ON p-COMPLETED CLASSIFYING SPACES OF DISCRETE GROUPS AND FINITE COMPLEXES

RAN LEVI

Abstract. We show that for certain discrete p-perfect groups G, in particular for all pperfect groups of finite cohomological dimension, the loop space on the p-completed classifying space ΩBG_{p}^{∧} is a retract of the loop spaces on a certain finite complex. For finite vcd groups we provide a bound on the the dimension of such a complex.

0. Introduction

It is customary to think, following the proof of the Sullivan conjecture, that there is not much in common between finite CW complexes and the family of classifying spaces of finite groups [13]. Indeed Miller proves that if G is a finite group and X a p-complete, nilpotent finite complex then the pointed mapping space Map_{∗}(BG_{p}^{∧}; X) is weakly contractible, where (−)_{p}^{∧} denotes the p-completion functor of Bousfield and Kan [2]. Miller’s theorem was in fact proven for a much larger family of groups, including many of those which are of classical interest in algebraic topology, and offers a point of view that is to regard classifying spaces of groups in this family and finite nilpotent spaces as two extremes in homotopy theory. In this note we suggest a different perspective on classifying spaces of finite groups. Many interesting infinite groups also fall within the scope of our discussion.

Recall that a group G is said to be p-perfect if the first mod-p homology group of BG vanishes. Every discrete group G contains a unique maximal p-perfect subgroup O^{p}G, which is normal in G. In particular if G is finite then G/O^{p}G is the maximal quotient of G, which is a p-group. If the p-completion functor preserves the mod-p homology of BG or, in other words, if BG is a p-good space, then B(O^{p}G)_{p}^{∧} is the universal cover of BG_{p}^{∧}. Classifying spaces BG of discrete groups are not generally p-good but they are, for instance for G finite or p-perfect [2].

We offer our viewpoint for a family of groups G, such that BG is p-good and such that B(O^{p}G)_{p}^{∧} is not contractible. In addition we require, in a sense, that G admits a finite dimensional faithful representation. The last condition is satisfied in particular for any group of finite vcd, where the first two restrictions mean that although p-completion preserves the homology of BG, the space BG_{p}^{∧} fails to be a classifying space of a discrete group. For such a group G, our main result states that ΩB(O^{p}G)_{p}^{∧} is, in fact, a retract of the loop space on some finite complex. For finite vcd groups we produce a bound on the dimension of such a complex, depending only on the vcd and the minimal dimension of a representation for the group under consideration. The main result is used to explain the failure of the homotopy of BG_{p}^{∧} to be detected by restrictions to subgroups, in certain cases. In particular we demonstrate that unstable algebraic K groups of finite fields can be very hard to detect by restriction to smaller general linear groups. We also deduce the existence of a bound on the order of the Freudenthal suspension map on BG_{p}^{∧}, if G has finite integral homology groups.

Date: February 20 1996.

1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55Q52.

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1. Definitions and Statement of results

Throughout this paper we fix a prime p. All spaces are assumed to have the homotopy type of a CW complex. For a space X, a ring R and an R[π_{1}(X)]-module M, define cd_{M}(X), the cohomological dimension of X mod-M, to be the largest integer n, such that H^{n}(X; M) ≠ 0, and say that X has infinite cohomological dimension mod-M if such an integer n does not exist. In particular we will abbreviate cd_{𝔽_{p}}(−) by cd_{p}(−), where 𝔽_{p} is assumed, of course, to have a trivial π_{1}(X) action. We call the number cd_{p}(X) the mod-p cohomological dimension of X.

Our statements use the language of p-compact groups, namely the homotopy theoretic analogue of compact Lie groups. We briefly recall the terminology we need. The reader is referred to [6] for details.

A space Y is said to be 𝔽_{p}-finite if its mod-p homology is a finite dimensional mod-p vector space. A p-compact group is an 𝔽_{p}-finite loop space X such that BX is p-complete. For example, every compact connected Lie group L gives rise to a p-compact group L_{p}^{∧}.

Definition 1.1. Let G be a discrete group and X a p-compact group. A map

f : BG ⟶ BX

is said to be a homotopy representation of G in X. If a homotopy representation of G in X is given, let X_{G} denote the homotopy fibre of f. A homotopy representation of G in X is said to be virtually faithful if cd_{p}(X_{G}) < ∞ and faithful if X_{G} is 𝔽_{p}-finite. The dimension of a homotopy representation f of G in X is defined to be cd_{p}(X).

The concept of a (faithful) homotopy representation is obviously a generalization of what we usually understand as the map induced on p-completed classifying spaces by a (faithful) group representation. Thus every finite group has faithful homotopy representations. But not every homotopy representation corresponds to a faithful group representation. For instance, under some mild restrictions, if G(q) is a group of Lie type over the finite field 𝔽_{q} and p is a prime not dividing q, then there is a faithful homotopy representation of BG(q) in the BG(ℂ)_{p}^{∧}, where G(ℂ) is the corresponding complex Lie group [7]. We shall observe below that every finite vcd group Γ such that H^{∗}(BΓ; 𝔽_{p}) is of finite type has faithful homotopy representations that do not arise from a faithful group representations. The reason for using p-compact groups rather then just compact Lie groups is that there are actually homotopy representations, where X is not of the homotopy type of a p-completed Lie group (see for instance [9, II. 6].

Notice that every virtually faithful homotopy representation of a group G is faithful if and only if G has mod-p cohomology of finite type. Also notice that the concept of dimension of a homotopy representation is different from the usual algebraic notion of dimension for group representations. For example an n-dimensional unitary representation of G induces an n^{2}-dimensional homotopy representation.

For a discrete group G and a given model BG for its classifying space, let B_{n}G denote the n-skeleton of BG (one can take, for example, the Milnor construction for BG and its n-th stage for B_{n}G [12]). Let α_{n} : B_{n}G ⟶ BG denote the natural inclusion.

Theorem 1.2. Let G be a p-perfect group, which admits a faithful homotopy representation in some p-compact group. Then for some n > 0 the map Ωα_{n} : ΩB_{n}G_{p}^{∧} ⟶ ΩBG_{p}^{∧} has a right homotopy inverse.

Several prior results suggested that there should be a relationship between spaces of the form ΩBG_{p}^{∧} for G finite and p-perfect and the family of finite complexes. For example, one can show that for such a group G, the loop space ΩBG_{p}^{∧} has an exponent in its reduced integral homology. It is known that the loop space on any finite torsion complex has an integral homology exponent [10]. Thus Theorem 1.2 provides a new proof for the existence of an exponent although, an upper bound is hard to determine by this method. This and some other general properties of ΩBG_{p}^{∧}, for G finite and p-perfect are discussed in [9].

To state our next result we first need to define the following numerical invariants for discrete groups.

Definition 1.3. Let G be a discrete group. Define μ_{p}(G) to be least non-negative integer n, such that there exists an n-dimensional virtually faithful homotopy representation of G. If G is p-perfect, define the p-essential dimension of G, ed_{p}(G) to be the least positive integer m such that the natural map α_{m} : B_{m}G_{p}^{∧} ⟶ BG_{p}^{∧} has a right homotopy inverse after looping.

Theorem 1.2 could be restated to say that every p-perfect group, which admits a faithful homotopy representation has a finite p-essential dimension.

Recall that a group Γ is said to have finite cohomological dimension if there is a finite dimensional model for BΓ. A group Γ is said to be of finite virtual cohomological dimension, or finite vcd, if it contains a normal subgroup Γ′ of finite cohomological dimension and finite index. The following result states that Theorem 1.2 applies to all finite vcd groups and also computes a, possibly rough, bound on the p-essential dimension of such a group.

Theorem 1.4. Let Γ be a p-perfect group of finite vcd. Let Γ′ be a normal torsion free subgroup of finite index and let G denote the quotient Γ/Γ′. Then

(1) μ_{p}(Γ) ≤ μ_{p}(G).

(2) If H^{∗}(Γ; 𝔽_{p}) is of finite type then ed_{p}(Γ) ≤ max(μ_{p}(Γ) + 1, μ_{p}(Γ) + vcd(Γ)).

(3) If H^{∗}(Γ; 𝔽_{p}) is of finite type and Γ̄ is a p-perfect subgroup then ed_{p}(Γ̄) ≤ ed_{p}(Γ).

In section 5 we give an example of a finite p-perfect group G and a p-perfect subgroup H, such that H contains the Sylow p-subgroup of G and so BG_{p}^{∧} is a stable retract of BH_{p}^{∧} but, on the other hand, the map induced by inclusion BH_{p}^{∧} ⟶ BG_{p}^{∧} carries no unstable homotopy information. This example has been known for a while but was considered by the author to be a coincidence. The following corollary of our observations explains why detection of homotopy by subgroups may fail. An example of a different nature from unstable algebraic K-theory is also given in section 5.

Corollary 1.5. Let G be a p-perfect group. Let f : BG ⟶ X be any map. If conn(X) > ed_{p}(G) then Ωf is null-homotopic. In particular if H is a p-perfect subgroup of G such that ed_{p}(H) < conn(BG_{p}^{∧}) then

ΩBρ_{p}^{∧} : ΩBH_{p}^{∧} ⟶ ΩBG_{p}^{∧}

is null-homotopic, where ρ is the inclusion map.

For a space X, let

E_{n}(X) : X ⟶ Ω^{n}Σ^{n}X

denote the Freudenthal suspension map. We use our results to bound the image of π_{∗}(BG_{p}^{∧}) under the map induced by E_{n}(BG_{p}^{∧}), for certain groups G.

Theorem 1.6. Let G be a p-perfect group of finite p-essential dimension and finite reduced integral homology groups. Then for any integer 1 ≤ n ≤ ∞, there exist a positive integer e, such that ΩE_{n}(BG_{p}^{∧}) has order dividing p^{e} in the group [ΩBG_{p}^{∧}; Ω^{n+1}Σ^{n}BG_{p}^{∧}].

This last claim is of course analogous to the corresponding fact for finite torsion complexes. Notice that every finite p-perfect group automatically satisfies the hypotheses of Theorem 1.6. We remark that for a finite p-perfect group G, it is known that the space ΩBG_{p}^{∧} has a stable homotopy exponent, bounded above by the order of the Sylow p-subgroup of G. On the other hand the stable homotopy of BG_{p}^{∧} contains elements of arbitrarily high order [9]. We regard Theorem 1.6 as an extra bit of evidence to the conjecture that BG_{p}^{∧} has a homotopy exponent for every finite group G (compare [9, 11]).

It is a pleasure to thank Fred Cohen, Brayton Gray, Chun-Nip Lee and Stewart Priddy for useful discussions. The author is also grateful to John Klein for pointing out to him an old theorem of T. Ganea, which provides a simpler proof of Theorem 1.2.

2. Proof of the main Theorem

In an earlier version of this note, Theorem 1.2 was proven independently of Ganea’s result, stated below. Using Ganea’s work [8] however makes the proof simpler and more elegant and so we start by recalling the relevant statement. For a space X, let CX denote the cone on X. For any two spaces X and Y let X ∗ Y denote the join product of X and Y. We remind the reader that the spaces X ∗ Y and ΣX ∧ Y are homotopy equivalent.

Theorem 2.1 (Ganea). Let F ⟶^{ι} E ⟶^{π} B be a fibration. Let ρ : E ∪_{ι} CF ⟶ B extend π and let F_{ρ} be the homotopy fibre of ρ. Then there is a homotopy equivalence

ω : F ∗ ΩB ⟶ F_{ρ}.

In the situation described by Theorem 2.1 the natural map ϕ : F ⟶ F_{ρ} is null-homotopic, implying that the principal fibration

ΩF_{ρ} ⟶ Ω(E ∪_{ι} CF) ⟶ ΩB

has a section and so there are homotopy equivalences

Ω(E ∪_{ι} CF) ≃ ΩB × ΩF_{ρ} ≃ ΩB × ΩΣ(F ∧ ΩB).

Let G be a discrete p-perfect group and let f be a faithful homotopy representation of G in some p-compact group X. Let X_{G} denote the homotopy fibre of f. We first observe that X may be assumed connected. Indeed, suppose that X is not connected. Then π_{1}(BX) is a finite p-group [6]. Thus, since G is p-perfect, the composite

BG ⟶^{f} BX ⟶^{γ} K(π_{1}(BX), 1)

is null-homotopic, where γ is the map classifying the universal cover for BX. Consequently f lifts to a map f̃from BG to the universal cover BY of BX, which is again the classifying space of a p-compact group. The homotopy fibre Y_{G} of f̃ obviously satisfies cd_{p}(Y_{G}) = cd_{p}(X_{G}), being a component of X_{G}.

Thus assume that the map f above is a faithful homotopy representation of G and that X is connected. Consider the fibre sequence

X ⟶^{ι} X_{G} ⟶^{π} BG ⟶^{f} BX.  (1)

Define S′_{f}(G) to be the mapping cone of ι,

S′_{f}(G) = (X_{G}) ∪_{ι} CX.

Lemma 2.2. The space S′_{f}(G) has a finite mod-p cohomological dimension, its fundamental group is isomorphic to G. Both X_{G} and S′_{f}(G) are p-good.

Proof. Since both X and X_{G} have finite mod-p cohomological dimension, so does S′_{f}(G). Since X is connected, the fundamental group of X_{G} is an extension of G by π_{1}(X). By the Van-Kampen theorem π_{1}(S′_{f}(G)) is the cokernel of this extension, namely G. Since G is p-perfect, S′_{f}(G) is p-good [2]. As for X_{G}, notice that it is the fibre in an orientable fibration with p-good base and total spaces. Hence it is itself p-good.

Since X is connected, BX is simply-connected and p-completion preserves the fibration (1). Thus Ganea’s theorem applies and we obtain the following

Proposition 2.3. There is a homotopy equivalence

Ω((X_{G})_{p}^{∧} ∪ι̂ CX_{p}^{∧}) ≃ ΩBG_{p}^{∧} × ΩΣ(X ∧ ΩBG_{p}^{∧}).

Notice that the space (X_{G})_{p}^{∧} ∪ι̂ CX_{p}^{∧} may fail to be p-complete However, since both X and X_{G} are p-good, the obvious map S′_{f}(G) ⟶ (X_{G})_{p}^{∧} ∪ι̂ CX_{p}^{∧} induces a mod-p homology isomorphism and hence a homotopy equivalence after completion [2].

Define S_{f}(G) to be the p-completion of S′_{f}(G). Having established that S′_{f}(G) is p-good and of finite mod-p cohomological dimension, the space S_{f}(G) is p-complete and cd_{p}(S_{f}(G)) < ∞. Since p-completion preserves products, the p-complete version of Proposition 2.3 takes the form

ΩS_{f}(G) ≃ ΩBG_{p}^{∧} × ΩΣ(X ∧ ΩBG_{p}^{∧})_{p}^{∧}.  (2)

One might like to assume that f is virtually faithful rather than faithful. In that case the resulting space S_{f}(G) will be of finite mod-p cohomological dimension but not generally 𝔽_{p}-finite. For the sake of simplicity however, we remain with the hypothesis of Theorem 1.2 and so S_{f}(G) is in fact 𝔽_{p}-finite since both X and X_{G} are.

To complete the proof of Theorem 1.2, notice that since S_{f}(G) is simply-connected and 𝔽_{p}-finite, it may be assumed to have the homotopy type of a finite dimensional CW -complex, up to p-completion. Thus for n ≥ cd_{p}(S_{f}(G)), the map α̂ : S_{f}(G) ⟶ BG_{p}^{∧} factors through B_{n}G_{p}^{∧} and we obtain a commutative diagram of fibrations

[figure: ΩBG_{p}^{∧}; =; (X_{ρ})_{p}^{∧}; S_{f}(G); BG_{p}^{∧}; ˆ∂; q̂; ρ̂; ΩBG_{p}^{∧}; P_{n}(BG_{p}^{∧}); B_{n}G_{p}^{∧}; BG_{p}^{∧}; δ̂; α̂_{n}]

where P_{n}(BG_{p}^{∧}) is the homotopy fibre of α̂_{n}. Since ˆ∂ is null-homotopic, so is δ̂, proving that Ωα̂_{n} has a right homotopy inverse and completing the proof of Theorem 1.2.

We proceed by proving Corollary 1.5. If G is a p-perfect group with ed_{p}(G) = m < ∞, then the map

Ωα̂_{m} : ΩB_{m}G_{p}^{∧} ⟶ ΩBG_{p}^{∧}

has a right homotopy inverse β. Let f : BG_{p}^{∧} ⟶ X be a map such that X is ed_{p}(G)- connected. Then the composition f ∘ α̂_{m} is null-homotopic. Thus

∗ ≃ Ω((f ∘ α̂_{m}) ∘ β) ≃ Ω(f ∘ (α̂_{m} ∘ β)) ≃ Ωf.

We end this section by pointing out that the result we proved is actually a bit stronger than our original claim. Indeed the fact that ΩBG_{p}^{∧} is a retract of ΩS_{f}(G) implies that it is a retract of ΩB_{n}G_{p}^{∧} but the converse does not automatically hold. One could wonder about the existence of a “minimal complex” M_{p}(G) such that ΩBG_{p}^{∧} is a retract of ΩM_{p}(G). It is not clear though what should minimality mean in this context.

3. groups of finite vcd

Groups of finite vcd form a rather interesting family of objects. In particular finite groups, amalgamated products of finite groups, the general linear group GL_{n}(ℤ) and its subgroups, braid groups and mapping class groups are included in this collection.

If f is a homotopy representation of a group G in some p-compact group X and ϕ : K ⟶ G is a group homomorphism, then, by abuse of terminology, we call the homotopy representation f ∘ Bϕ of K in X, the restriction of f to K via ϕ, even if ϕ is not a monomorphism of groups. Let f̃ denote the restriction of f to K via ϕ. Then we denote the homotopy fibre of f̃ by X_{K}, provided of course that confusion of notation cannot occur. If ϕ is a group monomorphism then there is a covering space X_{K} ⟶ X_{Γ} with fibre Γ/ϕ(K).

Throughout this section, let Γ denote a fixed p-perfect group of finite vcd. Choose a normal, torsion free subgroup of finite index Γ′ in Γ and let G denote the quotient Γ/Γ′. Let π denote the Sylow p-subgroup of G.

Part (1) of Theorem 1.4 claims that the minimal dimension μ_{p}(Γ) of a virtually faithful homotopy representation of Γ is bounded from above by μ_{p}(G). To prove this choose any faithful homotopy representation g : BG ⟶ BX of G in X and consider the restriction g̃ of g to Γ. We will show that g̃ is virtually faithful.

Indeed, one has the following commutative diagram of fibrations.

[figure: BΓ′; BΓ′; =; X_{Γ}; BΓ; Bq; BX; =; g̃; X_{G}; BG; BX.; g]  (3)

The space X_{G} is 𝔽_{p}-finite by the choice of g, whereas cd_{p}(BΓ′) ≤ vcd(Γ). By inspection of the Serre spectral sequence associated to the left column of Diagram (3) one has

cd_{p}(X_{Γ}) ≤ cd_{p}(X_{G}) + vcd(Γ).

Hence g̃ is virtually faithful and the result follows. Notice that (1) automatically implies that every finite vcd group with mod-p cohomology of finite type has finite mod-p essential dimension.

The proof of part (2) of the theorem is based upon the simple observation, that if S is any 1-connected p-complete 𝔽_{p}-finite space and f : S ⟶ BG_{p}^{∧} is a map, such that Ωf has a right homotopy inverse, then ed_{p}(G) ≤ cd_{p}(S).

The proof of Theorem 1.2 gives a procedure, associating the space S_{f}(Γ), which satisfies the necessary requirements, to any faithful homotopy representation f of Γ in a p-compact group X. If vcd(Γ) = 0, i.e. if Γ is finite, then X ⟶^{ι̂} X_{Γ} is a rational equivalence. It is not hard to show by inspection of the integral homology Serre spectral sequence for the fibration

X ⟶ (X_{Γ})_{p}^{∧} ⟶ BΓ_{p}^{∧}

that ι̂^{∗} must carry the top class in H_{∗}(X; ℤ) to a multiple of the generator of torsion free summand in H_{∗}(X_{Γ}; ℤ) by a power of p. Thus in that case we have cd_{p}(S_{f}(Γ)) = cd_{p}(X)+1. In any case cd_{p}(S_{f}(Γ)) ≤ cd_{p}(X_{Γ}). Thus we need to prove the following

Proposition 3.1. Let Γ be a finite vcd group with finite type mod-p cohomology. Let a faithful homotopy representation f of Γ in a p-compact group X be given. Then

cd_{p}(X_{Γ}) ≤ cd_{p}(X) + vcd(Γ).

Letting X in Proposition 3.1 be of minimal dimension μ_{p}(Γ), we obtain claim (2).

The main fact needed to prove Proposition 3.1 is contained in the following

Proposition 3.2. Let Γ be a finite vcd group with finite type mod-p cohomology. Let f be a faithful homotopy representation of Γ in some p-compact group. Let Γ̄ < Γ be an arbitrary subgroup. Then the restriction of f to Γ̄ is virtually faithful.

Assuming Proposition 3.2 we now prove Proposition 3.1. Let Γ(p) < Γ be the subgroup given by the pull-back of the system

Γ ⟶ G ⟵ π.

Then the index|Γ : Γ(p)| = |G : π| is prime to p. Thus the usual transfer argument for the covering X_{Γ(p)} ⟶ X_{Γ} gives that cd_{p}(X_{Γ}) ≤ cd_{p}(X_{Γ(p)}). Notice that vcd(Γ(p)) = vcd(Γ). Thus it suffices to show that

cd_{p}(X_{Γ(p)}) ≤ cd_{p}(X) + vcd(Γ).

The next lemma reduces this problem to Proposition 3.2 and thus completes the proof of 3.1.

Lemma 3.3. Let f be a virtually faithful homotopy representation of Γ(p) in X. Then

cd_{p}(X_{Γ(p)}) ≤ cd_{p}(X) + vcd(Γ(p)).

Proof. The group Γ(p) is an extension of the p-group π by Γ′. Restriction to Γ′ gives a principal fibration

X ⟶ X_{Γ′} ⟶ BΓ′.

Since Γ′ has finite cohomological dimension, it follows by inspection of the associated Serre spectral sequence that

cd_{p}(X_{Γ′}) = cd_{p}(X) + cd_{p}(BΓ′) ≤ cd_{p}(X) + vcd(Γ).

The Shapiro lemma for the regular covering X_{Γ′} ⟶ X_{Γ(p)} gives

H_{∗}(X_{Γ′}; 𝔽_{p}) ≃ H_{∗}(X_{Γ(p)}; 𝔽_{p}[π]),

so cd_{p}(X_{Γ′}) = cd_{𝔽_{p}[π]}(X_{Γ(p)}). The vector space 𝔽_{p}[π] becomes a π_{1}(X_{Γ(p)})-module via the projection of Γ(p) onto π and the obvious action of π on 𝔽_{p}[π]. Since π is a finite p-group, this action is nilpotent. Assuming cd_{p}(X_{Γ(p)}) < ∞, an easy inductive argument, as in [6], gives that

cd_{p}(X_{Γ(p)}) = cd_{𝔽_{p}[π]}(X_{Γ(p)})

and completes the proof of the lemma.

Next we turn to the proof of Proposition 3.2. Notice that Γ is not assumed p-perfect. Let Γ̄ be an arbitrary subgroup of Γ. Then the composition

Γ̄ ⟶ Γ ⟶ G

can be factored through an epimorphism from Γ̄ to a subgroup Ḡ < G followed by the inclusion of Ḡ in G. Let Γ̄′ denote the kernel of the projection Γ̄ ⟶ Ḡ. Then Γ̄′ is a subgroup of Γ′ and is therefore torsion free and, obviously, of finite index in Γ̄. Thus cd(Γ̄′) = vcd(Γ̄) ≤ vcd(Γ) < ∞. Let π̄ denote the Sylow p-subgroup of Ḡ.

Assuming |π̄| = p^{n}, choose a composition series for π̄

π̄ = π_{n} ⊳ π_{n−1} ⊳ ⋯ ⊳ π_{1} ⊳ π_{0} = {1},

such that each quotient π_{i}/π_{i−1} is isomorphic to ℤ/pℤ. Such a composition series gives rise to a sequence of subgroups

Γ̄(p) = Γ_{n} ⊳ Γ_{n−1} ⊳ ⋯ ⊳ Γ_{1} ⊳ Γ_{0} = Γ̄′,

where Γ_{i} is defined inductively as the pull-back of the system

π_{i} ⟶ π_{i+1} ⟵ Γ_{i+1}.

Restriction of the homotopy representation f to each of the subgroups Γ_{i} gives a homotopy representations f_{i} of Γ_{i} in X. We proceed by showing inductively that each f_{i} is a virtually faithful representation.

Let X_{i} denote the homotopy fibre of f_{i}. Notice that X_{0} = XΓ̄′, so cd_{p}(X_{0}) < ∞ and it can be used as the base of our induction. The following lemma records some fibration sequences, which will be involved in the calculation.

Lemma 3.4. There are the following homotopy fibre sequences (1) X_{0} ⟶ X_{i} ⟶ Bπ_{i}.

(2) X_{i−1} ⟶ X_{i} ⟶ Bℤ/pℤ.

Proof. Fibration (1) is induced from the fibration X_{Γ̄(p)} = X_{n} ⟶ Bπ̄, which has X_{0} = XΓ̄′ as fibre.

The existence of fibration (2) follows at once from the fact that Γ_{i} is an extension of ℤ/pℤ by Γ_{i−1}.

Consider the cohomology class of the composite

BΓ_{i} ⟶^{q_{i}} Bℤ/pℤ ⟶^{u} K(ℤ, 2),

where u denotes the generator of H^{2}(Bℤ/pℤ; 𝔽_{p}). Then there is a central extension F_{i} of Γ_{i} by ℤ, for which uq_{i} is the extension class. The projection F_{i} ⟶ Γ_{i} gives a homotopy representation of F_{i} in X with fibre X_{F_{i}}. Furthermore, since the extension defining F_{i} is central, the induced fibration

K(ℤ, 1) ⟶ X_{F_{i}} ⟶ X_{i}  (4)

is principal.

Lemma 3.5. For 1 ≤ i ≤ n if cd_{p}(X_{i−1}) < ∞ then the same holds for X_{F_{i}}.

Proof. For i ≥ 1 the group F_{i} may also be defined as the pull-back of the system

ℤ ⟶^{ρ} ℤ/pℤ ⟵ Γ_{i},

where ρ is the reduction map. Thus F_{i} can be regarded as an extension of ℤ by Γ_{i−1}. One now observes immediately that the inclusion of Γ_{i−1} in Γ_{i} factors through F_{i}. As a consequence we obtain a commutative diagram of fibrations

[figure: X_{i−1}; BΓ_{i−1}; BX; =; X_{F_{i}}; BF_{i}; BX; K(ℤ, 1); K(ℤ, 1); =]  (5)

The lemma now follows by inspection of the Serre spectral sequence for the left column of Diagram 5.

We now 0 by analyzing the principal fibre sequence (4) above, assuming cd_{p}(X_{F_{i}}) < ∞. Indeed we shall eventually make an induction hypothesis that cd_{p}(X_{i−1}) < ∞, which would imply the same for X_{F_{i}}.

First set some notation. Consider the sequence

X_{i} ⟶^{j_{i}} BΓ_{i} ⟶^{q_{i}} Bℤ/pℤ ⟶^{u} K(ℤ, 2).  (6)

Let ū = u^{∗}(ι_{2}), γ = q_{i}^{∗}(u) and v = j_{i}^{∗}(γ) denote the respective mod-p cohomology classes, where ι_{2} is the generator of H^{2}(K(ℤ, 2); 𝔽_{p}). Notice that (4) is induced by the composite given in (6).

The Serre spectral sequence associated to (4) has only two non-trivial lines and E_{3} = E_{∞}. Let ι_{1} denote the generator of H^{1}(K(ℤ, 1); 𝔽_{p}). Then d_{2} in the spectral sequence is determined by d(ι_{1}) = v.

Lemma 3.6. The inequality

cd_{p}(X_{i}) < cd_{p}(X_{F_{i}})

holds if and only if there exists some non-negative integer k such that v^{k} = 0 in H^{2k}(X_{i}; 𝔽_{p}).

Proof. If the inequality of the lemma holds then in particular cd_{p}(X_{i}) ≤ ∞. Thus v must be a nilpotent element.

Assume v^{k} = 0 for some k ≥ 0 and consider the mod-p cohomology Serre spectral sequence for (4). Let y ∈ H^{j}(X_{i}; 𝔽_{p}) be any element with j ≥ cd_{p}(X_{F_{i}}). Then d_{2}(ι_{1} ⊗ y) = vy and vy ≠ 0 or otherwise ι_{1} ⊗ y would survive to E_{∞}, thus giving a non-zero element over the mod-p cohomological dimension of X_{F_{i}}. The same argument implies to show inductively that d_{2}(ι_{1} ⊗ v^{j}y) = v^{j+1}y ≠ 0 for every j ≥ 0. But v^{k} = 0 by hypothesis, which implies a contradiction. This completes the proof.

Lemma 3.6 reduces the problem of bounding cd_{p}(X_{i}) to showing that v is a nilpotent element. Since v = j_{i}^{∗}(γ), the only non-trivial case we have to consider is when γ is an element of infinite hight. We will show that in this case a power of γ is in the ideal generated by

Im(f_{i}^{∗} : H^{∗}(BX; 𝔽_{p}) ⟶ H^{∗}(BΓ_{i}; 𝔽_{p})).

The first step in doing so is the following

Lemma 3.7. If γ ∈ H^{2}(BΓ_{i}; 𝔽_{p}) is an element of infinite hight then some power of γ is in the ideal in H^{∗}(BΓ_{i}; 𝔽_{p}) generated by

Im(g^{∗} : H^{∗}(BΓ; 𝔽_{p}) ⟶ H^{∗}(BΓ_{i}; 𝔽_{p})).

Proof. The projection q_{i} is defined as the composite

Γ_{i} ⟶^{δ_{i}} π_{i} ⟶^{ϵ_{i}} ℤ/pℤ.

Thus γ = q_{i}^{∗}(u) = δ_{i}^{∗}ϵ_{i}^{∗}(u) and γ′ = ϵ_{i}^{∗}(u) is an element of infinite height in H^{∗}(Bπ_{i}; 𝔽_{p}). By construction, there is a commutative diagram of groups

[figure: Γ_{i}; δ_{i}; Γ; δ; ϕ; π_{i}; G; ϕ_{i}]

Since π_{i} is a p-subgroup of G, its mod-p cohomology is a finitely generated module over H^{∗}(BG; 𝔽_{p}) via the restriction map ϕ_{i}. Hence the algebra cokernel

H^{∗}(Bπ_{i}; 𝔽_{p})//ϕ_{i}^{∗}(H^{∗}(BG; 𝔽_{p})) = H^{∗}(Bπ_{i}; 𝔽_{p}) ⊗_{ϕ_{i}^{∗}(H^{∗}(BG;𝔽_{p}))} 𝔽_{p}

is a finite dimensional vector space. Consequently some power of γ′ is in the ideal generated by Im(ϕ_{i}^{∗}). Commutativity of the diagram

[figure: H^{∗}(BG; 𝔽_{p}); =; H^{∗}(Bπ_{i}; 𝔽_{p}); δ_{i}^{∗}; H^{∗}(Bπ_{i}; 𝔽_{p})//ϕ_{i}^{∗}(H^{∗}(BG; 𝔽_{p})); ϕ_{i}^{∗}; H^{∗}(BG; 𝔽_{p}); H^{∗}(BΓ_{i}; 𝔽_{p}); H^{∗}(BΓ_{i}; 𝔽_{p})//δ_{i}^{∗}ϕ_{i}^{∗}(H^{∗}(BG; 𝔽_{p})); δ_{i}^{∗}ϕ_{i}^{∗}]

now implies that a certain power of γ is in the ideal generated by δ_{i}^{∗}ϕ_{i}^{∗}(H^{∗}(BG; 𝔽_{p})) in H^{∗}(BΓ_{i}; 𝔽_{p}). But we have

δ_{i}^{∗}ϕ_{i}^{∗}(H^{∗}(BG; 𝔽_{p})) = ϕ^{∗}δ^{∗}(H^{∗}(BG; 𝔽_{p})) ⊆ ϕ^{∗}(H^{∗}(BΓ; 𝔽_{p})).

Hence a power of γ is in the ideal generated by ϕ^{∗}(H^{∗}(BΓ; 𝔽_{p})), which gives the result.

Lemma 3.8. If H^{∗}(BΓ; 𝔽_{p}) is of finite type then some power of γ is in the ideal generated by f_{i}^{∗}(H^{∗}(BX; 𝔽_{p})) ⊆ H^{∗}(BΓ_{i}; 𝔽_{p}).

Proof. Let ϕ : Γ_{i} ⟶ Γ denote the inclusion as before. By Lemma 3.7 some power of γ, say γ^{p^{r}} has the form

γ^{p^{r}} = ∑_{j} ϕ^{∗}(a_{j})b_{j}, a_{j} ∈ H^{∗}(BΓ; 𝔽_{p}), b_{j} ∈ H^{∗}(BΓ_{i}; 𝔽_{p}).

By taking a sufficiently large r, we may in fact assume that a_{j} is a non-nilpotent element for each j.

Consider the fibration

X_{Γ} ⟶ BΓ ⟶^{f} BX.  (7)

By assumption H^{∗}(BΓ; 𝔽_{p}) is of finite type so X_{Γ} is 𝔽_{p}-finite. It follows that the E_{2} page of the Serre spectral sequence for (7) is a finitely generated module over H^{∗}(BX; 𝔽_{p}). The standard argument in this situation shows that H^{∗}(BΓ; 𝔽_{p}) is a finitely generated H^{∗}(BX; 𝔽_{p})- module via f^{∗}. Hence given an element a ∈ H^{∗}(BΓ; 𝔽_{p}) of infinite height, a sufficiently large power of a is in the ideal generated by Im(f^{∗}) in H^{∗}(BΓ; 𝔽_{p}). Thus there is a sufficiently large integer s such that for each a_{j}

a_{j}^{p^{s}} = ∑_{n} f^{∗}(x_{j,n})c_{j,n}, x_{j,n} ∈ H^{∗}(BX; 𝔽_{p}), c_{j,n} ∈ H^{∗}(BΓ; 𝔽_{p}).

Allowing n to be the same for all j’s by adding zeroes if necessary, we can combine the equations above and get

γ^{p^{r+s}} = ∑_{j} ϕ^{∗}(∑_{n} f^{∗}(x_{j,n})c_{j,n})b_{j}^{p^{s}} = ∑_{j,n} ϕ^{∗}f^{∗}(x_{j,n})ϕ^{∗}(c_{j,n})b_{j}^{p^{s}} = ∑_{j,n} f_{i}^{∗}(x_{j,n})ϕ^{∗}(c_{j,n})b_{j}^{p^{s}}.

This completes the proof.

We are now ready to prove Proposition 3.2. As in the proof of Proposition 3.1, it suffices to show that the homotopy representation of Γ̄(p) in X obtained by restricting f is virtually faithful. By our induction hypothesis cd_{p}(X_{i−1}) is finite. Thus Lemma 3.5 implies that the same holds for X_{F_{i}}. By Lemma 3.6 it suffices to show that v ∈ H^{2}(X_{i}; 𝔽_{p}) is a nilpotent element. As we observed, the only non-trivial case is when the corresponding element γ ∈ H^{2}(Γ_{i}; 𝔽_{p}) is non-nilpotent. But by Lemma 3.8, if this is the case then some power of γ is in the ideal in H^{∗}(Γ_{i}; 𝔽_{p}) generated by Im(f_{i}^{∗}). Hence some power of v = j_{i}^{∗}(γ) vanishes.

Thus it follows that cd_{p}(X_{i}) is finite for all 0 ≤ i ≤ n. But X_{n} = X_{Γ̄(p)} and Proposition 3.2 follows.

Finally we prove part (3) of Theorem 1.4. Choose a faithful homotopy representation f of Γ in a p-compact group X, with cd_{p}(X) = μ_{p}(Γ). Let Γ̄ ≤ Γ be a p-perfect subgroup. Let f̄ denote the restriction of f to Γ̄. Proposition 3.2 implies that f̄ is virtually faithful and so μ_{p}(Γ̄) ≤ μ_{p}(Γ). Since Γ̄ is also a finite vcd group with vcd(Γ̄) ≤ vcd(Γ), part (2) implies the inequality of (3).

4. The image of the Freudenthal suspension for BG_{p}^{∧}

Let X be some finite dimensional complex and let a loop space ΩY, whose homotopy groups are finite through the dimension of X be given. Then any map f : X ⟶ ΩY followed by a suitable power map on ΩY is null-homotopic. The property of finite complexes that is needed to show this is that a map from an n-dimensional space to an n-connected one is null-homotopic. Spaces of the form BG_{p}^{∧} are generally infinite dimensional, however, under the hypotheses of Theorem 1.2, they have finite p-essential dimension, which for the purpose of proving Theorem 1.6 is just as good. The following lemma is elementary.

Lemma 4.1. Let X[N] be a simply-connected, p-local finite Postnikov section with precisely n non-vanishing homotopy groups π_{i_{j}} in dimensions 1 < i_{1} < i_{2} ⋯ < i_{n} = N. Suppose that π_{i_{j}} is annihilated by p^{r_{j}}. Let P = p^{∑_{j=1}^{n} i_{j}}. Then the P -th power map on ΩX[N] is null-homotopic.

Theorem 1.6 is a consequence of Lemma 4.1 and Corollary 1.5. Indeed Let G be a pperfect group of p-essential dimension m and such that the integral homology groups of BG are finite in positive degrees. Then by Serre C-class theory the spaces Σ^{n}BG_{p}^{∧} have finite homotopy groups for all n ≥ 0. Consider (Ω^{n}Σ^{n}BG_{p}^{∧})[m], the m-th Postnikov section of Ω^{n}Σ^{n}BG_{p}^{∧}. Let (Ω^{n}Σ^{n}BG_{p}^{∧})⟨m⟩ denote the m-connected cover of Ω^{n}Σ^{n}BG_{p}^{∧}, namely the fibre in the fibration sequence

(Ω^{n}Σ^{n}BG_{p}^{∧})⟨m⟩ ⟶^{γ_{m}} Ω^{n}Σ^{n}BG_{p}^{∧} ⟶^{ρ_{m}} (Ω^{n}Σ^{n}BG_{p}^{∧})[m].

Since Ω^{n}Σ^{n}BG_{p}^{∧} is a space with finite homotopy groups it follows from Lemma 4.1 that some power map ⟨p^{e_{n}}⟩ is null-homotopic on (Ω^{n}Σ^{n}BG_{p}^{∧})[m]. Thus the composition

BG_{p}^{∧} ⟶^{E_{n}} Ω^{n}Σ^{n}BG_{p}^{∧} ⟶^{⟨p^{en}⟩} Ω^{n}Σ^{n}BG_{p}^{∧} ⟶ (Ω^{n}Σ^{n}BG_{p}^{∧})[m]

is null-homotopic. Therefore there is a map l_{n} : BG_{p}^{∧} ⟶ (Ω^{n}Σ^{n}BG_{p}^{∧})⟨m⟩ such that

⟨p^{e_{n}}⟩ ∘ E_{n} ≃ γ_{m} ∘ l_{n}.

By Corollary 1.5, Ωl_{n} is null-homotopic. Hence Ω(⟨p^{e_{n}}⟩ ∘ E_{n}) is null-homotopic and our assertion is proven.

5. Examples

We give two examples for the failure of subgroups to detect homotopy. The first one is rather elementary and probably well known independently of our methods, but demonstrates the sharp dichotomy between stable and unstable homotopy for BG_{p}^{∧}. Consider the cyclic group ℤ/pℤ for an odd prime p. For a positive integer n, dividing p − 1, define T_{n} to be the semidirect product of ℤ/pℤ by ℤ/nℤ, where ℤ/nℤ operates as a subgroup of Aut(ℤ/pℤ) ≅ ℤ/(p − 1)ℤ. Obviously if n|m then T_{n} < T_{m}. It is well known that

H^{∗}(BT_{n}; 𝔽_{p}) ≅ P [u_{2n}] ⊗ E[x_{2n−1}].

Thus (BT_{n})_{p}^{∧} is (2n − 2)-connected. For n|p − 1, the p-completed sphere (S^{2n−1})_{p}^{∧} is a pcompact group and it is easy to construct a map f : (BT_{n})_{p}^{∧} ⟶ B(S^{2n−1})_{p}^{∧}, picking up the polynomial generator in mod-p cohomology (see for instance [9]). An easy Serre spectral sequence calculation shows that the homotopy fibre of f is equivalent to (S^{2n−1})_{p}^{∧}, and so it follows that S_{f}(T_{n}) ≃ P^{2n}(p). Here P^{2n}(p) is the 2n-dimensional mod-p Moore space, the cofibre of a degree p map on S^{2n−1}. Notice that B_{2n}(T_{n})_{p}^{∧} could be a considerably bigger space than P^{2n}(p). Nevertheless, the connectivity of B(T_{n})_{p}^{∧} being 2n−2 implies that ed_{p}(T_{n}) = 2n. It follows from Corollary 1.5 that if n|m the map induced by inclusion

ΩB(T_{n})_{p}^{∧} ⟶ ΩB(T_{m})_{p}^{∧}

is null-homotopic unless n = m. This is in spite of the fact that, before looping, this map induces a monomorphism in mod-p cohomology and, furthermore, that (ΣBT_{m})_{p}^{∧} is a retract of (ΣBT_{n})_{p}^{∧} [4].

Another fact seems to be worth mentioning. Namely, for the example above, the decomposition given by equation (2) above reads

ΩP^{2n}(p) ≃ Ω(BT_{n})_{p}^{∧} × ΩΣ(S^{2n−1} ∧ Ω(BT_{n})_{p}^{∧}) ≃ S^{2n−1}{p} × ΩΣ^{2n}(S^{2n−1}{p}) ≃

S^{2n−1}{p} × Ω( ⋁_{k≥0} P^{4n+(2n−2)k−1}(p)),

which is precisely the Cohen-Moore-Neisendorfer decomposition for the loops on an even dimensional Moore space [5].

Next consider the following examples. For a commutative ring R with a unit, consider Quillen’s unstable algebraic K-groups

K_{i,n}(R) := π_{i}(BGL_{n}(R)^{+}).

Restrict attention to the case where R is a finite field 𝔽_{p^{r}} and consider the p-primary part of K_{i,n}(𝔽_{p^{r}}) given by π_{i}(BGL_{n}(𝔽_{p^{r}})_{p}^{∧}), which we denote by K_{i,n}(𝔽_{p^{r}})_{(p)}. There are natural maps

f_{n,k} : K_{i,n}(𝔽_{p^{r}})_{(p)} ⟶ K_{i,n+k}(𝔽_{p^{r}})_{(p)},

induced by the inclusion of GL_{n}(𝔽_{p^{r}}) in GL_{n+k}(𝔽_{p^{r}}) and

g_{r,s} : K_{i,n}(𝔽_{p^{r}})_{(p)} ⟶ K_{i,n}(𝔽_{p^{r+s}})_{(p)},

induced by the inclusion of 𝔽_{p^{r}} as a subfield of 𝔽_{p^{r+s}}.

It is well known that H^{i}(BGL_{n}(𝔽_{p^{r}}); 𝔽_{p}) = 0 for 0 < i < r(p−1). It is also well known that H^{i}(BGL_{n}(𝔽_{q}); 𝔽_{p}) = 0 through a range which increases with n [16]. Thus the connectivity of BGL_{n}(F_{p^{r}})_{p}^{∧} rises with both n and r. Consequently it is obvious that for a fixed i and sufficiently large k and s both f_{n,k} and g_{r,s} are zero on K_{i,n}(𝔽_{p^{r}}). However our result implies that integers K and S can be chosen such that if k ≥ K and s ≥ S then f_{n,k} and g_{r,s} are the zero map for all values of i.

Indeed, since GL_{n}(𝔽_{p^{r}}) is finite, it has a finite p-essential dimension. By Corollary 1.5 and the connectivity results, quoted above, if k and s are sufficiently large, the maps

ΩF_{n,k} : ΩBGL_{n}(𝔽_{q})_{p}^{∧} ⟶ ΩBGL_{n+k}(𝔽_{q})_{p}^{∧}

and

ΩGr, s : ΩBGL_{n}(𝔽_{p^{r}})_{p}^{∧} ⟶ ΩBGL_{n}(𝔽_{p^{r+s}})_{p}^{∧}

are null-homotopic. Here F_{n,k} and G_{r,s} are the maps inducing f_{n,k} and g_{r,s} respectively, whence our claim.

Finally we discuss an example of the bound provided by Theorem 1.4. Consider the general linear group SL_{n}(ℤ). It is well known that vcd(SL_{n}(ℤ)) = \binom{n}{2}. Fix a prime p and let q be a power of a prime different from p. Then there is a faithful homotopy representation of SL_{n}(𝔽_{q}) in the p-compact group SL_{n}(ℂ)_{p}^{∧} [16] whose homotopy fibre is a space whose p-completion is equivalent to SU(n)_{p}^{∧}. Now consider the natural inclusion of SL_{n}(ℤ) in SL_{n}(ℂ). The proof of Theorem 1.4 implies that

cd_{p}(SL_{n}(ℂ)/SL_{n}(ℤ)) ≤ cd_{p}(SU(n)) + vcd(SL_{n}(ℤ)) = (n^{2} − 1) + \binom{n}{2}.

On the other hand the topological dimension of SL_{n}(ℂ), and hence of any quotient space of SL_{n}(ℂ) by a discrete subgroup is 2n^{2} − 2, which is larger than (n^{2} − 1) + \binom{n}{2} for n ≥ 2.

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Department of mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, U.S.A.

E-mail address: ran@math.nwu.edu