Iterating series of localization functors

José L. Rodríguez and Dirk Scevenels ^{∗}

Abstract We use localizations with respect to group homomorphisms in order to study under which conditions homotopical localizations with respect to given maps commute. In this context we introduce the iterating series of two localizations, thus obtaining a description of the effect of a localization with respect to a wedge of maps. In particular this enables a better understanding of nullifications of nilpotent GEM spaces. Finally we consider nullifications with respect to torsion-free abelian groups of rank 1, and we show that even in this relative simple case, the length of an iterating series can be any integer.

0 Introduction

From the very beginning of homotopical localization with respect to a given map, the problem to what extent these functors commute with other homotopical constructions, such as the loop space functor, played a central role (see e.g. [2, 3, 10]). In this paper we start the study of whether homotopical localizations with respect to given maps commute among themselves. Our approach will be based on the analogous notion of localization with respect to a given group homomorphism in the category Grp of groups; a technique which has proved its usefulness in e.g. [1, 5, 6, 9, 8, 14, 15, 16]. We give various examples of localization functors which do not commute in Grp (see Examples 1.3, 1.5, 4.1), and we also provide a way to realize them in the homotopy category. On the positive side we show that nullifications with respect to finitely generated abelian groups (that is, localizations with respect to homomorphisms from a finitely generated abelian group to the trivial group), commute between them when restricted to the category of nilpotent groups (see Proposition 3.2). We then use this result to obtain in Theorem 3.4 a better understanding of nullifications of nilpotent GEM spaces, which are relevant in the theory of homotopical localization (cf. [4, 10]).

∗The first author was partially supported by DGES grant PB97–0202 and the Swiss National Science Foundation.

The question whether two localization functors L_{f} and L_{g} associated to given maps commute, is closely related to the problem of L_{f}L_{g} being homotopy idempotent (see Proposition 1.1). In fact, iterating the process of applying L_{f} and then L_{g} – we call this the iterating series of L_{f} and L_{g} – yields a homotopy idempotent functor (see Theorem 2.1). This result provides also a method for calculating the effect of a localization functor if this functor is equivalent to localization with respect to a wedge of maps. Of course, analogously one can describe the effect of a localization in Grp with respect to a free product of group homomorphisms. Furthermore, we show that in Grp any localization with respect to a homomorphism is equivalent to a localization with respect to a free product of a monomorphism and an epimorphism (see Theorem 2.6).

We finally consider iterating series of nullifications with respect to torsion-free abelian groups of rank 1. In particular, we give examples to show that one may need any number of steps in such an iterating series before it actually stabilizes (see Theorem 4.2).

Acknowledgements: we warmly appreciate discussions with Carles Casacuberta, Jérôme Scherer and Rüdiger Göbel. The first-named author also thanks the Institute of Mathematics of the University of Lausanne for its kind hospitality.

1 Commuting localization functors

Given any map f : W → V between CW-complexes, there exists for each space X a map η : X → L_{f}X which is homotopy initial among all maps from X into f-local spaces, that is, spaces Y for which the induced map f^{∗} : map_{∗}(W, Y) → map_{∗}(V, Y) is a weak homotopy equivalence. The space L_{f}X is called the f-localization of X (see e.g. [2, 3, 10]). A map h is called an f-equivalence if L_{f}h is a weak homotopy equivalence. In the special case that f is of the form f : V → ∗, the f-localization of X is also denoted P_{V} X, and it is called the V -nullification of X.

As in [4] we define the localization class of f, denoted by ⟨f⟩, as the collection of all maps g such that L_{g} and L_{f} are equivalent functors. When f is of the form f : V → ∗, the class of f is simply denoted by ⟨V ⟩ and it is called the nullity class of V.

We say that ⟨f⟩ ≤ ⟨g⟩ ([4]) if and only if every g-local space is f-local. This is equivalent to every f-equivalence being a g-equivalence, to f being a g-equivalence and to the existence of a natural transformation of localization functors L_{f} → L_{g} (see [6]). This relation defines a partial order on the collection of localization functors with respect to maps. Recall from [4] that each (small) set of localization classes {⟨f_{α}⟩} has a least upper bound, which we denote by ∨⟨f_{α}⟩, since the wedge ∨f_{α} is a representative of this least upper bound.

We feel free to use the analogous notions for localizations with respect to a group homomorphism in Grp (see e.g. [5, 8]) without restating the exact definitions. We just point out that the nullification of a group G with respect to a group A is also denoted with G//A, and it is sometimes called the A-reduction of G. We recall further (cf. [14]) that a representative for the least upper bound of a set of localization classes of homomorphisms is given by the free product of these homomorphisms.

If two localization functors with respect to given maps f and g are comparable, that is, ⟨f⟩ ≤ ⟨g⟩, then L_{f}L_{g} ≃ L_{g}L_{f} ≃ L_{g} ≃ L_{f∨g}. This is a particular case of two localization functors that commute and their composition being again a homotopy idempotent functor. In general, these two problems are closely related, as we next establish.

Proposition 1.1 Let f and g be maps between CW -complexes. The following are equivalent:

(i) There exists a natural transformation L_{g}L_{f} ⟶^{ν} L_{f}L_{g}.

(ii) L_{f}L_{g} is homotopy idempotent.

(iii) L_{f}L_{g} ≃ L_{f∨g}.

(iv) L_{f}L_{g}X is g-local for all spaces X.

Proof. First we show that (iv) implies (iii). Clearly L_{f}L_{g}X is f-local for any space X and, by hypothesis, it is g-local. Hence L_{f}L_{g}X is (f ∨g)-local. Moreover, since X → L_{g}X is a g-equivalence and L_{g}X → L_{f}L_{g}X is an f-equivalence, their composition X → L_{f}L_{g}X is an (f ∨ g)-equivalence. Hence, L_{f}L_{g}X ≃ L_{f∨g}X. Obviously, (iii) implies (ii). To show that (ii) implies (i), observe that by the universal property of a homotopical localization, there are unique natural maps L_{g}L_{f}(X) → L_{g}L_{f}(L_{g}X) → L_{f}(L_{g}L_{f}(L_{g}X)) for any X. However, by hypothesis, this last space is homotopy equivalent to L_{f}L_{g}X. To show that (i) implies (iv), note that there are unique natural maps L_{f}L_{g}X → L_{g}(L_{f}L_{g}X) ≃ L_{g}L_{f}(L_{g}X) ⟶^{ν} L_{f}L_{g}(L_{g}X) ≃ L_{f}L_{g}X for any X. Thus this composition is actually homotopic to the identity. Hence, L_{f}L_{g}X is g-local as homotopy retract of the g-local space L_{g}L_{f}(L_{g}X). □

Corollary 1.2 Let f and g be maps between CW -spaces. Then the following are equivalent:

(i) L_{f}L_{g} ≃ L_{g}L_{f}.

(ii) L_{f}X is g-local for all g-local spaces X, and L_{g}Y is f-local for all f-local spaces Y.

Moreover, both of the above statements imply that L_{f}L_{g} ≃ L_{f∨g} ≃ L_{g}L_{f}. □

We next exhibit some examples showing that in general two localization functors do not commute, even not in the case of nullifications. We state these examples for group localization, but also provide a way of realizing them as examples of homotopical localization.

Example 1.3 Let s be a fixed positive integer. For a given prime p, let φ_{r} : C_{p^{r}} → C_{p^{s}} with r > s be the canonical projection, where C_{n} denotes the cyclic group of order n. The effect of L_{φ_{r}} on a group G is to reduce to p^{s} the order of any element of order p^{k} with k ≥ s. Now consider the canonical projections φ : C_{p^{2}} → C_{p} and ψ : C_{q^{2}} → C_{q} for two different primes p and q. Let G = ⟨x, y, z | x^{q^{2}} = z^{p}y^{p}, z^{p^{2}} = y^{p^{2}} = 1⟩. Then L_{ψ}(L_{φ}G) ≅ C_{q} ∗ C_{p} ∗ C_{p}, while L_{φ}(L_{ψ}G) ≅ C_{q^{2}} ∗ C_{p} ∗ C_{p}.

This situation can be translated into homotopical localization as follows. Choose maps f and g between 2-dimensional complexes, such that π_{1}f = φ and π_{1}g = ψ. Denote by X the Eilenberg–Mac Lane space K(G, 1), where G is the group of Example 1.3. Then the following result will imply that L_{f}L_{g}X ≄ L_{g}L_{f}X.

Proposition 1.4 ([15]) Let f : V → W be a map with source a 2-dimensional CW - complex. Suppose that φ = π_{1}(f) : π_{1}V → π_{1}W is an epimorphism. Then f is π_{1}- compatible, that is,

π_{1}(L_{f}X) ≅ L_{φ}(π_{1}X). □

Example 1.5 If φ : C_{p} → 1 for some prime p, then L_{φ}G = P_{C_{p}}G = G//C_{p} is the quotient of G by its p-radical (also called its p-isolator). Let

G = ⟨x, y, z | x^{3} = 1, y^{3} = 1, xy = z^{2}⟩.

Clearly (G//C_{2})//C_{3} = G//C_{3} = ⟨z | z^{2} = 1⟩ ≅ C_{2}, while (G//C_{3})//C_{2} = 1. The counterpart of this example in homotopical localization can be given similarly as for Example 1.3. Indeed, considering Moore spaces M = M(C_{2}, 1) and M′ = M(C_{3}, 1), the nullification functors P_{M} and P_{M′} do not commute.

One might be led to think that nullifications with respect to abelian groups do commute when restricted to the category of nilpotent groups. However, in Example 4.1 we will provide nullifications with respect to abelian groups that do not commute, even when restricted to the category of abelian groups.

2 Iterating series of localization functors

Observe that, with the notations of Example 1.3, L_{g}L_{f}L_{g}G ≅ L_{g}L_{f}G ≅ C_{q} ∗ C_{p} ∗ C_{p}. Analogously, in Example 1.5, we have that ((G//C_{2})//C_{3})//C_{2} = (G//C_{3})//C_{2} = 1. These observations, together with Proposition 1.1 motivate the following construction. Let f and g be maps between CW -complexes. For any space X, consider the iterating series of L_{f} and L_{g} defined through transfinite induction by (L_{f}L_{g})^{0}X = X and

(L_{f}L_{g})^{α}X = \{ \begin{matrix}L_{f}L_{g}(L_{f}L_{g})^{α−1}X & if α is a succesor ordinal \\ hocolim_{β<α}(L_{f}L_{g})^{β}X & if α is a limit ordinal. \end{matrix}  (2.1)

Analogously we can define the iterating series for L_{g} and L_{f} (starting with L_{f}, then L_{g}). (Of course, these definitions have obvious analogues for group theoretical localizations.) We say that the iterating series of L_{f} and L_{g} for X has length κ if κ is the minimal ordinal for which (L_{f}L_{g})^{κ}X ≃ (L_{f}L_{g})^{κ+1}X.

We will show that this length always exists. Recall that the cardinality of a map f : V → W between CW -complexes is the sum of the cardinalities of cells of V and W.

Theorem 2.1 For any two maps f, g between CW -complexes and for any space X, the lengths κ_{1} and κ_{2} of the iterating series of L_{f} and L_{g}, resp. L_{g} and L_{f}, for X are smaller than the first limit ordinal whose cardinality is bigger than the sum of the cardinalities of f and g. In fact, we have

L_{f∨g}(X) ≃ hocolim_{α}(L_{f}L_{g})^{α}X ≃ hocolim_{α}(L_{g}L_{f})^{α}X ≃ (L_{f}L_{g})^{κ_{1}}X ≃ (L_{g}L_{f})^{κ_{2}}X.

Proof. Set L = (L_{f}L_{g})^{λ}X, where λ is the first limit ordinal whose cardinality is bigger than the sum of the cardinalities of f and g. We show that L is g-local. Suppose that g : V → W. As explained in [7], it suffices to show that g^{∗} : [W ∧ S_{+}^{k}, L] → [V ∧ S_{+}^{k}, L] is a bijection for all k ≥ 0, where S_{+}^{k} denotes the k-dimensional sphere with a disjoint point. Let h : W ∧S_{+}^{k} → L be any map. By the choice of λ, the map h factorizes up to homotopy through some X_{i} := L_{g}(L_{f}L_{g})^{α_{i}}X, that is, h is just the composition W ∧S_{+}^{k} ⟶^{h′} X_{i} → L. Since X_{i} is a g-local space, there exists a map h̃′ : V ∧ S_{+}^{k} → X_{i}, unique up to homotopy, such that h′ ≃ h̃′ ∘g. But then the composition V ∧S_{+}^{k} ⟶^{h̃′} X_{i} → L is the map (unique up to homotopy) corresponding to h under g^{∗}. Hence, L is g-local (and f-local by definition). Moreover, the map X → L is clearly an (f ∨ g)-equivalence as it is the composition of (f ∨g)-equivalences. This shows that L_{f∨g}X ≃ (L_{f}L_{g})^{λ+1}X ≃ (L_{f}L_{g})L ≃ L ≃ (L_{f}L_{g})^{λ}X, concluding the proof. □

Example 2.2 Let C_{p} and C_{q} denote the cyclic groups of order p and q, for two different primes p and q. For any integer n, let G_{n} be generated by {x_{1}, x_{2}, …, x_{N}, y_{1}, y_{2}, …, y_{2N}}, where N = 1 + 2^{2} + 2^{4} + 2^{6} + … + 2^{2n}, subject to the relations

x_{i}^{p} = y_{2i−1}y_{2i} for i = 1, …, N
y_{i}^{q} = x_{2i}x_{2i+1} for i = 1, …, 2N − 2^{n+2}
y_{i}^{q} = 1  for i = 2N − 2^{n+2} + 1, …, 2N.

(Observe that the group G of Example 1.5 is in fact G_{0} for p = 2 and q = 3.) Then (P_{C_{p}}P_{C_{q}})^{n+1}G_{n} = 1, while (P_{C_{p}}P_{C_{q}})^{n}G_{n} ≠ 1. Thus, the iterating series of P_{C_{p}} and P_{C_{q}} for G_{n} has length n + 1, and P_{C_{p}∗C_{q}}G = (P_{C_{p}}P_{C_{q}})^{n+1}G_{n} = 1.

We thank Gemma Bastardas for pointing out to us the following interesting example.

Example 2.3 Let p : ℤ → ℤ be multiplication by a prime p. Then L_{p} is p′-localization, where p′ denotes the complementary set of primes. One can show that, for two different primes p and q, L_{p} and L_{q} commute when restricted to nilpotent groups. However, in general, the length of an iterating series can be any positive integer n, as we can for example check using the groups G_{n} of Example 2.2. Moreover, if F is a free group on two generators then the length is ℵ_{0}.

The result of Theorem 2.1 gives a description of the least upper bound of two localization classes. Moreover, it is also a way of calculating the effect of a localization functor L_{h} if we know a decomposition of its localization class, say ⟨h⟩ = ⟨f⟩ ∨ ⟨g⟩. In fact, as we now show, every localization class can effectively be decomposed.

Proposition 2.4 Given a map f : A → B between CW -complexes, then

⟨f⟩ = ⟨l_{A}⟩ ∨ ⟨l_{B}⟩

where l_{A} and l_{B} are the co-augmentation maps l_{A} : A → L_{f}A and l_{B} : B → L_{f}B.

Proof. This is due to the fact that l_{A} and l_{B} are f-equivalences, while f is an (l_{A} ∨ l_{B})- equivalence. □

Note that the above results have obvious analogues for group theoretical localization with respect to homomorphisms. Moreover, for group theoretical localizations we can even say more. Indeed, the localization class of any group homomorphism can be decomposed as the class of an epimorphism and the class of a monomorphism. We first need the following result (cf [14]).

Proposition 2.5 Let f : A → B be any epimorphism and g : B → C be any group homomorphism. Then

⟨g ∘ f⟩ = ⟨f⟩ ∗ ⟨g⟩.

Proof. Clearly g ∘ f is an (f ∗ g)-equivalence. Hence, ⟨g ∘ f⟩ ≤ ⟨f⟩ ∗ ⟨g⟩. Conversely, let G be any (g ∘f)-local group, then it is straightforward to show that, under the hypothesis that f is an epimorphism, G is both f-local and g-local. □

Of course, the statement of Proposition 2.5 is not necessarily true if f is not an epimorphism, as is seen by considering the homomorphisms f : 1 → ℤ and g : ℤ → 1.

As a consequence of Proposition 2.5 we obtain the following decomposition theorem.

Theorem 2.6 Let f : A → B be any group homomorphism. Then

⟨f⟩ = ⟨π⟩ ∗ ⟨j⟩,

where f : A → →^{π} im f ↪^{j} B. □

Of course the decomposition of a class ⟨f⟩ of a group homomorphism as the class of an epimorphism and the class of a monomorphism is not unique. If we factor the co-augmentation l_{G} : G → L_{f}G of a given group G as π_{G} : G→ →im l_{G} followed by j_{G} : im l_{G} ↪ L_{f}G, then Proposition 2.4 and Theorem 2.6 also imply that

⟨f⟩ = ⟨π_{A} ∗ π_{B}⟩ ∗ ⟨j_{A} ∗ j_{B}⟩.

In [14] basic properties of localizations with respect to an epimorphism are discussed; see also [9]. However, localizations with respect to monomorphisms may be complicated and unexpected situations may occur (see e.g. [13], [11]).

3 Nullifications of GEM spaces

A space X is called a generalized Eilenberg–Mac Lane space, briefly a GEM space, if X is homotopy equivalent to a (possibly infinite) product of Eilenberg–Mac Lane spaces. As explained in [4] nullifications of nilpotent GEM spaces can be computed in terms of nullifications with respect to wedges of Moore spaces. More precisely, given a connected CW -complex W, let MHW = ∨_{k}M(H_{k}(W), k), where M(H_{1}(W), 1) is chosen so that there is an H_{1}-equivalence M(H_{1}(W), 1) → W inducing an epimorphism on the fundamental group. Then, for any nilpotent GEM space X,

P_{W} X ≃ P_{MHW} X.

Theorem 3.1 Let M = M(G, n) and M′ = M(G′, m) be Moore spaces of finite type with n, m ≥ 1. Then

P_{M∨M′}X ≃ P_{M}P_{M′}X ≃ P_{M′}P_{M}X.

for all nilpotent spaces X.

The proof of the above is based on the next proposition and Theorem 7.5 in [4], where the homotopy groups of such nullifications are described, see also Theorem 5.3 in [2].

Proposition 3.2 Let A and B be groups such that A_{ab} and B_{ab} are finitely generated. Then

(G//A)//B ≅ (G//B)//A

for any nilpotent group G.

Proof. For any group H, the H-nullification of a nilpotent group G coincides with its H_{ab}-nullification (cf. Proposition 7.3 of [4]). Hence, without loss of generality we may assume that A and B are finitely generated abelian groups. Note that, if A has an infinite cyclic summand, then ⟨A⟩ = ⟨ℤ⟩ and hence (G//A)//B ≅ (G//B)//A = 1 for all groups G. Hence, we may assume that A is finite abelian. The proof is now completed by the following lemma and by observing that for any nilpotent group G and for any prime p, the C_{p}-nullification of G is G//C_{p} ≅ G/(T G)_{p}, where (T G)_{p} denotes the p-torsion subgroup of G. □

Lemma 3.3 Let T be a finite abelian group and denote by P the set of primes p such that T has p-torsion. Then ⟨T ⟩ = ∗_{p∈P} ⟨C_{p}⟩.

Proof. From [16] we know that ⟨f ∗ g⟩ = ⟨f × g⟩ for any two homomorphisms f and g. The proof is now completed by observing that, if T_{p} denotes the p-torsion part of T, then ⟨T_{p}⟩ = ⟨C_{p}⟩, since T_{p} is a finite abelian p-group. □

It could be interesting to know for which groups A the associated A-nullification functor commutes with C_{p}-nullification onnilpotent groups.

As a consequence of the previous results we have

Theorem 3.4 Let W be a connected CW -complex of finite type and let X be a nilpotent GEM. Then

P_{W} X ≃ ⋯ P_{M(H_{3}(W),3)}P_{M(H_{2}(W),2)}P_{M(H_{1}(W),1)}X. □

4 Nullifications with respect to torsion-free abelian

groups

The statement of Proposition 3.2, and consequently Theorem 3.4, is not true in general if neither G nor G′ is finitely generated. This is illustrated by the following example due to Casacuberta, which shows that there are even nullifications with respect to abelian groups that do not commute when restricted to the category of abelian groups.

Example 4.1 Let {p_{1}, p_{2}, p_{3}, ⋯ } be the ordered sequence of all primes. Consider the torsion-free abelian groups of rank 1 given by

A = grp \{\frac{1}{p_{1}} , \frac{1}{p_{4}} , \frac{1}{p_{7}} , ⋯ \}
C = grp \{\frac{1}{p_{2}} , \frac{1}{p_{5}} , \frac{1}{p_{8}} , ⋯ \}
D = grp \{\frac{1}{p_{3}} , \frac{1}{p_{6}} , \frac{1}{p_{9}} , ⋯ \}
B = grp \{\frac{1}{p_{2}} , \frac{1}{p_{3}} , \frac{1}{p_{5}} , \frac{1}{p_{6}} , \frac{1}{p_{8}} , \frac{1}{p_{9}} , ⋯ \} ≅ C ⊗ D.

Let {x, y} be any two linearly independent elements of ℚ^{2}, and consider the torsion-free abelian group of rank 2

G = grp {Ax, Cy, D(x + y)}.

Then G/grp {Ax} ≅ B, so that there is an exact sequence of abelian groups

0 → A → G → B → 0.

Since Hom(A, B) = 0, we have G//A ≅ B. However, since Hom(B, G) = 0, we infer that G//B ≅ G. Concluding, (G//A)//B = 0, while (G//B)//A ≅ B. Observe further that the length of the iterating series of P_{A} and P_{B} is 2 for G, while the length of the iterating series of P_{B} and P_{A} is 1.

This example suggests that we still have a certain control on the associated iterating series. More generally, we can show the following result.

Theorem 4.2 Let A and B be two torsion-free abelian groups of rank 1. Then the iterating series of A and B for any torsion-free abelian group G of rank n has length ≤ \frac{n}{2}. Moreover, if Hom(A, B) = Hom(B, A) = 0, then there is a torsion-free abelian group G of rank 2n for which the length is exactly n.

Proof. The first statement is immediate, since if G is not A-null then rank G//A < rank G. Now suppose that Hom(A, B) = Hom(B, A) = 0. We shall construct a series of nullifications G = G_{2n} → G_{2n−1} → ⋯ G_{2} → G_{1} = A, where rank (G_{k}) = k, and the kernel of G_{k} → G_{k−1} is B if k is even, and A otherwise. Moreover, we want Hom(A, G_{2}) = 0, Hom(B, G_{3}) = 0, etc., so that P_{B}G_{2} = A, P_{A}G_{3} = G_{2}, P_{B}G_{4} = G_{3}, etc. This will then show that the length of the iterating series of A and B for G is precisely n.

Observe that there is a split short exact sequence

Hom(A, ℚ) ≅ ℚ ↣ Hom(A, ℚ/ℤ) ↠^{δ} Ext(A, B).

In other words, any extension B ↣ G_{2}→ →A is induced by a homomorphism g_{2} : A → ℚ/ℤ via a pull-back diagram:

\begin{aligned}B & ↣ & G_{2} & → → & A \\ ∥ & & ↓ & & ↓ g_{2} \\ B & ↣ & ℚ & → → & ℚ/ℤ. \end{aligned}  (4.1)

On the other hand, (cf. [12])

Hom(A, ℚ/ℤ) ≅ V ⊕ ℚ/ℤ ⊕ Hom(A, ℚ),  (4.2)

where V is a ℚ-vector space of uncountable dimension. Applying Hom(A, −) to (4.1) we obtain the following diagram where the rows are exact:

Hom(A, G_{2}) ↣ Hom(A, A) ≅ ℤ → Ext(A, B)
↓ ↓ g_{2}_{∗} ∥
Hom(A, ℚ) ≅ ℚ ↣ Hom(A, ℚ/ℤ) → → Ext(A, B)

Therefore Hom(A, G_{2}) = 0 if and only if g_{2} ∉ Hom(A, ℚ) and g_{2}_{∗} is a monomorphism,

that is, g_{2}_{∗}(m) = mg_{2} ≠ 0, for any 0 ≠ m ∈ ℤ. Hence, under the isomorphism (4.2)

the extensions G_{2} satisfying Hom(A, G_{2}) = 0, correspond to elements g_{2} ∈ V. Hence

it suffices to choose any g_{2} ∈ V and to consider the corresponding extension. A similar

argument allows to construct groups G_{i}, for i ≤ 2n, having the desired properties. □

As in the examples of Section 1, we can use Proposition 1.4 to realize at the level of spaces

the situation described in Theorem 4.2. We guess that a more general result involving

torsion-free abelian groups of higher rank must also be true. We further think that there

exist examples of (abelian) groups where this iterating series has length any given infinite

cardinal (even ordinal) number.

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[15] J. L. Rodríguez and D. Scevenels, Homology equivalences inducing an epimorphism on the fundamental group, preprint (1998).

[16] J. L. Rodríguez and D. Scevenels, Localizations associated to semidirect products, Topology Appl., to appear.

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

Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B–3001 Heverlee, Belgium,

e-mail: dirk.scevenels@wis.kuleuven.ac.be