On towers approximating

homological localizations

Carles Casacuberta and José L. Rodríguez ^{∗}

Abstract

Our object of study is the natural tower which, for any given map f: A → B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E_{∗}, yielding e.g. an analogue of Quillen’s plus-construction for each E_{∗}. We discuss in detail the case of ordinary homology with coefficients in Z/p or Z[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f: A → pt), which allows us, among other things, to generalize Neisendorfer’s observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(Z[1/p], 1).

0 Introduction

If E_{∗} is any generalized homology theory, then, as shown by Bousfield in [4], one can construct an idempotent functor ( )_{E} on the pointed homotopy category of CW-complexes Ho, which renders invertible precisely the class of E-equivalences. Thus, a map X → Y induces an isomorphism E_{∗}(X) ≅ E_{∗}(Y) if and only if it induces a homotopy equivalence X_{E} ≃ Y_{E}. This functor is referred to as (unstable) E-localization, and it is fully understood only in a few special cases, often restricted to suitable subcategories of Ho.

∗The authors are supported by DGICYT grant PB91–0467 1991 Mathematics Subject Classification: 55P60

In the case when E = HR, ordinary homology with coefficients in a commutative ring R with unit, the HR-localization functor is closely related with the R-completion functor R_{∞} defined by Bousfield and Kan in [9]. Specifically, there is a natural map X_{HR} → R_{∞}X which is a homotopy equivalence in many cases, e.g. whenever X is nilpotent or, more generally, whenever X_{HR} is nilpotent. In fact, HR-localization is the best idempotent approximation to R-completion, in the sense that they render invertible precisely the same class of maps.

The present paper originated from the observation, made from various sources, that inverting one single easy map often produces the same effect —or almost the same effect— as inverting a whole family of homology equivalences. With the phrase “inverting one map” we allude to the recently developed theory of homotopical localization, which started from the work of Bousfield [6], [7] and Dror Farjoun [13]. One first instance is Neisendorfer’s remark [21] that inverting a map of the form M(Z[1/p], 1) → pt (where “pt” denotes a one-point space and the letter M stands throughout for Moore spaces) has the same effect as HZ/p-localization on simply-connected spaces. Similarly, by inverting the pth power map p: S^{1} → S^{1}, one obtains a functor which is closely related to HZ[1/p]-localization, and in fact agrees with HZ[1/p]-localization on nilpotent spaces [11]. Almost the same effect is obtained by killing the homotopy cofibre of p, that is, by inverting the map M(Z/p, 1) → pt; see [2], [10]. Now, by inverting the suspension Σp one obtains exactly the same result as by inverting all HZ[1/p]-equivalences which are suspensions, namely a functor which preserves the fundamental group and tensors with Z[1/p] the higher homotopy groups. If we continue this process of successively suspending p and killing the corresponding homotopy cofibres, we obtain a tower of localizations with increasingly weaker effect and converging to X. Of course, this makes sense for every map f: A → B, and we call the resulting tower the f-tower associated to each space X. In the special case when f is the map S^{0} → pt, the f-tower is precisely the Postnikov tower. Sufficient motivation for the study of such f-towers was offered in [7] and [15]. Note that, if we choose f to be a v_{n}-map for n ≥ 1, then the outcome is a family of approximations to localization with respect to certain Morava K-theories. In particular, it would be interesting to decide how closely inverting a v_{1}-map approximates localization with respect to mod p complex K-theory; see [13].

It is important to retain the above suggested duality between Z/p and Z[1/p], which is very much in the spirit of some parts of the book by Bousfield and Kan [9]. Indeed, the most substantial part of our discussion involves reducing the study of certain localization functors to the study of their effect on the fundamental group of spaces. Thus we are led to comparing, for specific classes of groups, the effect of dividing out p-torsion (resp. all infinitely p-divisible elements) with the effect of dividing out the Z[1/p]-perfect radical (resp. the Z/p-perfect radical).

Acknowledgements: We are indebted to Emmanuel Dror Farjoun, Ian Leary, Ran Levi, and Joe Neisendorfer for several enlightening discussions.

1 Preliminaries

We assume that all spaces are pointed, and all maps and homotopies preserve basepoints. The space of based maps from X to Y is denoted by map_{∗}(X, Y).

Let f: A → B be any map. We follow the terminology of [7], [13], by calling f-local those spaces X for which the induced map

f^{∗}: map_{∗}(B, X) → map_{∗}(A, X)

is a weak homotopy equivalence. Likewise, a map h: X → Y is an f-equivalence if

h^{∗}: map_{∗}(Y, Z) → map_{∗}(X, Z)

is a weak homotopy equivalence for every f-local space Z.

We recall from [6], [13] that, replacing if necessary the map f with a cofibration, one may construct a functor L_{f} on the pointed category of CW-complexes, together with a natural transformation l: id → L_{f}, such that, for every space X, the map l_{X}: X → L_{f}X is an f-equivalence and the space L_{f}X is f-local. Moreover, L_{f} also defines a functor on the pointed homotopy category Ho, and, if so viewed, then it is idempotent. As such, L_{f} is left adjoint to the inclusion of the full subcategory of f-local spaces in Ho. It also follows that a map h: X → Y is an f-equivalence if and only if the induced map L_{f}h: L_{f}X → L_{f}Y is a homotopy equivalence, and this is the same as asserting that f^{∗}: [Y, Z] → [X, Z] is bijective for all f-local spaces Z; cf. [1].

For simplicity, we use the term “idempotent functor” (in any category) to denote a functor L which is part of an idempotent monad (L, l), as in [1]. All natural transformations θ: L_{2} → L_{1} between idempotent functors are implicitly assumed to satisfy θ ·l_{2} = l_{1}. If L is an idempotent functor, we call L-local those objects X such that X ≅ LX and L-equivalences those morphisms φ such that Lφ is invertible. There is a natural transformation θ: L_{2} → L_{1} of idempotent functors if and only if the class of L_{2}-equivalences is contained in the class of L_{1}-equivalences, or, equivalently, the class of L_{1}-local objects is contained in the class of L_{2}-local objects; in this case, the natural transformation θ is unique. As a special case of this general situation, we infer the following.

Proposition 1.1 Let f_{1}, f_{2} be any two maps between spaces. Then there is a natural transformation θ: L_{f_{2}} → L_{f_{1}} in Ho if and only if f_{2} is an f_{1}-equivalence. Moreover, in this case, the natural transformation θ is unique. □

We emphasize another elementary fact which will be useful in the analysis of diagrams (4.1) and (4.2) below.

Proposition 1.2 Let L_{3} →^{β} L_{2} →^{α} L_{1} be natural transformations of functors, where at least L_{3} and L_{2} are idempotent. Suppose that, for a certain object X, the composite arrow L_{3}X → L_{1}X is an isomorphism. Then both α_{X} and β_{X} are isomorphisms.

Proof. ¿From the assumption made, it follows that L_{3}X is a retract of L_{2}X and therefore L_{3}X is L_{2}-local. Hence the arrow β_{X} : L_{3}X → L_{2}X is an isomorphism, and so is also α_{X}: L_{2}X → L_{1}X. □

If the target space B of the map f : A → B is contractible, then f-local spaces have been called A-periodic [7] or A-null [8]. The corresponding f-localization functor is referred to as A-periodization or A-nullification, and denoted by P_{A}. Thus, a space X is A-null if and only if map_{∗}(A, X) is weakly contractible, or, equivalently, if [Σ^{k}A, X] consists of a single element for all k ≥ 0.

Proposition 1.3 [7], [14] Let F → E → X be a homotopy fibration and A any space. If X is A-null and connected, then P_{A}F → P_{A}E → X is a homotopy fibration. □

Most of the above machinery can be paralleled in the category of groups. In fact, as explained in [6], homotopy localization with respect to a map (or a set of maps —there is no increase in generality in saying this, provided that coproducts exist in our category) makes sense in every cocomplete closed simplicial model category [22] satisfying certain mild conditions. Thus the category of groups (endowed with the discrete simplicial model structure) fits into this framework very well. Specifically, if φ: G → K is any group homomorphism, we say that a group L is φ-local if the induced map

φ^{∗}: Hom(K, L) → Hom(G, L)

is a bijection of sets. Then, by [6] or [12], one may construct, for every group π, a homomorphism l_{π} : π → L_{φ}π which is initial among all homomorphisms from π into φ-local groups. Thus (L_{φ}, l) may be viewed as an idempotent monad on the category of groups. We shall refer to it as φ-localization. In the special case when the target K of φ: G → K is the trivial group, we preferably use the term G-reduction instead of φ-localization, and call the corresponding φ-local groups G-reduced, motivated by the special case considered in [7, § 5]. The G-reduction map will be denoted, as proposed in [7], by π → π//G. This is always an epimorphism [10, Theorem 3.2], and its kernel is called the G-radical of π. Thus, the G-radical of a group π is the (unique) maximal subgroup of π among those whose own G-reduction is trivial. The basic examples are the standard p-radical, where p is a prime (this case corresponds to G = Z/p, and coincides with the p-torsion subgroup whenever such a subgroup exists), and the perfect radical (for which G can be chosen as the free product of a set of representatives of all isomorphism classes of countable perfect groups [3]).

The analogue of Proposition 1.1 is also true in the category of groups. Moreover, the following stronger result holds in the case of G-reductions:

Proposition 1.4 Given two groups G_{1}, G_{2}, consider the homomorphisms φ_{i}: G_{i} → {1}, i = 1, 2. Then the following assertions are equivalent:

(a) There is a natural transformation θ: L_{φ_{2}} → L_{φ_{1}}.

(b) φ_{2} is a φ_{1}-equivalence.

(c) G_{2}//G_{1} is the trivial group.

(d) For every group π, the G_{2}-radical of π is contained in the G_{1}-radical of π.

(e) The class of groups annihilated by L_{φ_{2}} is contained in the class of groups annihilated by L_{φ_{1}}.

Moreover, if these equivalent assertions hold, then the natural transformation θ is unique, and θ_{π} is surjective for every group π. □

2 On π_{1}-compatible nullifications

Let f : A → B be any map between connected spaces, and let f_{∗}: π_{1}A → π_{1}B be the induced homomorphism of fundamental groups. Then, as shown in [10, § 3], for every connected space X the natural homomorphism π_{1}X → π_{1}(L_{f}X) is an f_{∗}-equivalence of groups and hence there is a natural transformation of functors

π_{1}L_{f} → L_{f_{∗}}π_{1}.  (2.1)

We say that L_{f} is π_{1}-compatible if (2.1) is an isomorphism for all connected spaces X. That is, L_{f} is π_{1}-compatible if and only if the group π_{1}(L_{f}X) is f_{∗}-local for every connected space X.

If both A and B are (possibly infinite) wedges of circles, then, for every connected space X, the sets [A, L_{f}X] and [B, L_{f}X] are in natural bijective correspondence with the sets Hom(π_{1}A, π_{1}(L_{f}X)) and Hom(π_{1}B, π_{1}(L_{f}X)), respectively. It follows that the group π_{1}(L_{f}X) is f_{∗}-local and therefore the functor L_{f} is π_{1}-compatible. This observation, together with Theorem 3.5 in [10], yields

Theorem 2.1 If f: A → B is any map between wedges of circles, then the functor L_{f} is π_{1}-compatible. If A is a CW-complex of dimension 1 or 2, then the A-nullification functor P_{A} is π_{1}-compatible. □

However, if A = K(Z/p, 1), then P_{A} is not π_{1}-compatible; see Example 3.4 in [10].

The properties of π_{1}-compatible nullification functors turn out to be particularly pleasant. For example, such functors are essentially determined by their effect on the fundamental group and their behaviour on simply-connected spaces. In the rest of this section, this claim is made more precise.

Theorem 2.2 Let L be any idempotent functor on Ho. Suppose given a π_{1}-compatible nullification functor P_{A} together with a natural transformation θ: P_{A} → L which is an equivalence on simply-connected spaces. Then θ is an equivalence on the class of spaces whose fundamental group is annihilated under π_{1}A-reduction.

Proof. Let X be any space such that π_{1}X//π_{1}A is trivial. Then P_{A}X is simplyconnected, and therefore the lower horizontal arrow in the commutative diagram

P_{A}X ⟶^{θ} LX
↓  ↓
P_{A}(P_{A}X) ⟶^{θ} L(P_{A}X)

is a homotopy equivalence. By the idempotence of P_{A}, the left-hand vertical arrow is also a homotopy equivalence. Finally, the existence of θ ensures that every P_{A}-equivalence is an L-equivalence, so that the right-hand vertical arrow is a homotopy equivalence as well, from which our claim follows. □

This provides us with a good method to decide whether two π_{1}-compatible nullification functors are isomorphic or not. Namely, we infer the following criterion, which will be used strongly in the special cases discussed in Section 4; cf. Theorem 4.5 below.

Theorem 2.3 Assume given two spaces A_{1}, A_{2} together with a natural transformation θ: P_{A_{2}} → P_{A_{1}}. Suppose further that both P_{A_{2}} and P_{A_{1}} are π_{1}-compatible, and that θ is an equivalence on simply-connected spaces. Then, for a space X, the map θ_{X}: P_{A_{2}}X → P_{A_{1}}X is a homotopy equivalence if and only if the π_{1}A_{1}-radical and the π_{1}A_{2}-radical of π_{1}X coincide.

Proof. Suppose that θ_{X} is a homotopy equivalence. Then the induced homomorphism π_{1}(P_{A_{2}}X) → π_{1}(P_{A_{1}}X) is an isomorphism. Since P_{A_{1}} and P_{A_{2}} are π_{1}-compatible, this means that π_{1}X//π_{1}A_{2} ≅ π_{1}X//π_{1}A_{1}, from which it follows that the π_{1}A_{1}-radical and the π_{1}A_{2}-radical of π_{1}X coincide.

Conversely, suppose that the radicals coincide. Let π = π_{1}(P_{A_{2}}X) ≅ π_{1}(P_{A_{1}}X) and let X̃ be the homotopy fibre of the natural map X → Bπ. Since Hom(π_{1}A_{1}, π) and Hom(π_{1}A_{2}, π) are trivial, the space Bπ is A_{1}-null and A_{2}-null. By Proposition 1.3, the rows of the commutative diagram

P_{A_{2}} X̃ ⟶ P_{A_{2}}X ⟶ Bπ
↓ ^{θX̃} ↓ ^{θ_{X}} ↓ ^{=}
P_{A_{1}} X̃ ⟶ P_{A_{1}}X ⟶ Bπ

are homotopy fibrations. By Theorem 2.2, θX̃ is a homotopy equivalence. Therefore, θ_{X} is also a homotopy equivalence. □

Essentially the same argument proves the next result. This is in fact a broad generalization of Theorem 4.4 of [10], according to which, for every connected space X and every prime p, there is a homotopy fibration (Z[1/p])_{∞}X̃ → P_{M(Z/p,1)}X → Bπ, where π is the quotient of π_{1}X by its p-radical, and X̃ is the homotopy fibre of the map X → Bπ. Another important instance of Theorem 2.4 is the case when L is ordinary homology localization and P_{A} is the associated plus-construction (see Section 3 below).

Theorem 2.4 Let L be any idempotent functor on Ho. Suppose given a π_{1}-compatible nullification functor P_{A} together with a natural transformation θ: P_{A} → L which is an equivalence on simply-connected spaces. For a given space X, write π = π_{1}X//π_{1}A, and let X̃ be the homotopy fibre of the natural map X → Bπ. Then there is a commutative diagram

X̃ ⟶ X ⟶ Bπ
↓ ↓ ↓ ^{=}
LX̃ ⟶ P_{A}X ⟶ Bπ

in which the lower row is also a homotopy fibration. □

3 Generalized Postnikov towers

Let f : A → B be any map (which we replace with a cofibration, if necessary). We consider the cofibre sequence

A →^{f} B → C → ΣA →^{Σf} ΣB → ΣC → Σ^{2}A → ….

Thus we may view

\begin{matrix}A & ⟶^{f} & B \\ ↓ & & ↓ \\ pt & ⟶^{g} & C \end{matrix}  \begin{matrix}C & ⟶^{g′} & pt \\ ↓ & & ↓ \\ ΣA & ⟶^{Σf} & ΣB \end{matrix}

as homotopy cofibre squares, showing that g is an f-equivalence and Σf is a g′-equivalence. In this situation, Proposition 1.1 yields natural transformations L_{Σf} → P_{C} → L_{f}. Therefore, one obtains, for every space X, an inverse system

L_{f}X ← P_{C}X ← L_{Σf}X ← P_{ΣC}X ← L_{Σ^{2}f}X ← …,  (3.1)

which we call the f-tower associated to X. It is very often convenient to regard it, not as a single tower, but as two towers whose terms alternate, namely

L_{f}X ← L_{Σf}X ← L_{Σ^{2}f}X ← …  (3.2)

and

P_{C}X ← P_{ΣC}X ← P_{Σ^{2}C}X ← ….  (3.3)

Of course, if the target B of f is contractible, then (3.3) is just a shift of (3.2). In the special case f: S^{0} → pt, one obtains, up to homotopy, the Postnikov tower of X. Indeed, if we consider the map σ_{n+1}: S^{n+1} → pt, then the class of σ_{n+1}-equivalences is precisely the class of maps X → Y inducing isomorphisms π_{k}X ≅ π_{k}Y for k ≤ n.

If f: A → B is any map between n-connected spaces, then f is a σ_{n+1}-equivalence. Therefore, according to our preliminary remarks in Section 1, every f-equivalence is a σ_{n+1}-equivalence. This gives an easy proof of the following statement (which is quite well-known, and could, in fact, be inferred from the construction of L_{f} described in [13] as well).

Proposition 3.1 Let f: A → B be any map between n-connected spaces, n ≥ 1. Then, for any connected space X, the f-localization map l: X → L_{f}X induces isomorphisms

π_{k}X ≅ π_{k}(L_{f}X)  for k ≤ n.  □

In addition, if A is n-connected and B is contractible, then we have an epimorphism

π_{n+1}X → → π_{n+1}(L_{f}X),

as deduced from the fact that the A-nullification of a space X is constructed by attaching copies of A and its suspensions Σ^{k}A to X; cf. [7, Proposition 2.9].

As a consequence of Proposition 3.1, for every space X and every map f : A → B, the f-tower of X converges to X, in the sense that the homotopy inverse limit of (3.1) is homotopy equivalent to X.

Theorem 3.2 Let f_{1}, f_{2} be two maps, and assume that f_{2} is an f_{1}-equivalence. Then there is a unique commutative diagram of idempotent functors

L_{f_{1}} ← P_{C_{1}} ← L_{Σf_{1}} ← P_{ΣC_{1}} ← L_{Σ^{2}f_{1}} ← …
↑ ↑ ↑ ↑ ↑
L_{f_{2}} ← P_{C_{2}} ← L_{Σf_{2}} ← P_{ΣC_{2}} ← L_{Σ^{2}f_{2}} ← …. □

Our main examples of f-towers involve homology localizations and certain “approximations” to these, as discussed in the next section. For a commutative ring R with unit and a space X, we denote by X_{HR} the localization of X with respect to ordinary homology with R coefficients; see [4]. More generally, let E_{∗} be any homology theory satisfying the limit axiom. Let c be the smallest infinite cardinal which is at least equal to the cardinality of E_{∗}(pt). Let h: A → B be the wedge of a set of representatives of all homotopy classes of CW-inclusions which are E-equivalences and for which the cardinality of the set of cells of B is less than or equal to c. Then L_{h} is precisely E-localization; see [4, § 11] or [15]. Moreover, if C is the cofibre of h, then the following statements are equivalent:

• X is E-acyclic.

• X_{E} is contractible.

• P_{C}X is contractible.

We shall use the notation X_{E}^{+} for P_{C}X, and call it a generalized plus-construction. Indeed, if E = HZ, then X_{E}^{+} is homotopy equivalent to the Quillen plus-construction on X; cf. [3]. More generally, if E = HR for a commutative ring R with unit, then X_{HR}^{+} is homotopy equivalent to the partial R-completion of X defined in [9, VII.6]; see also [19], [20]. Recall that a group π is called R-perfect if H_{1}(π; R) = 0, and that every group G contains a unique maximal R-perfect normal subgroup P^{R}G, which we call the R-perfect radical of G. The map X → X_{HR}^{+} is an HR-equivalence and induces the projection π_{1}X → π_{1}X/P^{R}(π_{1}X) on the fundamental group [9, VII.6.3].

Proposition 3.3 For every commutative ring R with unit, the functors ( )_{HR} and ( )_{HR}^{+} are π_{1}-compatible.

Proof. Let h: A → B be any map for which L_{h} is HR-localization. Then h_{∗}: π_{1}A → π_{1}B induces an isomorphism on H_{1}( ; R) and an epimorphism on H_{2}( ; R), so that from [4, Lemma 7.3] it follows that π_{1}(X_{HR}) is h_{∗}-local for every X. This means precisely that the functor ( )_{HR} is π_{1}-compatible. Similarly, let C be the homotopy cofibre of h. Then π_{1}C is R-perfect. Hence, for every space X, the image of any homomorphism π_{1}C → π_{1}X is contained in P^{R}(π_{1}X), which implies that π_{1}(X_{HR}^{+}) is π_{1}C-reduced, as required. □

It would be very interesting to decide if ( )_{E} and ( )_{E}^{+} are π_{1}-compatible for other homology theories E_{∗}, in particular for complex K-theory. Another challenging, closely related problem, is to find a purely group-theoretical description of the homomorphism π_{1}X → π_{1}(X_{E}^{+}) in general.

In what follows, we shall use the notation

X_{E} ← X_{E}^{+} ← X_{ΣE} ← X_{ΣE}^{+} ← X_{Σ^{2}E} ← …  (3.4)

for the tower (3.1) in the case of E-localization.

4 Approximating ordinary homology towers

For any abelian group G, we may pick a space M(G, 1) as in [2, § 2]. Thus we choose a free abelian presentation

0 → F_{0} → F_{1} → G → 0

together with a map w: W_{0} → W_{1} between suitable wedges of circles inducing the inclusion F_{0} → F_{1} on homology, and define M(G, 1) to be the homotopy cofibre of w. Of course, if n > 1, then Σ^{n−1}M(G, 1) is a Moore space M(G, n) in the usual sense. If we apply the functor [ , X] to the cofibre sequence associated with w, we obtain the following result; cf. [2] or [7, § 5].

Proposition 4.1 Let G be any abelian group, and choose w: W_{0} → W_{1} as above. Then, for a connected space X and n > 1, we have:

(a) X is Σ^{n−1}w-local if and only if Ext(G, π_{m}X) = Hom(G, π_{m}X) = 0 for m ≥ n.

(b) X is M(G, n)-null if and only if Ext(G, π_{m}X) = Hom(G, π_{m}X) = 0 for m > n and π_{n}X is G-reduced.

Moreover, assertion (b) is true for n = 1 if π_{1}X is abelian, and assertion (a) is true for n = 1 if X is simply-connected. □

Let M(Z/p, 1) be the homotopy cofibre of the standard map f: S^{1} → S^{1} of degree p. On the other hand, by writing

Z[1/p] = ⟨x_{1}, x_{2}, …, x_{n}, … | x_{i} = x_{i+1}^{p} for all i⟩,

we have presented Z[1/p] as the cokernel of a specific homomorphism between free groups. Let g: ∨_{i}S^{1} → ∨_{i}S^{1} induce this homomorphism, and denote by M(Z[1/p], 1) the homotopy cofibre of g. Observe that the fundamental group of this space M(Z[1/p], 1) is precisely Z[1/p], so that it is a “true” Moore space in the sense of [24]. ¿From now on, the letters f and g will consistently denote the two maps which we have just considered.

Remark 4.2 When G = Z/p, the condition that an abelian group A be Z/p-reduced amounts to A being p-torsionfree, and the condition Ext(Z/p, A) = Hom(Z/p, A) = 0 says precisely that A is a p′-local group [17], where p′ denotes the complement of p (thus, a p′-local abelian group is just a Z[1/p]-module). In the case G = Z[1/p], for an abelian group A, the condition Ext(Z[1/p], A) = Hom(Z[1/p], A) = 0 says precisely that A is Ext-p-complete; see [9, VI.3.4] or [23].

Taking into account the fact that, in any category, two idempotent functors with the same image class are necessarily isomorphic, we have just shown the following.

Theorem 4.3 Assume that X is connected and n > 0, or that X is simply-connected and n ≥ 0. Then, if f and g denote the maps above defined, we have natural homotopy equivalences

(a) L_{Σ^{n}f}X ≃ X_{Σ^{n}HZ[1/p]}.

(b) L_{Σ^{n}g}X ≃ X_{Σ^{n}HZ/p}.

(c) P_{M(Z/p,n+1)}X ≃ X_{Σ^{n}HZ[1/p]}^{+}.

(d) P_{M(Z[1/p],n+1)}X ≃ X_{Σ^{n}HZ/p}^{+}. □

This result can be conveniently depicted as follows. Since

H̃_{∗}(M(Z/p, 1); Z[1/p]) = 0,

Theorem 3.2 yields, for every space X, a homotopy commutative diagram of towers

Z[1/p]_{∞}X ← X_{HZ[1/p]} ← X_{HZ[1/p]}^{+} ← X_{ΣHZ[1/p]} ← X_{ΣHZ[1/p]}^{+} ← …
↑ _{θ} ↑ _{θ^{+}} ↑ _{≃} ↑ _{≃}
L_{f}X ← P_{M(Z/p,1)}X ← L_{Σf}X ← P_{M(Z/p,2)}X ← ….

(4.1)

Similarly, the fact that H̃_{∗}(M(Z[1/p], 1); Z/p) = 0 gives

(Z/p)_{∞}X ← X_{HZ/p} ←  X_{HZ/p}^{+}  ← X_{ΣHZ/p} ← X_{ΣHZ/p}^{+} ← …
↑ _{ζ}  ↑ _{ζ^{+}}  ↑ _{≃}  ↑ _{≃}
L_{g}X ← P_{M(Z[1/p],1)}X ← L_{Σg}X ← P_{M(Z[1/p],2)}X ← ….

(4.2)

According to Theorem 4.3, all vertical arrows in (4.1) and (4.2) are homotopy equivalences if X is simply-connected. In fact, in order to infer that θ^{+} and ζ^{+} are homotopy equivalences it suffices to impose that π_{1}X be abelian.

The extent to which L_{f}X approximates X_{HZ[1/p]} when X is not simply-connected was carefully discussed in [11]. Among other things, it was shown that if X is nilpotent (or, more generally, if L_{f}X is nilpotent) then θ is a homotopy equivalence [11, Proposition 8.1]. It seems very likely that similar results could be proved for the arrow ζ in (4.2), but this has not been done so far.

Theorem 4.4 Let X be a connected space.

(a) If π_{1}X//(Z/p) = {1}, then the map P_{M(Z/p,1)}X → Z[1/p]_{∞}X is a homotopy equivalence.

(b) If π_{1}X//Z[1/p] = {1}, then the map P_{M(Z[1/p],1)}X → (Z/p)_{∞}X is a homotopy equivalence.

Proof. We give the argument only for (b). By Theorem 2.1, the functor P_{M(Z[1/p],1)} is π_{1}-compatible. Hence, from Theorem 2.2 it follows that the arrows P_{M(Z[1/p],1)}X → L_{g}X and ζ^{+} : P_{M(Z[1/p],1)}X → X_{HZ/p}^{+} are homotopy equivalences. Since, by assumption, the space P_{M(Z[1/p],1)}X is simply-connected, so is also X_{HZ/p}^{+}, and hence X_{HZ/p}^{+} → X_{HZ/p} is also a homotopy equivalence. This ensures that X is Z/p-good, so that the arrow X_{HZ/p} → (Z/p)_{∞}X is also a homotopy equivalence. □

This result improves substantially Lemma 1.2 in [21]. We devote the rest of this paper to giving other sufficient conditions on X under which θ^{+} or ζ^{+} are homotopy equivalences, and explicit counterexamples showing that sometimes they are not. The analysis turns out to be of a purely group-theoretical nature, due to the following facts. By Theorem 2.1 and Proposition 3.3, all idempotent functors in (4.1) and (4.2) are π_{1}-compatible. Hence, by Theorem 2.3, θ^{+} and ζ^{+} will be homotopy equivalences if and only if they induce isomorphisms on the fundamental group. In other words,

Theorem 4.5 The map θ^{+} is a homotopy equivalence if and only if the Z/p-radical and the Z[1/p]-perfect radical of π_{1}X coincide. The map ζ^{+} is a homotopy equivalence if and only if the Z[1/p]-radical and the Z/p-perfect radical of π_{1}X coincide. □

The assumptions of Theorem 4.5 are satisfied in many cases. We record the following instances.

Theorem 4.6 Suppose that π is free or nilpotent. Then:

(a) For every prime p, the Z/p-radical and the Z[1/p]-perfect radical of π coincide.

(b) For every prime p, the Z[1/p]-radical and the Z/p-perfect radical of π coincide.

Proof. In case (a), if π is free, then both radicals are trivial; if π is nilpotent, then both radicals are equal to the p-torsion subgroup of π. In case (b), if π is free, then both radicals are again trivial. Now suppose that π is nilpotent. By Proposition 1.4, the existence of ζ^{+} ensures that the Z[1/p]-radical of π is contained in the Z/p-perfect radical. Conversely, suppose that π contains a nontrivial subgroup G such that H_{1}(G; Z/p) = 0. We shall prove that G//Z[1/p] is trivial. Suppose the contrary, and write Q = G//Z[1/p] for shortness. Since Q is an epimorphic image of G, we have H_{1}(Q; Z/p) = 0, and this implies that H_{1}(Q) is a p-divisible abelian group. If c is the nilpotency class of Q and Γ stands for the lower central series, then Γ^{c−1}Q is nonzero and it is an epimorphic image of ⊗^{c}H_{1}(Q); see [17, p. 44]. This implies that Γ^{c−1}Q is p-divisible, in contradiction with the fact that Q is Z[1/p]-reduced. □

It follows, of course, that θ^{+} and ζ^{+} are homotopy equivalences whenever the fundamental group of X is free or nilpotent. Note that, if π is nilpotent, then the Z[1/p]-radical of π is precisely the kernel of the Ext-p-completion map π → Ext(Z/p^{∞}, π), as defined in [9, VI.3.7].

Theorem 4.7 Suppose that π is finite. Then:

(a) For every prime p, the Z[1/p]-radical and the Z/p-perfect radical of π coincide.

(b) The Z/2-radical and the Z[1/2]-perfect radical of π coincide.

Proof. In case (a), By Proposition 1.4, it suffices to prove that a finite group π is Z[1/p]-reduced (i.e., Hom(Z[1/p], π) = 0) if and only if it contains no nontrivial Z/pperfect subgroup. But both conditions are obviously equivalent to π being a p-group.

In case (b), the existence of θ^{+} ensures that the Z/p-radical of π is contained in the Z[1/p]-perfect radical for all primes p (cf. Proposition 1.4). To prove the converse for p = 2, let Q be the quotient of π by its Z/2-radical. Then Q has odd order. Suppose that Q contains a subgroup H which is Z[1/2]-perfect. Then, since H has odd order, it is in fact Z-perfect. On the other hand, by the Feit–Thompson theorem [16], H is solvable. This forces H to be trivial. Therefore, the Z/2-radical of π contains all Z[1/2]-perfect subgroups of π. □

Example 4.8 Let π be any torsionfree perfect group. Then, for any prime p, the Z/pradical of π is trivial, while the Z[1/p]-perfect radical is the whole of π. If p ≠ 2, then we may even find examples of finite groups for which the Z/p-radical and the Z[1/p]-perfect radical are distinct, hence showing that part (b) of Theorem 4.7 fails for primes other than 2 in general. Namely, it suffices to pick a finite simple group whose order is not divisible by p (such groups are indeed available for all primes p ≠ 2).

Example 4.9 For any fixed prime p, let π be an infinite perfect group of exponent p^{r} for some r ≥ 1. Since π cannot contain any infinitely p-divisible element, it is Z[1/p]-reduced, and hence the Z[1/p]-radical of π is trivial. However, the Z/p-perfect radical of π is the whole of π.

One might ask if such groups actually exist! We next indicate a source, which was suggested to us by Ian Leary. Let

B(m, n) = F_{m}/(F_{m})^{n}

be a free Burnside group of exponent n, where F_{m} denotes a free group of rank m. For any prime p and any m ≥ 2, we may choose n = p^{r} big enough so that B(m, n) is infinite [18]. After this choice, let π be the minimal normal subgroup of finite index in B(m, n); see [25], [26]. Then π does not contain any proper normal subgroup of finite index, and hence the commutator subgroup of π has to be π itself. This shows that π is a perfect group of exponent a power of p, as desired.

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

e-mail CC: casac@mat.uab.es; e-mail JLR: jlrodri@mat.uab.es