

Take an arbitrary infinite countable 𝔉-set (N, ψ) and form the direct product (M, φ) × (N, ψ) = M × N. Let F_{M×N} denote the free right O_{𝔉}G-module on M × N, π : F_{M×N} → M the canonical projection and K = ker(π).

Now define E to be the category which has as objects pairs (I, S), where I is a finite 𝔉-subset of M × N and S is a finitely generated O_{𝔉}G-submodule of K. There is a unique arrow (I, S) → (I′, S′) in E if and only if I is an 𝔉-subset of I′ and S is an O_{𝔉}G-submodule of S′. E is a filtered category as (I, S) ∈ Obj(E) and (I′, S′) ∈ Obj(E) both have an arrow to (I ∪ I′, S + S′) ∈ Obj(E).

Consider e = (I, S) ∈ E and the right O_{𝔉}G-module M_{e} = F_{I}/S. Since S ⊂ K, the canonical injection F_{I} ↪ F induces a morphism Θ_{e} : M_{e} → M. Similarly one sees that there are induced morphisms φ_{e′}^{e} : M_{e} → M_{e′} for each arrow e → e′. Hence this construction gives us a functor Λ : E → Mod_{𝔉}-G, which sends e ↦ M_{e}. By the universal property for filtered colimits we obtain a unique morphism

Θ : \underrightarrow{colim}M_{e} → M.

Θ is obviously surjective, take, for each m ∈ M the element e = ({m}, 0), where {m} is viewed as an 𝔉-set.

To show that Θ is injective we use the explicit description of colimits (4). Every element y in \underrightarrow{colim}M_{e} is image of some y_{e} ∈ M_{e} (e = (I, S)). In case y ∈ ker(Θ), it follows that Θ_{e}(y_{e}) = 0. We shall show that there is an e′ ∈ E such that φ_{e′}^{e}(y_{e}) = 0. Let π_{e} : F_{I} → M_{e} and hence y_{e}π(f) for some f ∈ F. But Θ_{e}(y_{e}) = 0 and hence f ∈ K. The desired e′ = (I, S + K′), where K′ is the O_{𝔉}G-submodule of K generated by the 𝔉-set {f}.

So far we have shown, that every right O_{𝔉}G-module is the directed colimit of finitely presented ones.

It remains to show that there is a cofinal subset Ẽ of E such that F_{J}/L is a (finitely generated) free right O_{𝔉}G-module for all (J, L) ∈ Ẽ. Let (I, S) ∈ E be arbitrarily chosen. By (ii) there exists a finitely generated free right O_{𝔉}G-module F such that the canonical map F_{I}/S → M factors through F. We shall show that F = F_{J}/L for some (J, L) ∈ Ẽ.

Let B be a finite 𝔉-set, which is a basis for F. We can find an 𝔉-set B′ ⊂ M × N such that B′ ∩ I = ∅, B′ and B′ have the same order as 𝔉-sets and π(B) = π(B′). We put J = B′ ∪ I and there is a natural surjection σ : F_{J} ↠ K with L = ker(σ) is a finitely generated O_{𝔉}Gsubmodule of K. Thus, for each e ∈ E there exists an e′ ∈ Ẽ and an arrow e → e′, which proves our claim.

(iii) ⇒ (i): Direct colimits commute with tensor-products over O_{𝔉}G, which follows from (4) and the fact, that coproducts commute with ⊗_{𝔉}. Also, projectives are flat.

□

It follows from the proof that every O_{𝔉}G-module is the filtered colimit of finitely presented modules. We also obtain the following.

Corollary 3.3. Every finitely presented flat right O_{𝔉}G-module is projective. □

Lemma 3.4. Let {P_{i}}_{i∈I} be a countable direct system of projectives in Mod_{𝔉}-G. Then \underrightarrow{colim}P_{i} = P has projective dimension ≤ 1.

Proof. Since I is countable, we can pick a cofinal System C ⊂ I, so that P_{c_{1}} → P_{c_{2}} → .... with \underrightarrow{colim}P_{c_{j}} = P. Hence, in the map in the explicit description of the colimit above is injective and we have a projective resolution

∐_{φ:i→j} P_{i} ↪ ∐_{i∈C} P_{i} ↠ P

of P of length 1. □

Analogously to the finitely presented case we say a right O_{𝔉}G-module M is countably presented if there is a short exact sequence N ↪ F ↠ M with F countably generated free and N countably generated, i.e. there is a countably generated free mapping onto N.

Proposition 3.5. Every countably presented flat module M ∈ Mod_{𝔉}-G has pd_{𝔉}M ≤ 1.

Proof. There exists an exact sequence of right O_{𝔉}G-modules

F_{1}⟶^{d}F_{0}⟶^{π}M → 0,

where F_{0} and F_{0} are countably generated free, i.e. there exist countable subfamilies 𝔉_{0} = {K_{1}, K_{2}, ...} and 𝔉_{1} = {L_{1}, L_{2}, ...} of 𝔉 such that

F_{1} = ∐_{L∈𝔉_{1}} P_{L} and F_{0} = ∐_{K∈𝔉_{0}} P_{K}.

By Theorem 3.2 M is the directed colimit \underrightarrow{colim}F_{λ} (λ ∈ Λ) of finitely generated free modules F_{λ}. Let μ_{λ} : F_{λ} → M denote the canonical maps. Hence there exist maps f_{λ} : F_{λ} → F_{0} such that πf_{λ} = μ_{λ} for all λ ∈ Λ. For arbitrary j ≥ 1 we denote by m_{j} = max{i | P_{K_{i}} ⊆ d(P_{L′_{j}}) some j′ ≤ j}. Furthermore, for each j > 0 there exists a λ(j) ∈ Λ such that for each i ≤ m_{j} there is a PK̄_{i}, a direct summand of F_{λ(j)}, with f_{λ(j)}(PK̄_{i}) = P_{K_{i}}. Therefore, for each coproduct we have π(∐ PK̄_{i}) = 0, i.e. PK̄_{i} ⊆ d(F_{1}).

Therefore M is a direct summand of \underrightarrow{colim}F_{λ(j)}, a countable colimit of free modules and has projective dimension 1 by Lemma 3.4. □

4. Homological and cohomological dimensions

From now on let 𝔉 denote the family of all finite subgroups of a group G. Analogously to ordinary homology and cohomology we define (co)homological dimensions via resolutions of the trivial module \underline{ℤ}, which denotes the constant functor \underline{ℤ}(G/K) = ℤ for all K ∈ 𝔉, where all maps \underline{ℤ}(G/K) → \underline{ℤ}(G/L) are the identity transformation. The obvious definitions are now \underline{cd}G = pd_{𝔉}\underline{ℤ} and \underline{hd}G = fld_{𝔉}\underline{ℤ}. We shall compare the two quantities and also make some remarks about their properties.

Similiarly to ordinary group-homology, a standard resolution plays an important rôle in Bredon-Homology. We shall begin this section with a description of this resolution.

Let Δ denote the 𝔉-set with Δ_{K} = {G/K} for all K ∈ 𝔉. As remarked in (1) the free module on Δ is of the form F_{Δ} = ∐_{K∈𝔉} P_{K}(−). Simultaneoulsy we shall further consider the G-set Δ̃ = ∐_{K∈𝔉} G/K. This set has the following property:

Δ̃^{H} ≠ ∅ ⟺ H ∈ 𝔉  (7)

When evaluating F_{Δ} at G/H ∈ O_{𝔉}G, we obtain the following module for the Weyl-group W H = N_{G}(H)/H:

F_{Δ}(G/H) = ∐_{K∈𝔉}P_{K}(G/H) = ⨁_{K∈𝔉}ℤ[G/K]^{H} = ℤΔ̃^{H}.  (8)

We shall consider free O_{𝔉}G-modules on cartesian products of 𝔉-sets.

\begin{aligned}F_{Δ×Δ} & = & ∐_{(δ,δ′)∈Δ×Δ} ℤmor(−, φ(δ, δ′)) \\ & = & ∐_{K,L∈𝔉} ℤ[mor(−, G/K) × mor(−, G/L)], \end{aligned}

by (1) and hence, after evaluating at G/H ∈ O_{𝔉}G:

F_{Δ×Δ}(G/H) = ⨁_{K,L∈𝔉}ℤ[(G/K)^{H} × (G/L)^{H}] = ℤ(Δ̃^{H} × Δ̃^{H}). (9)

We now define a map of 𝔉-sets

d_{n} : Δ^{(n+1)} = \underbrace{Δ × ... × Δ}_{n times} → \underbrace{Δ × ... × Δ}_{n−1 times} = Δ^{(n)}.

as follows: for each K ∈ 𝔉 put d_{n}(g_{0}K, ..., g_{i}K, ..., g_{n}K) = ∑_{i=0}^{n}(−1)^{i}(g_{0}K, ..., ĝ_{i}K, ..., g_{n}K) for each 0 ≤ i ≤ n, where ĝ_{i}K means leaving this term out. This now gives us a complex of free O_{𝔉}G-modules

... → F_{Δ^{(n+1)}} → F_{Δ^{(n)}} → ... → F_{Δ}⟶^{ε}\underline{ℤ} → 0.  (10)

The map ε : F_{Δ} → \underline{ℤ} is defined by ε(gK) = 1.

When evaluating (10) at G/H ∈ O_{𝔉}G, we obtain as sequence of W H-modules

→ ℤ(Δ̃^{H})^{(n+1)} → ℤ(Δ̃^{H})^{(n)} → ... → ℤΔ̃^{H} → ℤ → 0,  (11)

where the induced connecting maps are as follows: d̃^{n}(δ_{0}, ..., δ_{i}, ..., δ_{n}) = ∑_{i=0}^{n}(−1)^{i}(δ_{0}, ..., δ̂_{i}, ..., δ_{n}), where the δ_{i} ∈ Δ̃^{H} for all 0 ≤ i ≤ n. It was shown in [19], Lemma 3.1 that the G sets Δ̃^{H} satisfy condition (7) and furthermore that the sequence (11) is an exact sequence of W Hmodules and hence the standard resolution (10) is indeed an exact sequence of right O_{𝔉}G-modules.

Theorem 4.1. Let G be an arbitrary group. Then (i) \underline{hd}G ≤ \underline{cd}G

(ii) Is G countable then also \underline{cd}G ≤ \underline{hd}G + 1

Proof. (i) follows directly from the fact that projectives are flat. Being countable, G has only countably many finite subgroups. Hence the standard resolution (10) is a countable free resolution and (ii) follows from Proposition 3.5. □

Following [16] we say a family 𝔉 is bounded, if each K ∈ 𝔉 is contained in a maximal M ∈ 𝔉.

Theorem 4.2. Let {G_{λ}}_{λ∈Λ} be a direct system of groups. Then, for the direct limit G = lim_{→}G_{λ} and 𝔉 a bounded family of finite subgroups of G,

(i) \underline{hd}G = max\{\underline{hd}G_{λ}\}
(ii) If Λ is countable, then \underline{cd}G ≤ max\{\underline{cd}G_{λ}\} + 1

Proof. Let ι_{λ} : G_{λ} → G and φ_{μ}^{λ} : G_{λ} → G_{μ} be the canonical maps for all λ, μ ∈ Λ. Let Δ_{λ} denote the Obj{O_{𝔉}G_{λ}}-set with Δ_{λ,K_{λ}} = {G_{λ}/K_{λ}} for all K_{λ} ∈ 𝔉_{λ}, the family of all finite subgroups of G_{λ}. These sets are in a one-to-one correspondence with the 𝔉-set Δ′_{λ}, which are defined as Δ_{λ,K} = {G/K} for all K ∈ 𝔉 with K = ι_{λ}(K_{λ}) and empty otherwise. By some abuse of notation we shall both denote with Δ_{λ}.

We have a directed system of 𝔉-sets given through:

\begin{matrix}Δ_{λ} & → & Δ_{μ} \\ G_{λ}/K_{λ} & ↦ & G_{μ}/K_{μ} \end{matrix}

for all φ_{μ}^{λ} : G_{λ} → G_{μ}. Since 𝔉 is bounded, the colimit \underrightarrow{colim}Δ_{λ} = Δ, the 𝔉-set defined by Δ_{K} = {G/K} for all K ∈ 𝔉.

It now follows that

F_{Δ^{(n)}} ≅ F_{\underrightarrow{colim}Δ_{λ}^{(n)}} ≅ \underrightarrow{colim}F_{Δ_{λ}^{(n)}}

for all n ≥ 0. Since colimits are exact (3.1) we have that ker(F_{Δ^{(n)}} → F_{Δ^{(n−1)}}) = K_{n} ≅ \underrightarrow{colim}K_{n,λ}, the n-th kernel in the respective standard resolutions for O_{𝔉}G_{λ}. (i) now follows directly and (ii) from Lemma 3.4. □

Proposition 4.3. For every G we have that \underline{hd}G ≥ hd_{ℚ}G.

Proof. Let \underline{hd}G = n. Take an arbitrary free resolution of \underline{ℤ}. Hence the n-th kernel is flat and by 3.2 a direct limit of free modules. When evaluating at G/1 and tensoring the result with ℚ, this gives us a ℚG-flat resolution of ℚ, where the n-th kernel is a direct limit of free modules, hence flat. □

5. Extensions

In this section we will be concerned with the Bredon homological and cohomological dimensions for extensions N ↪ G ↠ Q of groups. Our results will be applications of Martinez’ spectral sequences [16]. We shall start by recalling her results and by adapting them to our setting.

Let us denote by 𝔉(G), 𝔉(N) and 𝔉(Q) the families of finite subgroups of G, H and Q respectively. Let ℌ = {S ≤ G : N ≤ S and S/N ∈ 𝔉(Q)}. We also have a map

\begin{matrix}π : 𝔉(G) & → & ℌ \\ L & ↦ & LN \end{matrix}

and there is an obvious equivalence of categories between Mod_{𝔉(Q)}-Q and Mod_{ℌ}-G and between Q-Mod_{𝔉(Q)} and G-Mod_{ℌ}. This implies in particular that

cd_{ℌ}G = \underline{cd}Q and hd_{ℌ}G = \underline{hd}Q.

An application of Martinez’ spectral sequence [16, 5.1] now yields the following result:

Theorem 5.1. [16, 5.2] Let N ↪ G ↠ Q be an extension of groups. Assume that there are integers m and n such that for every S ∈ ℌ, \underline{cd}S ≤ n and \underline{hd}S ≤ m. Then

\underline{cd}G ≤ m + \underline{cd}Q

and

\underline{hd}G ≤ m + \underline{hd}Q.

Example 5.2. The conditions of theorem 5.1 are obviously satisfied when Q is torsion-free. Here the theorem says that \underline{cd}G ≤ \underline{cd}N + cdQ and \underline{hd}G ≤ \underline{hd}N + hdQ.

Example 5.3. Theorem 5.1 can also be applied when N is polycyclic. See for example [15, Example 5.25], where it is shown that for a polycyclic-by-finite group there is always a model for an \underline{E}G of dimension equal to the Hirsch-rank if its polycyclic subgroups of finite index. Hence, for all S ∈ ℌ, \underline{cd}S = \underline{hd}S = hN.

Example 5.4. We can, however, not hope to get a result like 5.1 for arbitrary group-extensions. Ian Leary and the author give examples of virtually torsion-free groups G, where \underline{cd}G is strictly greater than cdH for H a torsion-free subgroup of finite index, see [12, Example 12]. Since H is of type F P_{∞}, the homological and cohomological dimensions are equal [1, Theorem 4.6]. Analogously to the proof of [12, Theorem 6] we can use the homology of \underline{E}G relative to its singular set to provide a lower bound for \underline{hd}G. Let N be a ℤG-module and let I_{1}N be the covariant O_{𝔉}G-module defined as I_{1}N(G/K) = 0 for all K ≠ {1} and I_{1}N(G/1) = N. It can be checked that H_{∗}^{G}(\underline{E}G, \underline{E}G^{sing}; N) ≅ T or_{∗}^{𝔉}(\underline{ℤ}, I_{1}N). For the above examples there is a model X for \underline{E}G such that the relative cellular chain complex C_{∗}(X, X^{sing}) contains as a direct summand a chain complex consisting of only one copy of ℤG in the top degree n = \underline{gd}G. Hence for our examples H_{n}^{G}(\underline{E}G, \underline{E}G^{sing}; ℤG) ≠ 0 and therefore \underline{hd}G = \underline{cd}G > cdH = hdH.

In the next section we shall take a closer look at nilpotent groups where we can prove some more detailed results. Also, 5.1 can be applied to the case when N is finite; but again, we have a stronger result, which follows next. This is an application of Martinez’ spectral sequence [16, 4.3].

Theorem 5.5. Let F ↪ G ↠ Q be an extension of groups where F is finite. Then

\underline{cd}G = \underline{cd}Q and \underline{hd}G = \underline{hd}Q.

Proof. Since F is finite, ℌ ⊆ 𝔉(G) and also, for each L ∈ 𝔉(G), there is S = LF ∈ ℌ such that L ≤ S. Hence the conditions of [16, 4.5] apply and H_{ℌ}^{n}(G, C) ≅ H_{𝔉}^{n}(G, C) and H_{n}^{ℌ}(G, D) ≅ H_{n}^{𝔉}(G, D) for all C ∈ Mod_{𝔉}-G and all D ∈ G-Mod_{𝔉}. □

6. Nilpotent groups

In this section we shall be concerned with nilpotent groups G. The elements of finite order in G form a charcteristic locally finite subgroup, denoted τ(G). And we write hG for the Hirsch length of G.

Theorem 6.1. Let G be a nilpotent group and 𝔉 be the family of finite subgroups of G. Then

\underline{hd}G = hG.

Proof. Obviously, by 4.3 and Stammbach’s result [23] it follows that \underline{hd}G ≥ hG. We shall show the other inequality.

Let G be of finite Hirsch length n. We have a group extension

τ(G) ↪ G ↠ H

with H torsion-free, see e.g. [20], page 132. Since H is torsion-free Stammbach’s result [23] implies that hH = hdH = \underline{hd}H.

Now we use a similar argument as in Theorem 5.5, but this time we apply it to 𝔉̂, the completion of 𝔉. Now ℌ = {τ(G)}, and the families ℌ and 𝔉̂ satisfy the conditions of [16, 4.5] and hence hd_{ℌ}G = hd𝔉̂G. Also, Theorem 4.8 of [16] implies that hd𝔉̂G ≥ \underline{hd}G. As the Hirsch length is additive for group-extensions we obtain the following chain of inequalities:

hG = hH = \underline{hd}H = hd_{ℌ}G = hd𝔉̂G ≥ \underline{hd}G ≥ hG

proving the theorem. □

We cannot hope to use the methods of 6.1 to prove a similar result for soluble groups as one of the conditions in [16, 4.5] is that ℌ is intersection-closed and we don’t have a unique maximal locally finite subgroup.

Remark 6.2. Theorem 4.1 now gives us bounds for the Bredon-co-homological dimension of nilpotent groups. For virtually torsion-free nilpotent groups the next result is of interest, linking \underline{cd}G to the virtual cohomological dimension (vcdG) of G.

Theorem 6.3. Let G be a virtually torsion-free nilpotent group. Then \underline{cd}G = vcdG.

Proof. Let N ↪ G ↠ Q be an extension with N torsion-free and Q finite. As N is torsion-free, the intersection τ(G) ∩ N is trivial. Therefore τ(G) is isomorphic to a subgroup of Q and hence finite. Now apply Theorem 5.5 to the extension τ(G) ↪ G ↠ H and use the fact that N is isomorphic to a subgroup of finite index in H. □

Remark 6.4. This theorem can be used to verify that the Brown conjecture holds for arbitrary virtually torsion-free nilpotent groups. It follows directly for groups of \underline{cd}G ≠ 2 from the results of Dunwoody [7] and Lück [13], as here we can always find a model for \underline{E}G of dimension equal to \underline{cd}G. We only have to make sure that an Eilenberg-Ganea type phenomenon cannot happen for virtually torsion-free nilpotent groups of \underline{cd}G = 2. Let N be a torsion-free subgroup of finite index in G. Then cdN = 2. Applying [1, Theorem 8.8] to N and using that the centre Z(N) ≠ {1} yields cdZ(N) = 1. Hence Z(N) is infinite cyclic. The extension Z(N) ↪ N ↠ Q now satisfies the assumptions of Fel’dman’s theorem, see Theorem 5.5 and Remark of [1]. Hence 2 = cdN = cdZ(N) + cdQ and cdQ = 1 implying that Q is free abelian of rank 1. Hence N is polycyclic and therefore admits a 2dimensional model for an Eilenberg-Mac Lane space EG. Applying this to τ(G) ↪ G ↠ H implies that G is an extension of finitely generated groups, hence finitely generated and in particular polycylic. And for a polycyclic group of \underline{cd}G = 2 we can always find a 2-dimensional model for \underline{E}G, see [15, Example 5.25].

Acknowledgements

The author thanks Conchita Martinez-Pérez for many interesting and fruitful discussions and also for making early versions of [16] available to her. Thank you also to Ian Leary for making useful remarks on earlier versions, and to the referee of an earlier version, whose comments led to improvements to Section 6 and the inclusion of Section 5.

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Departement Mathematik, ETH-Zentrum, CH8092 Zürich

  E-mail address: bean@math.ethz.ch