Cellular approximations using Moore spaces

José L. Rodríguez and Jérôme Scherer ^{∗}

Abstract

For a two-dimensional Moore space M with fundamental group G, we identify the effect of the cellularization CW_{M} and the fiber \overline{P}_{M} of the nullification on an Eilenberg–Mac Lane space K(N, 1), where N is any group: both induce on the fundamental group a group theoretical analogue, which can also be described in terms of certain universal extensions. We characterize completely M-cellular and M-acyclic spaces, in the case when M = M(Z/p^{k}, 1).

0 Introduction

Let M be a pointed connected CW -complex. The nullification functor P_{M} and the cellularization functor CW_{M} have been carefully studied in the last few years (see e.g. [8], [17], [18], [14]). These are generalizations of Postnikov sections and connective covers, where the role of spheres is replaced by a connected CW -complex M and its suspensions. This list of functors also includes plus-constructions and acyclic functors associated with a homology theory, for which M is a universal acyclic space ([2], [13], [22], [24]). Recall that a connected space X is called M-cellular if CW_{M}X ≃ X, or, equivalently, if it belongs to the smallest class C(M) of spaces which contains M and is closed under homotopy equivalences and pointed homotopy colimits. Analogously, X is called M-acyclic if P_{M}X ≃ ∗ or, equivalently, \overline{P}_{M}X ≃ X. It was shown in [14] that the class of M-acyclic spaces is the smallest class \overline{C(M)} of spaces which contains M and is closed under homotopy equivalences, pointed homotopy colimits, and extensions by fibrations.

∗The first author was partially supported by DGICYT grants PB94–0725 and PB97-0202. The second by the grant 81LA-51213 of the Swiss National Foundation for Science and the CRM.

Very interesting examples are given by the family of Moore spaces M(Z/p, n), the homotopy cofiber of the degree p self-map of S^{n}. For n ≥ 2, these spaces are the “building blocks” for simply-connected p-torsion spaces. More precisely, it is shown in [3] (see also [11]) that the M(Z/p, n)-cellular spaces are exactly the (n − 1)-connected spaces X such that p · π_{n}X = 0 and π_{k}X is p-torsion for k > n. However the methods used in those papers can not handle the case n = 1. In this paper we introduce the group theoretical tools that are necessary to deal with this case. They apply to the more general situation when M is a two-dimensional CW -complex with fundamental group G. As we will see in Proposition 2.10 and in the introduction of Section 3, the interesting phenomena occur when H_{2}(M; Z) = 0. In that case we say that M is a Moore space of type M(G, 1), and we shortly write M = M(G, 1).

The G-socle of a group N, which we denote by S_{G}N, is the subgroup of N generated by the images of all homomorphisms from G into N. We introduce the class C(G) for any group G. It is the smallest class of groups containing G which is closed under isomorphisms and colimits. We construct explicitely the right adjoint C_{G} to the inclusion of C(G) in the category of groups and show the following (see also Theorem 2.3).

Theorem 2.9 Let M be a two dimensional CW -complex with fundamental group G. Let X = K(N, 1) where N is any group. Then we have a natural isomorphism

π_{1}(CW_{M}X) ≅ C_{G}N.

Moreover, the action of C_{G}N on the higher homotopy groups of CW_{M}X is trivial.

We further prove the existence of a central extension

0 → A → C_{G}N → S_{G}N → 1

which is universal in the sense explained in Theorem 2.7.

The proof of such results uses a description of Chachólski, exhibiting CW_{M}X as the fibre of a map X → LX, where LX is obtained from X by first killing all maps from M, and then applying ΣM-nullification.

This leads us, in the case when G = Z/p, to the following result. We must note that our proof is also valid for M(Z/p, n) with n ≥ 2, cases which were previously dealt with in [3] or [11].

Theorem 6.2 Let M = M(Z/p, 1) be the cofiber of the degree p self-map of S^{1} and X be a connected space. Then X is M-cellular if and only if π_{1}X is generated by elements of order p and H_{n}(X; Z) is p-torsion for n ≥ 2.

In particular, a nilpotent space X is M(Z/p, 1)-cellular if and only if π_{1}X is generated by elements of order p and π_{n}(X) is p-torsion for n ≥ 2.

Of course, the homotopy groups of non-nilpotent M-cellular spaces need not be pgroups. For instance, the universal cover of M(Z/2, 1) is S^{2}. Likewise, a space all whose homotopy groups are p-groups need not be M-cellular, as shown by Example 6.4, where we compute the M(Z/2, 1)-cellularization of K(Σ_{3}, 1).

Nullifications with respect to Moore spaces are better understood. Our aim here is to investigate the homotopy fibre of such nullifications.

Recall that the G-radical of a group N, which we denote by T_{G}N as in [10] or [8], is the smallest subgroup of N such that Hom(G, N/T_{G}N) = 0. It is known [10] that when M is a two dimensional CW -complex with fundamental group G, then π_{1}P_{M}X ≅ π_{1}X/T_{G}(π_{1}X). If in addition M is a M(G, 1), the space P_{M}X can be viewed as the fibrewise R-completion, in the sense of Bousfield and Kan [9], of a covering fibration associated to the G-radical subgroup (for a suitable coefficient ring R); see [9], [10], and [12].

We introduce the class \overline{C(G)}, for any group G. It is the smallest class of groups containing G which is closed under isomorphisms, colimits, and extensions. We show in Proposition 3.11 that the fundamental group of any M-acyclic space belongs to this class. We define then D_{G}N as the fundamental group of \overline{P}_{M}K(N, 1), note that the action of D_{G}N on the higher homotopy groups of \overline{P}_{M}K(N, 1) is trivial, and prove:

Theorem 3.13 Let M = M(G, 1) be a two-dimensional Moore space. Then D_{G} is right adjoint to the inclusion of \overline{C(G)} in the category of groups.

As in the case of cellularization, there exists a central extension

0 → B → D_{G}N → T_{G}N → 1  (0.1)

which is universal in the sense explained in Theorem 3.3.

A very enlightening example is given by the acyclic space described by Berrick and Casacuberta in [2, Example 5.3], which turns out to be an M(G, 1) for some acyclic group G. In this case M(G, 1)-nullification is equivalent to Quillen’s plus-construction and the G-radical of any group N is its largest perfect subgroup. Thus, in this case, the central extension (0.1) is the usual universal central extension of T_{G}N.

Similar results have been obtained by Mislin and Peschke in [22] in the case when P_{M} is the plus construction associated to a generalized homology theory. In all these cases CW_{M} and \overline{P}_{M} coincide.

Finally, we obtain the following result for G = Z/p (compare with [3] or [11]). Theorem 6.1 Let M = M(Z/p, 1). Then X is M-acyclic if and only if π_{1}X coincides with its Z/p-radical and H_{n}(X; Z) is p-torsion for n ≥ 2.

In particular, a nilpotent space X is M(Z/p, 1)-acyclic if and only if π_{1}X coincides with its Z/p-radical and π_{n}(X) is p-torsion for n ≥ 2.

Acknowledgements: This paper was mainly elaborated during the 1998 Topology Semester at the CRM. We are especially indebted to Jon Berrick and Wojciech Chachólski for several enlightening discussions. We warmly appreciate Carles Casacuberta’s advice and encouragement. The second author would like to thank the CRM for making the life of a post-doctoral fellow so pleasant.

1 Preliminary results

We give here a short review of the terminology involved in the theory of homotopical localization. We also remind the reader some of the results needed in this paper. More details can be found in [17], [18], [14], [7].

By an idempotent augmented functor in the category of spaces, we mean a functor E from the category of pointed spaces to itself. It preserves weak homotopy equivalences and is equipped with a natural transformation c : E → Id from E to the identity functor, such that c_{EX} : EEX ≃ EX for all spaces X.

The most important examples for us are CW_{M} and \overline{P}_{M}. Let M be a connected CW - complex. A map Y → X is an M-cellular equivalence if it induces a weak equivalence on pointed mapping spaces

map_{∗}(M, Y) ⟶^{∼} map_{∗}(M, X) .

There exists then, for each connected space X, a map CW_{M}X → X, called M-cellular approximation, which is universal (initial) among all M-cellular equivalences to X. The spaces for which CW_{M}X ≃ X are called M-cellular. The class of M-cellular spaces has been identified as the smallest class C(M) of spaces containing M and closed under weak equivalences and pointed homotopy colimits; see [14, Theorem 8.2] and [18, 2.D]. A connected space Z is said to be M-null if map_{∗}(M, Z) is weakly contractible, that is, [Σ^{k}M, X] = ∗ for all k ≥ 0. There exists a map X → P_{M}X, called M-nullification, which is universal (terminal) among all maps from X to an M-null space. Finally denote by \overline{P}_{M}X → X the homotopy fibre of X → P_{M}X. A connected space X is called Macyclic if P_{M}X ≃ ∗, i.e., \overline{P}_{M}X ≃ X. The class of M-acyclic spaces has been identified in [14, Theorem 17.3] as the class \overline{C(M)}. In addition to being closed under weak equivalences and pointed homotopy colimits, it is also closed under extensions by fibrations.

So, every M-cellular space is M-acyclic and furthermore, it is known that each ΣM-acyclic space is M-cellular ([18, 3.B.3]). Hence by universality we have natural maps

\overline{P}_{ΣM}X ⟶^{α} CW_{M}X ⟶^{β} \overline{P}_{M}X.  (1.2)

Thus CW_{M}X can be thought of as the fiber of a mixing process between M- and ΣMnullification. More precisely:

Theorem 1.1 ([14, Theorem 20.5]) Let M be any connected CW -complex. There is a fibration

CW_{M}X ⟶ X ⟶^{η} LX

where η is the composition of the inclusion X → X′ of X into the homotopy cofibre of the evaluation map ∨_{[M,X]}M → X, followed by X′ → P_{ΣM}(X′).

Note that the inclusion X → X′ is in fact functorial in the homotopy category, and it is universal (initial) among all maps X → Z such that M → X → Z is homotopically trivial (compare with [2, Corollary 2.2]). Hence, X → LX is also functorial in the homotopy category.

The fundamental groups of P_{M}X and LX have a group theoretical meaning in the case when M is a two-dimensional CW -complex, as we next explain. The G-socle of a group N, which we denote by S_{G}N, is the subgroup of N generated by the images of all homomorphisms from G into N. The G-radical of N, which we denote by T_{G}N as in [10] or [8], is the smallest subgroup of N such that Hom(G, N/T_{G}N) = 0. The G-radical of N can be constructed as a (possibly transfinite) direct limit of subgroups T_{i} where T_{1} is the G-socle of N and T_{i}/T_{i−1} = S_{G}(N/T_{i−1}). In other words, the groups N such that T_{G}N = N are precisely the groups which have a normal series whose factors coincide with their G-socles. The first link between the topological nullification and their discrete analogues is given by the following result. Its first part is [10, Theorem 3.5] (see also [6, Theorem 5.2]).

Lemma 1.2 If M is a two-dimensional CW -complex with fundamental group G and X is any space, we have isomorphisms

π_{1}(P_{M}X) ≅ π_{1}X/T_{G}(π_{1}X) and π_{1}X′ ≅ π_{1}LX ≅ π_{1}X/S_{G}(π_{1}X),

where X′ and LX are as in Theorem 1.1. □

Such a two-dimensional CW -complex M is called a Moore space if H_{2}(M; Z) = 0. It has type M(G, 1) if π_{1}M ≅ G. Using Theorem 1.1 it is easy to see that two Moore spaces of type M(G, 1) determine the same cellularization and nullification functors.

Bousfield computed in [8, Section 7] the effect of the nullification functor with respect to a two-dimensional Moore space M(G, 1) on nilpotent spaces; see also [10, Theorem 4.4] for G = Z/p, and [12, Theorem 2.4] when G = Z[1/p]. Let J be the set of primes p for which G_{ab} is uniquely p-divisible. Define R = Z_{(J)}, the integers localized at J, if G_{ab} is torsion, and R = ⊕_{p∈J}Z/p otherwise. Let R_{∞}X be the Bousfield-Kan R-completion of X; see [9]. Then P_{M}X can be obtained as the fibrewise R-completion of the covering fibration associated to the G-radical of π_{1}X. That is, we have a diagram of fibrations

\begin{matrix}X̃ & ⟶ & X & ⟶ & K(π_{1}X/T_{G}(π_{1}X), 1) \\ ↓ & & ↓ & & ↓_{id} \\ R_{∞}X̃ & ⟶ & P_{M}X & ⟶ & K(π_{1}X/T_{G}(π_{1}X), 1), \end{matrix}  (1.3)

where R_{∞}X̃ is simply connected, as the fundamental group of X̃ is R-perfect, i.e., H_{1}(X̃; R) = H_{1}(T_{G}(π_{1}X); R) = 0. Hence R_{∞}X̃ coincides with X̃_{HR}^{+}, the plus-construction with respect to ordinary homology with coefficients in R; see [9, VII.6] and [12]. The universal cover of P_{M}X is thus equivalent to the three following spaces:

P_{M} X̃ ≃ R_{∞}X̃ ≃ X̃_{HR}^{+}.

Let A_{R}X denote the R-acyclic functor, that is, the homotopy fiber of X → X_{HR}^{+} (cf. [9, VII, 6.7]). Then, a connected space X is HR-acyclic, i.e., H̃_{∗}(X; R) = 0, if and only if A_{R}X ≃ X. The above remark immediately implies the following.

Proposition 1.3 Let M = M(G, 1) be a two-dimensional Moore space, and X be any connected space. If X̃ denotes the covering of X corresponding to the subgroup T_{G}(π_{1}X), then

\overline{P}_{M}X ≃ A_{R}X̃ ≃ \overline{P}_{M} X̃,

where R is the ring associated to G as above. □

Corollary 1.4 Let M = M(G, 1) be a two-dimensional Moore space, and X be any connected space. Then, X is M-acyclic if and only if

T_{G}(π_{1}X) = π_{1}X and H_{k}(X; R) = 0 for k ≥ 2 . □

2 The fundamental group of M(G, 1)-cellular spaces

In this section we define algebraically a G-cellularization functor C_{G} in the category of groups. We show that C_{G}N coincides with the fundamental group of the M(G, 1)- cellularization of K(N, 1). This yields a characterization of C_{G}N as a certain universal central extension of the G-socle of N. We also prove that the action of C_{G}N is trivial on the higher homotopy groups of CW_{M(G,1)}K(N, 1).

As suggested by Dror-Farjoun, we introduce the closed class of groups C(G). It is the smallest class of groups containing G, and closed under isomorphisms and taking colimits. In other words, if F : I → Groups is a diagram with F (i) ∈ C(G) for any i ∈ I, then colim_{I}F should again belong to C(G).

The following proposition gives the explicit construction of a G-cellularization functor. The existence of such a functor is also ensured by [5, Corollary 7.5].

Proposition 2.1 Let G be a group. The inclusion C(G) ⊂ Groups has a right adjoint C_{G} : Groups → C(G).

Proof. For any group N, the map C_{G}N → N is constructed by induction as follows. First define C_{0} = ∗_{h:G→N}G, the free product of as many copies of G as there are morphisms from G to N, and let h_{0} : C_{0} → N be the evaluation (so that h_{0}(C_{0}) = S_{G}N). Now take the free product ∗_{(h′,h′′)}G, where h′, h′′ : G → C_{0} is any pair of morphisms coequalized by h_{0}. Define C_{1} as the coequalizer of ∗_{(h′,h′′)}G ⇉ C_{0}, and repeat this process (maybe transfinitely). Notice that this inductive construction of C_{G}N shows that we have a natural epimorphism C_{G}N→ →S_{G}N for any group N. The group C_{G}N is in C(G) and the morphism c : C_{G}N → N is universal (terminal), i.e., c induces a bijection of sets Hom(G, c) : Hom(G, C_{G}N) ≅ Hom(G, N). □

By analogy to the case of spaces, a group N in C(G) is called G-cellular.

Lemma 2.2 Let M be a two-dimensional CW -complex with fundamental group G. A group homomorphism N → N′ induces an isomorphism C_{G}N ≅ C_{G}N′ if and only if CW_{M}K(N, 1) ≃ CW_{M}K(N′, 1).

Proof. The pointed mapping space map_{∗}(M, K(N, 1)) is weakly equivalent to the discrete set Hom(G, N). □

Since π_{1} commutes with homotopy colimits, the fundamental group of any M-cellular space is π_{1}M-cellular, for any M. Furthermore, the following holds (this could also have been taken as definition of C_{G}):

Theorem 2.3 Let M be a two-dimensional CW -complex with fundamental group G. Let X = K(N, 1) where N is any group. Then we have a natural isomorphism

π_{1}(CW_{M}X) ≅ C_{G}N.

Proof. By the previous observation, π_{1}(CW_{M}X) is G-cellular. It thus only remains to prove that c : CW_{M}X → X induces a bijection of sets π_{1}(c)_{∗} = Hom(G, π_{1}(c)). Consider the following commutative diagram (of sets)

[M, CW_{M}X] ⟶^{e} Hom(G, π_{1}(CW_{M}X))
↓_{c_{∗}}  ↓_{π_{1}(c)_{∗}}
[M, X] ⟶^{e′} Hom(G, N).

Since M is two-dimensional, e is surjective and X being a K(N, 1), e′ is bijective. On the other hand, c_{∗} is also bijective by the universal property of CW_{M}. Thus, π_{1}(c)_{∗} is bijective, as desired. □

Lemma 2.4 Let X̃ denote the covering of X corresponding to the subgroup S_{G}(π_{1}X) and let M be a two-dimensional CW -complex with fundamental group G. Then

(i) (X̃)′ is the universal cover of X′,

(ii) LX̃ is the universal cover of LX, where X′ and LX are defined in Theorem 1.1.

Proof. The covering fibration X̃ → X →^{p} K(π_{1}X/S_{G}(π_{1}X), 1) induces a bijection [M, X̃] ≅ [M, X] because the map [M, p] is trivial. Apply now Mather-Puppe theorem (see [16, Proposition 6.1]) saying that “the fiber of the push-out is the push-out of the fibers” when the base space is fixed, to get a fibration X̃′ → X′ → K(π_{1}X/S_{G}(π_{1}X), 1). This is the universal cover fibration by Lemma 1.2. So part (i) holds. For part (ii) we note that the previous fibration has a ΣM-null base, and is therefore preserved under ΣM-nullification. □

The following corollary could also have been proved directly by checking that the covering X̃ → X is indeed an M-equivalence.

Corollary 2.5 Let X̃ denote the covering of X corresponding to the subgroup S_{G}(π_{1}X) and let M be a two-dimensional CW -complex with fundamental group G. We have a homotopy equivalence CW_{M}X ≃ CW_{M} X̃. □

Lemma 2.6 Let G be any group, and 0 → A → E → N → 1 be a central extension of groups. Then C_{G}E ≅ C_{G}N if and only if Hom(G_{ab}, A) = 0 and the natural map Hom(G, N) ⟶ H^{2}(G; A) is trivial.

Proof. Apply map_{∗}(K(G, 1), −) to the fibration

K(E, 1) → K(N, 1) → K(A, 2) .

This gives a new fibration, whose homotopy sequence

0 → Hom(G, A) → Hom(G, E) → Hom(G, N) → H^{2}(G; A)

is exact as in [9, IX, 4.1]). The lemma is proved. □

We now know enough to describe our first universal central extension. It is nothing but a universal central G-cellular equivalence.

Theorem 2.7 Let G be any group. Then, for each group N, there is a central extension

0 → A → C_{G}N → S_{G}N → 1

such that Hom(G_{ab}, A) = 0 and the natural map Hom(G, N) ⟶ H^{2}(G; A) is trivial. Moreover, this extension is universal with respect to these two properties.

Proof. Let M be a two-dimensional CW -complex with fundamental group G and let X = K(S_{G}N, 1). The space LX is 1-connected by Lemma 1.2 and define then

A = π_{2}LX ≅ π_{2}X′/T_{G_{ab}}(π_{2}X′)

(this follows from [8, Theorem 7.5]). The long exact sequence in homotopy of the fibration CW_{M}X → X → LX produces now the desired central extension, where we identify C_{G}(S_{G}N) with C_{G}N. This can be deduced from Corollary 2.5. The G-cellularization of C_{G}N → S_{G}N is also an isomorphism since Hom(G, C_{G}N) ≅ Hom(C, S_{G}N). Hence Hom(G_{ab}, A) = 0 and the natural map Hom(G, N) ⟶ H^{2}(G; A) is trivial by Lemma 2.6. The universal property is a direct consequence of the same lemma. □

Corollary 2.8 Let M be a two-dimensional CW -complex with fundamental group G. Then S_{G}N = N if and only if LK(N, 1) is 1-connected, and C_{G}N ≅ N if and only if LK(N, 1) is 2-connected. In particular

(1) π_{2}LK(N, 1) ≅ H_{2}LK(S_{G}N, 1);

(2) π_{3}LK(N, 1) ≅ H_{3}LK(C_{G}N, 1). □

In the next theorem we identify the universal cover of CW_{M}K(N, 1) and remark that the action of the fundamental group is trivial. This could also be seen as a particular case of [20, Proposition A.1] or the even more general [22, Corollary 7.7].

Theorem 2.9 Let M be a two-dimensional CW -complex with fundamental group G. Then CW_{M}K(N, 1) ≃ CW_{M}K(C_{G}N, 1) and the universal cover fibration is given by

ΩLK(C_{G}N, 1) ⟶ CW_{M}K(N, 1) ⟶ K(C_{G}N, 1).

Moreover the action of C_{G}N on π_{n}CW_{M}K(N, 1) for n ≥ 2 is trivial.

Proof. The first part follows from Lemma 2.2. The proof of the second part has been suggested by Carles Casacuberta. The fibration

CW_{M}K(N, 1) → K(C_{G}N, 1) → LK(C_{G}N, 1)

induces a long exact sequence of C_{G}N-modules in homotopy. But the space LK(C_{G}N, 1) is 2-connected by Corollary 2.8, so that the action of C_{G}N on the higher homotopy groups of CW_{M}K(N, 1) is trivial. □

Proposition 2.10 Let M be a two-dimensional CW -complex with fundamental group G. Assume that H_{2}(M; Z) ≠ 0. Then

CW_{M}K(N, 1) ≃ K(C_{G}N, 1) .

Proof. Choose a presentation ϕ : ∗Z → ∗Z of G, and realize it as a map f between wedges of circles having M as its homotopy cofiber. Note that a simply connected space Y is ΣM-null if and only if the G-radical of π_{2}Y is trivial, and π_{k}Y is ϕ_{ab}-local for any k ≥ 3, i.e., Hom(ϕ_{ab}, π_{k}Y) is bijective (see [23, Theorem 4.3.6]). When H_{2}(M; Z) ≠ 0, the homomorphism ϕ_{ab} is not injective and any ϕ_{ab}-local group is trivial. So LK(C_{G}N, 1) is the trivial space, as it is already 2-connected (see [23, Corollary 4.3.9]). □

Example 2.11 If G = C_{2} and N is nilpotent then C_{G}N = S_{G}N (by Corollary 6.3 below). However, C_{G}N ≅ S̸_{G}N in general, as shown by the following example, which was suggested by Alejandro Adem:

N = ⟨a, b, c, d | a^{2} = b^{2} = c^{2} = d^{2} = 1, abab = cdcd⟩ .

In other words N is the push-out of the diagram C_{2} ∗ C_{2} ← Z → C_{2} ∗ C_{2} where both arrows send the generator of Z to the commutator. The Mayer–Vietoris sequence shows then that H_{2}N ≅ Z ≅ A. Thus C_{G}N is an extension of N by Z. This also provides an example of a quotient of a free product of copies of G which is not cellular.

3 The fundamental group of M(G, 1)-acyclic spaces

We imitate now the preceding section, replacing CW_{M} by \overline{P}_{M}. First we change our closed class. In addition to being closed under isomorphisms and colimits, the class \overline{C(G)} is assumed to be closed under taking arbitrary extensions. That is, if N ↪ E→ →Q is an extension with N, Q ∈ \overline{C(G)}, then E belongs to \overline{C(G)} as well. The right adjoint of the inclusion of \overline{C(G)} in the category of groups is denoted by D_{G} and we construct it by topological means, namely as the fundamental group of \overline{P}_{M(G,1)}K(N, 1). It could be interesting to have an algebraic description of D_{G}N, similar to that of C_{G}N, in terms of colimits and extension by short exact sequences.

A well known topological proof of the existence of the universal central extension over a perfect group N uses Quillen’s plus-construction. We will follow exactly the same line of proof here, the plus-construction being replaced by a nullification functor with respect to a Moore space. This is a true generalization of this old result in light of [2], where it is proven that the plus-construction is indeed the nullification with respect to a Moore space. Another approach is taken in [22], where the plus-construction associated to any homology theory determines a universal central extension.

When M is a two-dimensional CW -complex with fundamental group G which is not a Moore space, i.e., H_{2}(M; Z) ≠ 0, the effect of P_{M} is drastic. As in the proof of Proposition 2.10, one shows that

P_{M}X ≃ K(π_{1}X/T_{G}(π_{1}X), 1).

Hence \overline{P}_{M}X is the covering of X corresponding to the subgroup T_{G}(π_{1}X). From now on, we will therefore only consider Moore spaces.

Define, for a two-dimensional Moore space M = M(G, 1),

D_{G}N := π_{1}(\overline{P}_{M}K(N, 1)).

This does not depend on the choice of M by the observation made after Lemma 1.2.

Lemma 3.1 Let M be a two-dimensional Moore space with fundamental group G and let B denote the group π_{2}P_{M}K(T_{G}N, 1). The space K(B, 2) is then M-null, or equivalently, Hom(G_{ab}, B) = 0 = Ext(G_{ab}, B).

Proof. The space P_{M}K(T_{G}N, 1) is 1-connected by Lemma 1.2. The second Postnikov section K(π_{2}X, 2) of a simply connected M-null space X is M-null as well, since π_{2}P_{M}X only depends on π_{2}X by [8, Theorem 7.5]. □

The groups B satisfying Hom(G_{ab}, B) = 0 = Ext(G_{ab}, B) can only be of the two following forms ([8, 7.1]):

Fact 3.2 Let J denote the set of primes p such that G_{ab} is uniquely p-divisible, and J′ the complementary set of primes. Then, either G_{ab} is J′-torsion and B is J-local, or G_{ab} is uniquely J-divisible and B is Ext-J-complete (in the sense of [9]). In other words, Hom(G_{ab}, B) = 0 = Ext(G_{ab}, B) if and only if Hom(H, B) = 0 = Ext(H, B) where H = ⊕_{p∈J′}Z/p if G_{ab} is torsion, or H = Z[J^{−1}] otherwise.

We are now ready to prove the existence of our second universal central extension.

Theorem 3.3 Let G be the fundamental group of a two-dimensional Moore space, which we denote by M. Then, for each group N, there is a central extension

0 → B → D_{G}N → T_{G}N → 1

such that Hom(G_{ab}, B) = 0 = Ext(G_{ab}, B). Moreover, this extension is universal (initial) with respect to this property.

Proof. The idea of the proof is analogous to that of Theorem 2.7. Let X = K(T_{G}N, 1). The fibration \overline{P}_{M}X → X → P_{M}X produces the desired extension using Proposition 1.3, where B = π_{2}P_{M}K(T_{G}N, 1) satisfies the property by Lemma 3.1.

We check now the universal property. Let B′ be an abelian group having the above property, and 0 → B′ → E → T_{G}N → 1 a central extension. Realize it as a fibration K(E, 1) ⟶ X ⟶ K(B′, 2), where the base space is M-null. There exists therefore a map P_{M}X → K(B′, 2), unique up to homotopy, inducing a map of fibrations. □

Example 3.4 Let G = C_{2} and N be the group described in Example 2.11. Then D_{G}N is an extension of N by Z[1/2].

Remark 3.5 The group T_{G}N is R-perfect and the central extension of Theorem 3.3 is the universal central extension of T_{G}N with coefficients in R. By this we mean the central extension induced from the fibration A_{R}K(T_{G}N, 1) → K(T_{G}N, 1) → K(T_{G}N, 1)_{HR}^{+}. These two extensions coincide by Proposition 1.3. Even though this “R universal central extension” seems to be classical, we do not know any other reference than [22].

Example 3.6 Let G = Z[1/p], so that R = Z/p. In [22, Proposition 5.4] Mislin and Peschke computed that

B ≅ Ext(Z(p^{∞}), H_{2}(T_{G}N; Z)) ⊕ Hom(Z(p^{∞}), H_{1}(T_{G}N; Z)) ,

where Z(p^{∞}) is the p-torsion subgroup of Q/Z. Let G = Z/p, so that R = Z[1/p]. Then B = H_{2}(T_{G}N; R).

An interesting consequence of the previous result is that the functor D_{G} is idempotent. It is worth noting that it seems rather difficult to prove this fact directly from the definition.

Theorem 3.7 Let M be as above. Then \overline{P}_{M}K(D_{G}N, 1) ≃ \overline{P}_{M}K(N, 1) and in particular the functor D_{G} is idempotent. The universal cover fibration is given by

ΩP_{M}K(D_{G}N, 1) ⟶ \overline{P}_{M}K(N, 1) ⟶ K(D_{G}N, 1).

Moreover the action of D_{G}N on π_{n}\overline{P}_{M}K(N, 1) for n ≥ 2 is trivial.

Proof. The functor P_{M} preserves the fibration

K(D_{G}N, 1) → K(T_{G}N, 1) → K(B, 2)

of Theorem 3.3 since K(B, 2) is M-null by Lemma 3.1. Thus so does the functor \overline{P}_{M}. That is, we have \overline{P}_{M}K(D_{G}N, 1) ≃ \overline{P}_{M}K(T_{G}N, 1). The later space is equivalent to \overline{P}_{M}K(N, 1) by Proposition 1.3. The statements about the universal cover fibration follow as in Theorem 2.9. □

Remember that the ring R is determined by the group G as follows: R = Z_{(J)} if G_{ab} is torsion, and R = ⊕_{p∈J}Z/p otherwise. We say that a group N is super R-perfect if H_{1}(N; R) = 0 = H_{2}(N; R).

Proposition 3.8 Let G be the fundamental group of a two-dimensional Moore space M. The following statements are equivalent:

(1) D_{G}N ≅ N.

(2) The space P_{M}K(N, 1) is 2-connected.

(3) H^{2}(N; B) = 0 for any B such that Hom(G_{ab}, B) = 0 = Ext(G_{ab}, B).

(4) T_{G}N = N and N is super R-perfect.

Proof. Theorem 3.3 implies that (1), (2), and (3) are equivalent. We only prove that (4) implies (2). Since N coincides with its G-radical, P_{M}K(N, 1) ≃ K(N, 1)_{HR}^{+} (see diagram (1.3)), and it is 1-connected. Thus K(N, 1)_{HR}^{+} ≃ K(N, 1)_{HR} the HR-homological localization by [22, Proposition 1.6]. Moreover, by [4, Theorem 5.5], π_{2}K(N, 1)_{HR} is an HR-local group. But H_{1}(π_{2}K(N, 1)_{HR}; R) = 0, so it has to be trivial. □

As a consequence, we obtain the following formulae for the low-dimensional homotopy groups of P_{M}K(N, 1); cf. [1, Corollary 8.6].

Corollary 3.9 Let M be a two-dimensional Moore space with fundamental group G. Then

(1) π_{2}P_{M}K(N, 1) ≅ H_{2}P_{M}K(T_{G}N, 1);

(2) π_{3}P_{M}K(N, 1) ≅ H_{3}P_{M}K(D_{G}N, 1). □

We end this section by proving that this topological construction gives nothing else but the right adjoint of the inclusion of \overline{C(G)} into the category of groups. We denote the class {N | D_{G}N ≅ N} by D(G).

Proposition 3.10 Let M be a two-dimensional Moore space with fundamental group G. The class D(G) is closed under arbitrary extensions and colimits.

Proof. The class of G-radical groups is closed under colimits and extensions, and so is the class of super R-perfect groups: An easy Hochschild–Serre spectral sequence argument shows that an extension of super R-perfect groups is again super R-perfect, and a Mayer– Vietoris argument shows it for a push-out. Since homology commutes with telescopes, the proposition is proved. □

Proposition 3.11 Let M be a two-dimensional Moore space with fundamental group G. Then \overline{C(G)} = D(G).

Proof. By Proposition 3.10, \overline{C(G)} ⊂ D(G). To show the converse we prove that the fundamental group of any space in \overline{C(M)} is in \overline{C(G)}. But \overline{C(M)} is the smallest class containing M which is closed under homotopy colimits and extensions by fibrations. Clearly the fundamental group of the homotopy colimit of a diagram all whose values have π_{1} in \overline{C(G)} is again in \overline{C(G)}. So assume we have a fibration F → E → B of connected spaces, where the fundamental groups of F and B are in \overline{C(G)}. We have to prove π_{1}E ∈ \overline{C(G)}. The cokernel of the boundary morphism π_{2}B → π_{1}F is isomorphic to the coinvariants (π_{1}F)_{π_{2}B} = colim_{π_{2}B}(π_{1}F) and thus belongs to \overline{C(G)}. Therefore π_{1}E is an extension of two groups of \overline{C(G)}. □

Corollary 3.12 Let G be the fundamental group of a two-dimensional Moore space M. A group N belongs then to D(G) if and only if there exists an M-acyclic space X with π_{1}X ≅ N. □

Theorem 3.13 Let M be a two-dimensional Moore space with fundamental group G. The augmented functor D_{G} is then right adjoint to the inclusion of \overline{C(G)} in the category of groups, i.e., we have an isomorphism Hom(L, D_{G}N) ≅ Hom(L, N) for any group L ∈ \overline{C(G)}.

Proof. The map K(D_{G}N, 1) → K(N, 1) induces a weak equivalence

\overline{P}_{M}K(D_{G}N, 1) ≃ \overline{P}_{M}K(N, 1)

by Theorem 3.7. Let L ∈ \overline{C(G)}. Then

map_{∗}(\overline{P}_{M}K(L, 1), K(D_{G}N, 1)) ≃ map_{∗}(\overline{P}_{M}K(L, 1), K(N, 1)) ,

i.e., Hom(L, D_{G}N) ≅ Hom(L, N). □

4 Acyclic spaces

We illustrate the preceding sections by the case when the Moore space M is acyclic. We identify the functors \overline{P}_{M} and CW_{M}. The motivating example is the universal acyclic group F of Berrick and Casacuberta [2, Example 5.3]. It satisfies S_{F}N = T_{F}N = PN, the maximal perfect subgroup of N. In this case C_{F}N = D_{F}N = P̃N the universal central extension of PN and the two central extensions coincide. If M = M(F, 1), the functors \overline{P}_{M} and CW_{M} coincide with Dror’s acyclic functor A, the fiber of Quillen’s plus-construction.

We want to consider now an arbitrary acyclic group G, and an acyclic complex M with fundamental group G. This space M is of course not determined by the group. Since ΣM ≃ ∗, the fibration of Theorem 1.1 has the form

CW_{M}X → X → X′

where X′ is the homotopy cofibre of the map ∨_{[M,X]}M → X. Let X_{N}^{+} denote the plusconstruction of X with respect to a perfect, normal subgroup N of π_{1}X, and let A_{N}X be the homotopy fibre of the natural map X → X_{N}^{+}. The universal property of the plusconstruction ensures that X′ ≃ X_{S}^{+}, where S = S(M, X) is the topological socle, i.e., the subgroup generated by the images of all homomorphisms π_{1}(M) → π_{1}(X) which are induced by maps M → X (see [2, Section 2]). This subgroup of π_{1}X is sometimes also called the subgroup swept by M. Arguing similarly with P_{M}X we deduce the following; cf. [2, Corollary 2.2].

Theorem 4.1 Let M be an acyclic CW -complex. Then the map β : CW_{M}X → \overline{P}_{M}X is equivalent to A_{S}X → A_{T} X where S is the subgroup swept by M and T is such that π_{1}P_{M}X ≅ π_{1}X/T. □

When M is two-dimensional, S = S_{G}N and T = T_{G}N for all X, and N denotes the fundamental group of X. We also have S = S_{G}N if M is any CW -complex and X = K(N, 1). Therefore, we deduce the following.

Corollary 4.2 Let X be a space with fundamental group N and M an acyclic CW - complex with fundamental group G. Suppose that M is of dimension two or X = K(N, 1). Then π_{1}(CW_{M}X) ≅ C_{G}N is the universal central extension of S_{G}N. □

5 Nilpotent spaces

When X is a nilpotent space, the homotopy long exact sequence associated to the fibration \overline{P}_{M}X → X → P_{M}X yields the homotopy groups of the M-acyclic part of X, as follows:

Proposition 5.1 Let n ≥ 1, M be a Moore space M(G, n), and let X be any connected space. Suppose that X is nilpotent if n = 1. Let J be the set of primes p such that G is uniquely p-divisible and J′ be the complementary set of primes. Then \overline{P}_{M}X is (n − 1)- connected and for k ≥ n

(I) if G_{ab} is torsion, then

π_{k}(\overline{P}_{M}X) ≅ \{ \begin{matrix}Π_{p∈J′}(Z(p^{∞}) ⊗ π_{k+1}X ⊕ Tor(Z(p^{∞}), π_{k}X))  if k ≥ n + 1, \\ Π_{p∈J′}(Z(p^{∞}) ⊗ π_{n+1}X ⊕ T_{G}(π_{n}X))  if k = n; \end{matrix}

(II) if G_{ab} is not torsion, then

π_{k}(\overline{P}_{M}X) ≅ \{ \begin{matrix}Π_{p∈J}(Ext(Z[1/p], π_{k+1}X) ⊕ Hom(Z[1/p], π_{k}X)) if k ≥ n + 1, \\ Π_{p∈J}(Ext(Z[1/p], π_{n+1}X) ⊕ D_{G}(π_{n}X) if k = n. \end{matrix}

Proof. We use from [8, Theorem 7.5] that in the first case we have

π_{k}(P_{M}X) ≅ \{ \begin{matrix}π_{k}(X) ⊗ Z_{(J′)}  if k ≥ n + 1, \\ π_{n}X/T_{G}(π_{n}X)  if k = n; \end{matrix}

In the second case we have:

π_{k}(P_{M}X) ≅ \{ \begin{matrix}Π_{p∈J}(Ext(Z(p^{∞}), π_{k}X) ⊕ Hom(Z(p^{∞}), π_{k−1}X)) if k ≥ n + 1, \\ π_{n}X/T_{G}(π_{n}X) if k = n. \end{matrix}

□

Example 5.2 Let G be a rational group of rank 1 of type (r_{2}, r_{3}, r_{5}, …). That is, G is the additive subgroup of Q generated by the fractions 1/p^{s}, for s ≤ r_{p} (we write r_{p} = ∞ if G is uniquely p-divisible). Note that if r_{p} < ∞ then the G-radical contains the Z/p-radical. Moreover, in the category of abelian groups, the G-socle coincides with the G-radical if and only if G = Z[J^{−1}] (see [19]).

This allows us to construct two subgroups of Q having the same set of primes for which they are uniquely p-divisible, but with distinct radical, and thus distinct acyclic approximation. Fix a prime p and define G by r_{p} = ∞ and r_{q} = 1 when q ≠ p, so that H = Z[1/p]. Let M = M(G, 1) and M′ = M(H, 1). Then we have \overline{P}_{M}K(Z[1/p], 1) ≃ ∗, while \overline{P}_{M′}K(Z[1/p], 1) ≃ K(Z[1/p], 1).

We next give a description of the class of nilpotent M-acyclic spaces; compare with Corollary 7.9 in [8], see also [21]. The case when n = 1 gives a less general result than Corollary1.4, but gives a characterization of M(G, 1)-acyclic spaces in terms of their homotopy groups rather than their homology groups.

Proposition 5.3 Let M be a Moore space M(G, n) with n ≥ 1, and let X be any connected space. Suppose that X is nilpotent if n = 1. Then, X is M-acyclic if and only if X is (n − 1)-connected, π_{n}X coincides with its G-radical and π_{k}(X) is J′-torsion for k ≥ n + 1 in the case when G is torsion, or π_{k}(X) is uniquely J-divisible for k ≥ n + 1 otherwise. □

6 The torsion case

In this section we only deal with the case of the Moore spaces M(Z/p^{k}, 1), for k ≥ 1. We give a characterization of M(Z/p^{k}, 1)-cellular spaces, which holds even for non-nilpotent spaces. The M(Z/p^{k}, 1)-acyclic spaces have been already identified in Corollary 1.4. The following reformulation only makes use of the fact that a group A is p-torsion if and only if A ⊗ Z[1/p] = 0.

Theorem 6.1 Let M = M(Z/p^{k}, 1), k ≥ 1. Then a space X is M-acyclic if and only if π_{1}X coincides with its Z/p-radical and H_{n}(X; Z) is p-torsion for n ≥ 2. □

Theorem 6.2 Let M = M(Z/p^{k}, 1), k ≥ 1. Then a space X is M-cellular if and only if π_{1}X is generated by elements of order p^{l} for l ≤ k and H_{n}(X; Z) is p-torsion for n ≥ 2.

Proof. We use again the fact that CW_{M}X can be obtained as the fiber of the map X → P_{M(Z/p^{k},2)}X′, where X′ is the cofiber of ∨M → X. So we have to find a necessary and sufficient condition for P_{M(Z/p,2)}X′ to be trivial. First X′ has to be 1-connected, and this is equivalent to π_{1}X coinciding with its Z/p^{k}-socle. Knowing that X′ and thus P_{M(Z/p,2)}X′ are 1-connected, the triviality of the latest is equivalent to its acyclicity. By Proposition 5.3 the homotopy, or equivalently the reduced integral homology of X′, has to be p-torsion. The long exact sequence in homology of the cofibration sequence defining X′ shows that this is equivalent to H̃_{∗}(X; Z) being p-torsion. □

Corollary 6.3 Let M = M(Z/p^{k}, 1). A nilpotent space X is M-cellular if and only if π_{1}X is generated by elements of order p^{l} for l ≤ k and π_{n}(X) is p-torsion for n ≥ 2. □

The characterization given in [14, 12.5] or [15] of M(Z/2, n)-cellular spaces (π_{n} is generated by involutions, and the higher homotopy groups are 2-torsion) is true for n = 1 if we work in the category of nilpotent spaces. An easy counter-example for non-nilpotent spaces is given by M(Z/2, 1) itself. It is of course an M(Z/2, 1)-cellular space, even though π_{2}M(Z/2, 1) ≅ Z. We finally consider the following example.

Example 6.4 The symmetric groups Σ_{n} are C_{2}-cellular for n ≥ 2. Indeed, using the presentation Σ_{n} = ⟨σ_{1}, ..., σ_{n−1} | σ_{i}^{2} = 1, σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}, σ_{i}σ_{i+j}σ_{i} = σ_{i+j} for j ≥ 2⟩, one can obtain Σ_{n} as a push-out of a family of homomorphisms between free products of C_{2}. For n − 1 ≥ j ≥ 1, let G_{j} be the coproduct of n − 1 − j copies of C_{2} ∗ C_{2}, and H_{j} the coproduct of as many copies of C_{2}. Define G_{j} → H_{j} to be the coproduct of the fold maps C_{2} ∗ C_{2} → C_{2}. Let K be the coproduct of n − 1 copies of C_{2}, and call the generators x_{1}, ⋯, x_{n−1}. Define a map G_{1} → K by sending the (2i − 1)st generator to x_{i}x_{i+1}x_{i} and the 2ith one to x_{i+1}x_{i}x_{i+1}, where n − 1 ≥ i ≥ 1. For j ≥ 2, the map G_{j} → K is defined by sending the (2i − 1)st generator to x_{i}x_{i+j}x_{i} and the 2ith one to x_{i+j}. The push-out of the diagram

(∗_{j=1}^{n−1}H_{j}) ← (∗_{j=1}^{n−1}G_{j}) → K

is then the symmetric group Σ_{n}.

The spaces K(Σ_{n}, 1) have therefore C_{2}-cellular fundamental group, and 2-torsion higher homotopy groups. They are however not M(Z/2, 1)-cellular by Theorem 6.2, since the integral homology of K(Σ_{n}, 1) contains 3-torsion. Actually, we can even compute the cellularization of K(Σ_{3}, 1). We know by Theorem 1.1 that it is the fiber of the map K(Σ_{3}, 1) → P_{M(Z/2,2)}K(Σ_{3}, 1)′. The later space is simply connected, and is 3-complete. Its mod 3 cohomology is that of K(Σ_{3}, 1), so it is by [9, VII, 4.4] the delooping of S^{3}{3}, the fiber of the degree 3 self-map of S^{3}. In other words CW_{M(Z/2,1)}K(Σ_{3}, 1) is a space whose fundamental group is Σ_{3} and whose universal cover is S^{3}{3}, i.e., the universal cover fibration is

S^{3}{3} → CW_{M(Z/2,1)}K(Σ_{3}, 1) → K(Σ_{3}, 1).

The action of Σ_{3} on S^{3}{3} is trivial by Theorem 2.9.

In the above example we could identify a certain 3-complete space as the delooping of S^{3}{3}. In general it is of course not to expect to find nice and well-known spaces as the fiber of the cellularization. However, the same argument as above proves the following proposition.

Proposition 6.5 Let π be a finite C_{p}-cellular group. The universal cover fibration of Theorem 2.9 is then

ΩK(π, 1)_{p′}ˆ → CW_{M(Z/p,1)}K(π, 1) → K(π, 1)

where X_{p′}ˆ denotes the completion away from p, i.e., X_{p′}ˆ = ∏_{q≠p} X_{q}ˆ . □

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain, e-mail: jlrodri@mat.uab.es

Centre de Recerca Matemàtica, Institut d’Estudis Catalans, E–08193 Bellaterra, Spain, e-mail: jerome@crm.es