Localizations associated to semidirect products

José Luis Rodríguez and Dirk Scevenels ^{∗}

August 19, 1998

Abstract For homotopical localization with respect to any continuous map, there are results describing the relations among the localization functors associated to the maps of a given fibration. Here we treat an analogous question in a group-theoretical context: we study localization functors associated to a short exact sequence of groups. We further specialize to a split short exact sequence of groups. In particular, we describe explicitely the localization functors associated to a semidirect product of finitely generated abelian groups.

0 Introduction

Localization with respect to a group homomorphism has proved to be a useful tool in understanding homotopical localization with respect to any continuous map, as developed by Bousfield, Dror Farjoun and others ([2], [4], [11]). This algebraic technique was introduced by Casacuberta in [5], [6], and led to some further interesting results in e.g. [1], [7], [8], [10], [12]. Thus, to any given group homomorphism f, there is associated a localization functor L_{f} on the category of groups, rendering f invertible in a universal way. The collection of all group homomorphisms g yielding a localization functor L_{g} which is naturally equivalent to L_{f}, is called the localization class ⟨f⟩ of f. In the special case where f is of the form A → 1, the associated localization functor is also denoted by P_{A}, and the class ⟨f⟩ is denoted by ⟨A⟩, which is called a nullification class. As we recall in Section 1, the collection Locs of all localization classes forms a small-complete lattice for an obvious partial order relation.

∗The first-named author was partially supported by DGICYT grant PB94–0725

In homotopy theory there are results describing the relations between the localization functors associated to the spaces and maps of a given fibration (cf. [11]). Here we study an analogous question in the group-theoretical context. More precisely, given a short exact sequence of groups

1 → H ⟶^{ϕ} N ⟶^{π} K → 1,  (0.1)

we relate the classes ⟨H⟩, ⟨N⟩, ⟨K⟩, ⟨ϕ⟩ and ⟨π⟩ under the partial order relation on Locs (cf. Proposition 2.2). In the case where the short exact sequence (0.1) splits, we show in Theorem 2.5 that ⟨ϕ⟩ = ⟨K⟩ ∗ ⟨ε⟩, where ε: H→ →H/[H, K] is the natural projection, and ⟨K⟩ ∗ ⟨ε⟩ denotes the least upper bound of ⟨K⟩ and ⟨ε⟩ in the lattice Locs. Moreover, the relations between the localization classes associated to a split short exact sequence (0.1) give rise to the following diagrams of localization functors:

Id → P_{H/[H,K]} → L_{π} → P_{H}
↓ ↓ ↓ ↓
P_{K} → P_{H/[H,K]×K} → P_{N} → P_{H×K}

and

Id → L_{ε} → P_{[H,K]}
↓ ↓ ↓
P_{K} → L_{ϕ} → P_{[H,K]×K},

where Id denotes the identity functor.

Finally, in Section 3 we illustrate the above results with explicit calculations in the case of a semidirect product of finitely generated abelian groups. We show in Theorem 3.1 that the localizations associated to a nilpotent semidirect product of these groups are particularly easy to describe. On the other hand, Proposition 3.4 shows that even for a semidirect product of cyclic groups for which the corresponding action is perfect, the situation can be more complicated.

Acknowledgements. We would like to thank Carles Casacuberta for his constant interest and invaluable advice. The second-named author also thanks the Centre de Recerca Matemàtica for its hospitality.

1 Preliminaries

Recall from [6] the definition of localization with respect to a given group homomorphism f: A → B. A group G is called f-local if the induced map of sets

f^{∗}: Hom(B, G) → Hom(A, G)

is a bijection. For every group G there is a homomorphism l_{G} : G → L_{f}G which is initial among all homomorphisms from G into f-local groups. L_{f} is called the localization functor with respect to f. A homomorphism ϕ is called an f-equivalence if L_{f}ϕ is an isomorphism. A group G is f-acyclic if L_{f}G = 1.

For a given homomorphism f, the collection of all homomorphisms g such that L_{g} is naturally equivalent to L_{f}, is called the localization class of f, and it is denoted by ⟨f⟩. There is an obvious partial order relation on the collection Locs of all localization classes: we say that ⟨f⟩ ≤ ⟨g⟩ if there exists a natural transformation L_{f} → L_{g} of localization functors. Note that this is equivalent to all f-equivalences being g-equivalences, to every g-local group being f-local, or to f being a g-equivalence. With this partial order, Locs forms a small-complete lattice (cf. [12]). Recall that the least upper bound of two classes ⟨f⟩ and ⟨g⟩, which is denoted by ⟨f⟩ ∗ ⟨g⟩, has the free product f ∗ g as a representative, i.e. ⟨f⟩ ∗ ⟨g⟩ = ⟨f ∗ g⟩.

Recall further that a class ⟨f⟩ is called an epireflection class if G → L_{f}G is surjective for all groups G. By Theorem 2.1 of [12], this is equivalent to ⟨f⟩ = ⟨g⟩ for some epimorphism g. In the case where g is of the form A → 1, the functor L_{g} is usually denoted by P_{A}, and it is called A-nullification (cf. [2], [6]). The localization class ⟨g⟩ is then called a nullification class (which is also denoted by ⟨A⟩), while the g-local groups are often refered to as A-null. Observe that a group G is f-acyclic if and only if ⟨G⟩ ≤ ⟨f⟩. For general properties of nullification and epireflection classes, in particular the relation with radicals in group theory, we refer to [9] and [12].

2 Localizations associated to a short exact sequence

We start by a preliminary result, giving another description of the least upper bound of two localization classes.

Lemma 2.1 Let f and g be any group homomorphisms. Then

⟨f⟩ ∗ ⟨g⟩ = ⟨f ∗ g⟩ = ⟨f × g⟩.

Proof. The first identity was explained in Section 1. Clearly, ⟨f ∗ g⟩ ≤ ⟨f × g⟩. Furthermore, f × g = (f × id) ∘ (id × g), where (f × id) is an f-equivalence and (id × g) is a g-equivalence. This implies that f × g is an (f ∗ g)-equivalence. □

Given a short exact sequence of groups of the form

1 → H ⟶^{ϕ} N ⟶^{π} K → 1,  (2.1)

we first clarify the relations among the localization functors L_{ϕ}, L_{π}, P_{H}, P_{N} and P_{K}. Observe that if N = H×K, then we have a complete description of all involved localization functors, since ⟨H × K⟩ = ⟨H⟩ ∗ ⟨K⟩, ⟨ϕ⟩ = ⟨K⟩ and ⟨π⟩ = ⟨H⟩. In general, we have the following relations.

Proposition 2.2 Consider the short exact sequence (2.1). Then the following relations hold:

⟨N⟩ = ⟨π⟩ ∗ ⟨K⟩, ⟨π⟩ ≤ ⟨H⟩ and ⟨K⟩ ≤ ⟨ϕ⟩ ≤ ⟨H⟩ ∗ ⟨K⟩.

In particular, we have ⟨N⟩ ≤ ⟨H⟩∗⟨K⟩. In other words, there is a diagram of localization functors, given by

\begin{aligned}Id & → & L_{π} & → & P_{H} \\ ↓ & & ↓ & & ↓ \\ P_{K} & → & P_{N} & → & P_{H×K} \\ ∥ & & & & ∥ \\ P_{K} & → & L_{ϕ} & → & P_{H×K}, \end{aligned}  (2.2)

where Id denotes the identity functor.

Proof. Applying the functor Hom(−, G) to the exact sequence given in (2.1), yields an exact sequence of sets (where π^{∗} is an inclusion)

∗ → Hom(K, G) ⟶^{π^{∗}} Hom(N, G) ⟶^{ϕ^{∗}} Hom(H, G),

which implies directly the result. □

An immediate consequence of the previous proposition is the following result (compare with Corollary 4.8 of [2] and Theorem 3.D.1 of [11]).

Corollary 2.3 Consider the exact sequence (2.1) and let f be any group homomorphism.

(i) If H is f-acyclic, then π is an f-equivalence.

(ii) If ϕ is an f-equivalence, then K is f-acyclic. □

However, as the next example shows, the converse of the statements of Corollary 2.3 are false in general.

Example 2.4 (i) Let f: ℤ ∗ ℤ → ℤ ⊕ ℤ be the canonical projection. Then L_{f}G ≅ G_{ab} for any group G, where G_{ab} denotes its abelianization. Consider the short exact sequence 1 → [N, N] ⟶^{ϕ} N ⟶^{π} N/[N, N] → 1. Clearly π is an f-equivalence, but the commutator subgroup [N, N] is only f-acyclic if N is soluble of derived length at most two.

(ii) Consider the short exact sequence 1 → C_{3} ⟶^{ϕ} Σ_{3} ⟶^{π} C_{2} → 1, where Σ_{3} denotes the permutation group of a set of three elements, and C_{2}, C_{3} are cyclic groups of order 2, resp. 3. Let f: ∗_{p}ℤ ⟶^{ϕ_{p}} ∗_{p}ℤ, where the free product is taken over all primes p which are different from 3, and the homomorphism ϕ_{p} is just multiplication by p. Then L_{f} is localization at the prime 3 (cf. [5]). We conclude that C_{2} is f-acyclic, but ϕ is not an f-equivalence, since L_{f}(C_{3}) = C_{3} and L_{f}(Σ_{3}) = 1.

We now specialize to the case of a split short exact sequence of groups:

1 → H ⟶^{ϕ} H ⋊ K ⟶^{π} K → 1.  (2.3)

We denote the action of an element x ∈ K on an element a ∈ H by x · a = s(x)as(x)^{−1}, where s: K → H ⋊ K is a splitting homomorphism for π. Let [H, K] denote the commutator subgroup, i.e. the normal subgroup of H generated by the elements of the form (x · a)a^{−1} = xax^{−1}a^{−1} (from now on we will omit writing the inclusion s). Denote by ε: H → H/[H, K] the canonical homomorphism. Observe that, for every group G, the set Hom(H ⋊K, G) is in bijective correspondence with the subset of Hom(H, G)×Hom(K, G) consisting of those pairs (ψ, ξ) such that

ψ(x · a) = ξ(x)ψ(a)ξ(x)^{−1},

for all a ∈ H and x ∈ K.

Theorem 2.5 Consider the split short exact sequence (2.3). Then ⟨ϕ⟩ = ⟨K⟩ ∗ ⟨ε⟩.

Proof. Suppose that G is ϕ-local. By Proposition 2.2, G is K-null. Moreover, we claim that any homomorphism ψ: H → G is trivial on [H, K]. Indeed, since Hom(K, G) is trivial and ϕ^{∗}: Hom(H ⋊ K, G) ≅ Hom(H, G), it is clear that ψ satisfies ψ(x · a) = ψ(a) for all a ∈ H, x ∈ K. This is equivalent to ψ being trivial on [H, K], which in turn implies that G is ε-local. Hence, G is both K-null and ε-local. Conversely, suppose that G is K-null and ε-local. Any homomorphism H ⋊ K → G then corresponds to (ψ, id) for some homomorphism ψ: H → G. Hence, ϕ^{∗} is clearly injective. Moreover, since G is ε-local, any homomorphism ψ: H → G is trivial on [H, K], so that ψ(x · a) = ψ(a) for all a ∈ H, x ∈ K. Hence, (ψ, id) defines a homomorphism from H ⋊ K to G, and clearly ϕ^{∗}((ψ, id)) = ψ. This shows that ϕ^{∗} is surjective, thereby concluding the proof. □

With the terminology introduced in [12], Theorem 2.5 thus states that ⟨ϕ⟩ is an epireflection class.

Theorem 2.6 Consider the split short exact sequence (2.3). Then a group G is π-local if and only if every homomorphism from H ⋊ K to G is trivial on H. Moreover,

⟨H/[H, K]⟩ ≤ ⟨π⟩ ≤ ⟨H⟩.

Proof. The first claim is obvious, and the fact that ⟨π⟩ ≤ ⟨H⟩ was proved in Proposition 2.2. Suppose now that G is π-local, so that any homomorphism (ψ, ξ): H ⋊ K → G is trivial on H, i.e. ψ = 1. Given any homomorphism f: H/[H, K] → G, the composition f ∘ ε is trivial on [H, K]. This means that (f ∘ ε, 1) is a homomorphism from H ⋊ K to G, so that f ∘ ε = 1. Since ε is an epimorphism, we infer that f is trivial, and hence that G is H/[H, K]-null. □

Summarizing the previous results, for a split short exact sequence (2.3), we have the following diagrams of localization functors:

\begin{aligned}Id & → & P_{H/[H,K]} & → & L_{π} & → & P_{H} \\ ↓ & & ↓ & & ↓ & & ↓ \\ P_{K} & → & P_{H/[H,K]×K} & → & P_{H⋊K} & → & P_{H×K} \end{aligned}  (2.4)

and

\begin{aligned}Id & → & L_{ε} & → & P_{[H,K]} \\ ↓ & & ↓ & & ↓ \\ P_{K} & → & L_{ϕ} & → & P_{[H,K]×K}, \end{aligned}  (2.5)

where Id denotes the identity functor.

At the beginning of this section, we already dealt with the special case of a semidirect product (2.3) where the action of K on H is trivial. Another “extremal” case is when the action is perfect, i.e. [H, K] = H.

Corollary 2.7 Consider the split short exact sequence (2.3). If [H, K] = H, then ⟨ϕ⟩ = ⟨H⟩ ∗ ⟨K⟩. Therefore, diagrams (2.4) and (2.5) collapse to

\begin{aligned}Id & → & L_{π} & → & P_{H} \\ ↓ & & ↓ & & ↓ \\ P_{K} & → & P_{H⋊K} & → & L_{ϕ} ≅ P_{H×K}. □ \end{aligned}  (2.6)

Note that the above result is a special case of the general fact that ⟨[H, K]⟩ ≤ ⟨ε⟩ if and only if ⟨[H, K]⟩ = ⟨ε⟩.

We close this section by exhibiting an example showing that in general the arrows in diagram (2.6) are not necessarily natural equivalences.

Example 2.8 Consider the split short exact sequence

1 → C_{3} ⟶^{ϕ} Σ_{3} ⟶^{π} C_{2} → 1.

The commutator subgroup is [C_{3}, C_{2}] = C_{3}. It is easy to check that all localizations in diagram (2.6) are different, by evaluating them on the groups C_{2}, C_{3}, C_{3} × C_{2} and Σ_{3}.

Other examples, generalizing Example 2.8, and illustrating that the arrows in diagrams (2.4) and (2.5) are not necessarily natural equivalences, will be exhibited in the next section, where we specialize to split short exact sequences of finitely generated abelian groups.

3 Localizations associated to semidirect products of

finitely generated abelian groups

¿From [4, Proposition 7.3] we know that P_{(H⋊K)_{ab}} = P_{H/[H,K]×K} and P_{H⋊K} are naturally equivalent when restricted to the category of nilpotent groups. In particular, if H ⋊ K is itself nilpotent, we infer that P_{H⋊K} is equivalent to P_{H/[H,K]×K} on the category of all groups. Here we show that, under the additional assumption that H and K are finitely generated abelian groups, these nullifications actually are equivalent to P_{H×K}.

Theorem 3.1 Consider the split short exact sequence (2.3), where H and K are finitely generated abelian groups. If H ⋊ K is nilpotent, then ⟨H/[H, K]⟩ = ⟨π⟩ = ⟨H⟩ and ⟨H ⋊ K⟩ = ⟨H × K⟩. Therefore, diagrams (2.4) and (2.5) collapse to

\begin{aligned}Id & → & L_{ε} & → & P_{[H,K]} & → & P_{H} ≅ L_{π} ≅ P_{H/[H,K]} \\ ↓ & & ↓ & & ↓ & & ↓ \\ P_{K} & → & L_{ϕ} & → & P_{[H,K]×K} & → & P_{H×K} ≅ P_{H⋊K} ≅ P_{H/[H,K]×K}. \end{aligned}  (3.1)

Proof. To prove the first claim, it suffices to show that ⟨H/[H, K]⟩ = ⟨H⟩. If H is a finite abelian group, then it follows easily from the fact that H ⋊ K is nilpotent, that the set T (H) of prime numbers p for which H has p-torsion coincides with the set T (H/[H, K]) of prime numbers p for which H/[H, K] has p-torsion. Hence, ⟨H/[H, K]⟩ = ⟨H⟩ = ⟨∏_{p∈T (H)} C_{p}⟩, where C_{p} denotes the cyclic group of order p. If H is not finite, then we can choose generators a_{1}, ..., a_{m} for the torsion-free part of H such that the torsion-free part of [H, K] is generated by t_{1}a_{1}, ..., t_{m}a_{m} for some integers t_{1}, ..., t_{m} with t_{1}|...|t_{m}. Since H ⋊K is nilpotent, we infer that t_{m} = 0. Hence, ⟨H⟩ = ⟨H/[H, K]⟩ = ⟨ℤ⟩. Finally, if ⟨π⟩ = ⟨H⟩, then the second claim follows immediately from Proposition 2.2 and Lemma 2.1. □

In the case of a semidirect product of finitely generated abelian groups, we can even go further in our description of the associated localization functors. We henceforth adopt the convention that the cyclic group of order m is written ℤ_{m} when we want to use additive notation, and that we write C_{m} when thought of this cyclic group as a multiplicative group. We allow m to be any positive integer or to be ∞, with the convention that C_{∞} = ℤ_{∞} = ℤ. Let

H = ℤ_{n_{1}} × ⋯ × ℤ_{n_{k}} and K = C_{m_{1}} × ⋯ × C_{m_{l}}  (3.2)

be finitely generated abelian groups (we do not exclude the case where some of the factors of H and/or K are infinite cyclic groups). Choose generators x_{i} for C_{m_{i}} and a_{j} for ℤ_{n_{j}}. Suppose further that, for this choice of generators, the action of x_{i} on H is given by a (k×k)-matrix P_{i} (representing the corresponding element of Aut(ℤ_{n_{1}} ×⋯×ℤ_{n_{k}})). Hence, the commutator subgroup [H, K] is the subgroup of H generated by the images of (P_{i} −I) for all i, where I denotes the identity matrix, i.e.,

[H, K] = grp {im (P_{1} − I), ⋯, im (P_{l} − I)}.

We restate the results of Theorem 2.5 and Theorem 2.6 in this particular case.

Proposition 3.2 Consider the split short exact sequence (2.3), where H and K are given by (3.2). Then

(i) ⟨ϕ⟩ = ⟨K⟩ ∗ ⟨ε⟩, where ε: H→ →H/grp {im (P_{1} − I), ⋯, im (P_{l} − I)}.

(ii) ⟨H/grp {im (P_{1} − I), ⋯, im (P_{l} − I)}⟩ ≤ ⟨π⟩ ≤ ⟨H⟩,

where P_{i} is the matrix corresponding to the action of x_{i} on H. □

For example, if every P_{i} is a diagonal matrix, i.e. if there exist integers t_{ij} such that x_{i} · a_{j} = t_{ij}a_{j} for all i, j, then ⟨ϕ⟩ = ⟨K⟩ ∗ ⟨ε⟩, where ε is the natural projection from ℤ_{n_{1}} × ⋯ × ℤ_{n_{k}} onto ℤ_{n_{1}}/grp {g_{1}a_{1}} × ⋯ × ℤ_{n_{k}}/grp {g_{k}a_{k}}, and, furthermore, ⟨ℤ_{n_{1}}/grp {g_{1}a_{1}} × ⋯ × ℤ_{n_{k}}/grp {g_{k}a_{k}}⟩ ≤ ⟨π⟩ ≤ ⟨ℤ_{n_{1}} × ⋯ × ℤ_{n_{k}}⟩, where the integer g_{j} = gcd{t_{ij} − 1 | t_{ij} − 1 ≠ 0 and i = 1, ..., k}.

We now confine our attention to

1 → ℤ_{n} = ⟨a⟩ ⟶^{ϕ} ℤ_{n} ⋊ C_{m} ⟶^{π} C_{m} = ⟨x⟩ → 1,  (3.3)

where n is a positive integer (and m is a positive integer or ∞).

Proposition 3.3 Consider the split short exact sequence (3.3). Let the corresponding action be given by x · a = ta with 0 ≤ t < n. Suppose that the prime decomposition of n is given by n = p_{1}^{r_{1}} ⋯ p_{k}^{r_{k}} (where r_{i} ≥ 1). Suppose further that t − 1 = p_{1}^{s_{1}} ⋯ p_{k}^{s_{k}}Q, where Q is coprime to p_{1}, ⋯, p_{k} and 0 ≤ s_{i} ≤ r_{i}. Then

(i) ⟨π⟩ = ⟨π_{1}⟩ ∗ ⋯ ∗ ⟨π_{k}⟩, where the exact sequence 1 → ℤ_{p_{j}^{r_{j}}} → ℤ_{p_{j}^{r_{j}}} ⋊ C_{m} ⟶^{π_{j}} C_{m} → 1

represents the induced action;

(ii) ⟨ε⟩ = ⟨ε_{1}⟩ ∗ ⋯ ∗ ⟨ε_{k}⟩, where ε_{j}: ℤ_{p_{j}^{r_{j}}} → →ℤ_{p_{j}^{s_{j}}} is induced by ε: ℤ_{n}→ →ℤ_{n}/[ℤ_{n}, C_{m}].

Proof. Clearly, [ℤ_{n}, C_{m}] = grp {(t−1)a} ≅ ℤ_{p_{1}^{r_{1}−s_{1}}} ×⋯×ℤ_{p_{k}^{r_{k}−s_{k}}}. Hence ℤ_{n}/[ℤ_{n}, C_{m}] ≅ ℤ_{p_{1}^{s_{1}}} × ⋯ × ℤ_{p_{k}^{s_{k}}}. Moreover, it is clear that a group G is π-local if and only if G is π_{j}-local for all j. □

The above result enables us to reduce the general case of a semidirect product (3.3) to the case where n is a power of a prime number.

Proposition 3.4 Consider the split short exact sequence (3.3) where n = p^{r} for a prime p and r ≥ 1. Suppose that the corresponding action is given by x·a = ta, where 0 ≤ t < p^{r}. Suppose further that t − 1 = p^{s}Q, where 0 ≤ s < r and Q is coprime to p. Then (i) ⟨ϕ⟩ = ⟨C_{m}⟩ ∗ ⟨ε⟩, where ε: ℤ_{p^{r}}→ →ℤ_{p^{s}}.

(ii) If ℤ_{p^{r}} ⋊ C_{m} is nilpotent (i.e., s ≥ 1), then ⟨π⟩ = ⟨ℤ_{p}⟩.

(iii) If the action is perfect (i.e., s = 0), then ⟨π⟩ ≤ ⟨ℤ_{p}⟩. □

To complete our description, we recall that P_{ℤ_{p}}G = G/I_{p}(G) for every group G, where I_{p}(G) denotes the p-radical (or p-isolator) of the group G. Furthermore, if f: ℤ_{p^{r}}→ →ℤ_{p^{s}} (with r > s) is the canonical projection, then the effect of L_{f} on a group G is to reduce the order of any element of order p^{k} (for k ≥ s) to p^{s}. Finally, L_{π} kills all the elements g of a group G for which g^{p^{r}} = 1 and for which there exist an element y in G such that y^{m} = 1 (if m is finite) and [y, g] = g^{t−1}.

Finally, to see that we may have ⟨π⟩ < ⟨ℤ_{p}⟩ in Proposition 3.5 (iii), it suffices to consider a perfect action of C_{q} on ℤ_{p}, where p and q are distinct primes (cf. Example 2.8). Indeed, ⟨π⟩ = ⟨ℤ_{p}⟩ would imply that ⟨ℤ_{p} ⋊ C_{q}⟩ = ⟨ℤ_{p} × C_{q}⟩, which is impossible, since P_{ℤ_{p}⋊C_{q}}(ℤ_{p} × C_{q}) ≠ 1.

References

[1] J. Berrick and C. Casacuberta, A universal space for plus constructions, Topology (to appear).

[2] A. K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), 831–873.

[3] A. K. Bousfield, Unstable localization and periodicity, in: Algebraic Topology: New Trends in Localization and Periodicity, Progress in Math. 136, Birkhäuser Verlag, Basel (1996), 33–50.

[4] A. K. Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997), 1321–1354.

[5] C. Casacuberta, Recent advances in unstable localization, in: The Hilton Symposium 1993; Topics in Topology and Group Theory, CRM Proceedings and Lecture Notes 6, Amer. Math. Soc., Providence (1994), 1–22.

[6] C. Casacuberta, Anderson localization from a modern point of view, in: The Čech Centennial, Contemporary Math. 181, Amer. Math. Soc., Providence (1995), 35–44.

[7] C. Casacuberta and J. L. Rodríguez, On towers approximating homological localizations, J. London Math. Soc. (2) 56 (1997), 645–656.

[8] C. Casacuberta, J. L. Rodríguez, and Jin-Yen Tai, Localizations of abelian Eilenberg– Mac Lane spaces of finite type, preprint (1998).

[9] C. Casacuberta, J. L. Rodríguez, and D. Scevenels, Singly generated radicals associated with varieties, in: Proceedings of Groups St Andrews 1997 in Bath, London Math. Soc. Lecture Note Series, Cambridge University Press, to appear.

[10] C. Casacuberta, D. Scevenels and J. H. Smith, Implications of large-cardinal principles in homotopical localization, preprint (1998).

[11] E. Dror Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Math. 1622, Springer-Verlag, Berlin Heidelberg New York (1996).

[12] J. L. Rodríguez and D. Scevenels, Universal epimorphic equivalences for group localizations, preprint (1998).

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