Homology equivalences inducing an epimorphism on

the fundamental group

José L. Rodríguez and Dirk Scevenels

Abstract

Quillen’s plus construction is a topological construction that kills the maximal perfect subgroup of the fundamental group of a space without changing the integral homology of the space. In this paper we show that there is a topological construction that, while leaving the integral homology of a space unaltered, kills even the intersection of the transfinite lower central series of its fundamental group. Moreover, we show that this is the maximal subgroup that can be factored out of the fundamental group without changing the integral homology of a space.

0 Introduction

As explained in [8], [9], Bousfield’s Hℤ-localization X_{Hℤ} of a space X ([2]) is homotopy equivalent to its localization with respect to a map Bf: BF_{1} → BF_{2} induced by a certain homomorphism f: F_{1} → F_{2} between free groups. This means that a space X is Hℤ-local if and only if the induced map Bf^{∗}: map(BF_{2}, X) → map(BF_{1}, X) is a weak homotopy equivalence. Moreover, the effect of Bf-localization on the fundamental group produces precisely the group-theoretical Hℤ-localization (i.e. f-localization) of the fundamental group, i.e. π_{1}L_{Bf}X ≅ L_{f}(π_{1}X) ≅ (π_{1}X)_{Hℤ} for all spaces X.

A universal acyclic space for Hℤ-localization (i.e. Bf-localization), in the sense of Bousfield ([4]), was studied by Berrick and Casacuberta in [1]. They show that nullification with respect to such a universal acyclic space coincides with Quillen’s plus construction X^{+} for a space X. Moreover, this universal acyclic space can be taken to be a two-dimensional Eilenberg–Mac Lane space K(A(f), 1), where A(f) is a locally free, universal f-acyclic group, in the sense of [12]. The effect of K(A(f), 1)-nullification on the fundamental group is precisely given by its A(f)-nullification. This A(f)-nullification factors out the perfect radical of a group, i.e. the maximal perfect subgroup, which can be obtained as the intersection of the (transfinite) derived series of the group ([7]).

The intersection ΓG of the transfinite lower central series of a group G is also a radical, and, as observed by Bousfield ([3]), it is in fact the maximal G-perfect normal subgroup of G (where a normal subgroup H of G is called G-perfect if H ≅ [H, G]). As explained in [7], there is an epireflection that corresponds to this radical ΓG. By adapting the methods used by Berrick and Casacuberta in [1], we describe in Theorem 2.6 an epimorphism g such that L_{g}G ≅ G/ΓG for all groups G.

We further construct a localization functor of topological spaces that is intermediate between Quillen’s plus construction and Bousfield’s homological localization, and induces localization with respect to g on the fundamental group. More precisely, we describe in Theorem 2.10 a map φ such that φ-localization of spaces factors out the intersection of the transfinite lower central series of the fundamental group, while preserving the integral homology of the space.

We furthermore show that this is in some sense the best possible result. Indeed, we show in Proposition 3.2 that the maximal subgroup that can be factored out of the fundamental group without altering the integral homology of a space, is precisely the intersection of the transfinite lower central series of the fundamental group.

Finally, we turn our attention to the question whether there exists a localization functor of spaces which induces localization with respect to a universal epimorphic Hℤequivalence of groups (in the sense of [12]) on the fundamental group, without changing the integral homology of a space. In fact, we show in Proposition 3.3 that the answer to this question is affirmative if and only if, for all groups G, the kernel of the Hℤ-localization homomorphism G → G_{Hℤ} coincides with ΓG.

1 Preliminaries

For the convenience of the reader, we recall some terminology and bacis facts about localization with respect to a given continuous map (see e.g. [4], [8]). Given a map f: A → B, a space X is called f-local if the induced map

f^{∗}: map(B, X) → map(A, X)

is a weak homotopy equivalence. For every space X there is a map l_{X}: X → L_{f}X, which is initial among all maps from X into f-local spaces. L_{f} is called the localization functor with respect to f. A map ϕ is called an f-equivalence if L_{f}ϕ is a homotopy equivalence. Further a space X is called f-acyclic if L_{f}X is contractible. In the special case where f is of the form f: A → ∗, the f-localization of a space X is also denoted by P_{A}X and it is called the A-nullification of X. The localization class of f, denoted by ⟨f⟩, is defined as the collection of all maps g such that L_{g} is naturally equivalent to L_{f}. When f is of the form f: A → ∗, the class of f is simply denoted by ⟨A⟩ and is called the nullification class of A. One says that ⟨f⟩ ≤ ⟨g⟩ if and only if there is a natural transformation of localization functors L_{f} → L_{g}. Recall that this is equivalent to every g-local space being f-local, to every f-equivalence being a g-equivalence, or to f being a g-equivalence. As shown by Bousfield in [4], the collection of localization functors with respect to maps is a small-complete lattice for this partial order relation. Note that a space X is f-acyclic if and only if ⟨X⟩ ≤ ⟨f⟩. Furthermore, in [4] Bousfield proved that every localization class has a best possible approximation by a nullification class. More precisely, for any map f, there is a maximal nullification class ⟨A(f)⟩ which is smaller than ⟨f⟩ in the lattice of localization classes. The space A(f) is called a universal f-acyclic space, since a space X is f-acyclic if and only if X is A(f)-acyclic.

An important algebraic tool in studying localization with respect to a given map is given by its discrete analogue in the category of groups (see e.g. [4], [5]). For 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. This allows to introduce the localization functor L_{f} with respect to f. In an obvious way, one speaks of f-equivalences and f-acyclic groups, and, if f is of the form f : A → 1, of A-nullification. It makes also sense to speak of the localization class of a homomorphism f, denoted by ⟨f⟩, which in the special case when f is of the form f : A → 1, is simply denoted by ⟨A⟩ and is called the nullification class of A. Furthermore, the collection of all localization classes of homomorphisms is again a small-complete lattice for an obvious partial order relation. In [12] it was proved that every localization class of a homomorphism has a best possible approximation by a nullification class and by the localization class of an epimorphism.

More precisely, for any homomorphism f, there is a maximal nullification class ⟨A(f)⟩ and a maximal class ⟨E(f)⟩ where E(f) is an epimorphism such that

⟨A(f)⟩ ≤ ⟨E(f)⟩ ≤ ⟨f⟩.

The group A(f) is called a universal f-acyclic group, since a group G is f-acyclic if and only if G is A(f)-acyclic. Accordingly, E(f) is called a universal epimorphic f-equivalence, since an epimorphism g is an f-equivalence if and only if g is an E(f)-equivalence.

It was shown in [12] (see also [7]) that to any localization class ⟨f⟩ of an epimorphism f, there is associated a radical R_{f} on the category of groups such that L_{f}G ≅ G/R_{f}G for all groups G. (Recall that a radical R is a functor assignating to every group G a normal subgroup RG in such a way that every homomorphism G → K restricts to RG → RK and such that R(G/RG) = 1.) In fact, there is a bijective correspondence between epireflections (i.e. idempotent functors L on the category of groups for which G → LG is an epimorphism for any group G) and radicals. Furthermore, if the class ⟨f⟩ is actually a nullification class, then the associated radical R_{f} is idempotent, meaning that R_{f}R_{f}G = R_{f}G for all groups G.

2 A universal epimorphic Hℤ-map

It is well known that Bousfield’s Hℤ-localization X_{Hℤ} of a space X ([2]) is homotopy equivalent to its localization with respect to a map Bf: BF_{1} → BF_{2} induced by a certain homomorphism f: F_{1} → F_{2} between free groups ([8], [9]). In fact, the homomorphism f can be taken to be the free product of a set of representatives of isomorphism classes of homomorphisms between countable, free groups inducing an isomorphism on the first integral homology group. Furthermore, the effect of Bf-localization on the fundamental group is to produce its f-localization, which coincides with the group-theoretical Hℤlocalization of the fundamental group. In [1, Proposition 4.2] the authors show that a universal f-acyclic group A(f) (in the sense of [12]) can be taken to be the free product of a set of representatives of all isomorphism classes of countable, locally free, perfect groups. The key lemma here is a result due to Heller ([10]), which states that for every element x in any perfect group P, there exists a countable, locally free, perfect group D and a homomorphism D → P containing x in its image. We first prove a “relative” version of [10, Lemma 5.7]. Recall that a normal subgroup H of a group G is called G-perfect if H ≅ [H, G].

Lemma 2.1 Let A be any group with an A-perfect normal subgroup K. Let F be a countable free group and let F′ be a subgroup. Let ψ : F → A be a homomorphism such that the restriction ψ|F′ is a homomorphism into K. Then there exists a countable, locally free group D with a D-perfect normal subgroup D′ such that ψ can be factorized as F → D → A and moreover this factorization restricts to a factorization of ψ|F′ as F′ → D′ → K.

Proof. Choose free generators x_{i}, y_{j} for F such that the generators x_{i} freely generate F′. For each generator x_{i} of F′, we can find finitely many elements a_{i,ℓ} of A and k_{i,ℓ} of K, such that ψ(x_{i}) = ∏_{ℓ}[a_{i,ℓ}, k_{i,ℓ}]. Let F_{1} be the free group generated by the set {x(i, ℓ, 1), x(i, ℓ, 2), y(j)}, and let F′_{1} be the free subgroup generated by the subset {x(i, ℓ, 1), x(i, ℓ, 2)}. Define a homomorphism ψ_{1}: F_{1} → A, by setting ψ_{1}(x(i, ℓ, 1)) = a_{i,ℓ}, ψ_{1}(x(i, ℓ, 2)) = k_{i,ℓ} and ψ_{1}(y(j)) = ψ(y_{j}). Define a homomorphism φ : F → F_{1} by φ(x_{i}) = ∏_{ℓ}[x(i, ℓ, 1), x(i, ℓ, 2)] and φ(y_{j}) = y(j). Then ψ_{1} ∘ φ = ψ and the image of ψ_{1} restricted to F′_{1} is contained in K. Iterating this construction by similarly factorizing ψ_{1}, we arrive at a sequence F → F_{1} → F_{2} → ... and a sequence of subgroups F′ → F′_{1} → F′_{2} → ..., whose colimits are the required D, resp. D′. □

This immediately implies the following result.

Lemma 2.2 Let A be any group with an A-perfect normal subgroup K. Let x be any element in K. Then there exist a countable, locally free group D with a D-perfect normal subgroup D′ and a homomorphism ψ: D → A such that x belongs to the image of the restriction ψ|D′. □

Recall ([3]) that any group G has a maximal G-perfect subgroup, which we denote by ΓG, and that can be obtained as the intersection of the transfinite lower central series of G. Lemma 2.2 enables us to prove the following result.

Proposition 2.3 Let G be any group. Then the following assertions are equivalent:

(i) For every group A and every A-perfect normal subgroup K of A, the restriction of any homomorphism A → G to K is trivial;

(ii) For every countable, locally free group A and every A-perfect normal subgroup K of A, the restriction of any homomorphism A → G to K is trivial; (iii) For every countable, locally free group A, the restriction of any homomorphism

A → G to ΓA is trivial. □

The subgroup ΓG actually defines a radical on the category of groups, and, hence, by [7], the assignation G → LG = G/ΓG is an epireflection. Our aim is to show that this epireflection is singly generated. More precisely, we want to exhibit a homomorphism g such that L_{g}G ≅ LG = G/ΓG for any group G. Moreover, by [12] we know that it is possible to choose g to be an epimorphism.

Let Λ be a set of representatives of isomorphism classes of countable, locally free groups. If D′ denotes the normal closure of ⋃_{Λ} ΓD in the free product D of all groups D in Λ, which we denote by Fr_{Λ}D, then we have a short exact sequence (cf. [11, Exercise 6.2.5])

D′ ↣ D = Fr_{Λ}D ↠ D/D′ ≅ Fr_{Λ}(D/ΓD),  (2.1)

where Fr_{Λ}(D/ΓD) denotes the free product of all groups D/ΓD for which (the isomorphism class of) D belongs to Λ.

Proposition 2.4 Localization with respect to the epimorphism

h: D ↠ D/D′

given in (2.1) satisfies L_{h}G ≅ G/ΓG for all groups G.

Proof. Observe that a group G is f-local for any given epimorphism f: A ↠ B if and only if the restriction of any homomorphism A → G to the kernel of f is trivial. The proof is now completed by using Proposition 2.3. □

Note that we can partition Λ into Λ_{1}, containing all the representatives of the isomorphism classes of countable, locally free, perfect groups, and its complement Λ_{1}^{c}. We then can write

⟨h⟩ = ⟨Fr_{Λ_{1}}(D → D/ΓD)⟩ ∗ ⟨Fr_{Λ_{1}^{c}}(D → D/ΓD)⟩
= ⟨Fr_{Λ_{1}}(D → 1)⟩ ∗ ⟨Fr_{Λ_{1}^{c}}(D → D/ΓD)⟩
= ⟨F⟩ ∗ ⟨Fr_{Λ_{1}^{c}}(D → D/ΓD)⟩,

where F is the universal Hℤ-acyclic group defined in [1]. (Here we denote by ⟨f_{1}⟩ ∗ ⟨f_{2}⟩ the least upper bound of the classes ⟨f_{1}⟩ and ⟨f_{2}⟩ in the lattice of localization classes, and we have used the fact that the free product f_{1} ∗ f_{2} is a representative of this least upper bound.)

However, it is possible to give another description of ⟨h⟩, which will be more useful later on. Indeed, note that, if f_{1} : A ↠ B is an epimorphism, and f_{2} : B → C is any homomorphism, then ⟨f_{2} ∘ f_{1}⟩ = ⟨f_{1} ∗ f_{2}⟩ = ⟨f_{1}⟩ ∗ ⟨f_{2}⟩. This enables us to prove the following preliminary result.

Lemma 2.5 Let f: A ↠ B be any epimorphism. Then ⟨f⟩ = ⟨l_{A}⟩, where l_{A}: A → L_{f}A denotes the localization homomorphism.

Proof. Since f is an epimorphism, we infer from [12, Theorem 2.1] that l_{A} is an epimorphism. Hence, ⟨f⟩ = ⟨f⟩ ∗ ⟨l_{B}⟩ = ⟨l_{B} ∘ f⟩ = ⟨L_{f}f ∘ l_{A}⟩ = ⟨L_{f}f⟩ ∗ ⟨l_{A}⟩ = ⟨l_{A}⟩. □

The above lemma now proves the following alternative description of ⟨h⟩.

Theorem 2.6 Consider the natural homomorphism g: D → D/ΓD, where D is as defined in (2.1). Then ⟨g⟩ = ⟨h⟩, where h is defined in (2.1). In other words, L_{g}G ≅ G/ΓG for all groups G. □

In fact, the homomorphism g is a “universal epimorphic Hℤ-map”, as we next show. Recall from [3] that a group homomorphism g is called an Hℤ-map if H_{1}g (i.e. the homomorphism induced by g on the first integral homology group) is an isomorphism and H_{2}g is an epimorphism. We first need a characterization of the epimorphisms that are Hℤ-maps (cf. [3]).

Lemma 2.7 Let h be an epimorphism h: A ↠ B with kernel K. Then h is an Hℤ-map if and only if K is A-perfect.

Proof. This is an obvious consequence of the 5-term exact sequence

H_{2}(A) → H_{2}(B) → K/[K, A] → H_{1}(A) → H_{1}(B) → 0. □

Proposition 2.8 Let G be any group and let g : D → D/ΓD, where D is as defined in (2.1). The homomorphism G → L_{g}G ≅ G/ΓG is terminal among all epimorphic Hℤ-maps going out of G. □

Observe that the epimorphism g that we have constructed is not a universal epimorphic Hℤ-equivalence (in the sense of [12]). Indeed, there are “more” epimorphic Hℤ-equivalences than epimorphic Hℤ-maps. However, if we denote by A(f), resp. E(f) a universal Hℤ-acyclic group, resp. a universal epimorphic Hℤ-equivalence, then there are natural homomorphisms

G → P_{A(f)}G → L_{g}G ≅ G/ΓG → L_{E(f)}G → G_{Hℤ},

for any group G, where G → L_{E(f)}G → G_{Hℤ} is an epi-mono factorization. Moreover, for many groups G (e.g. finite groups, nilpotent groups, or more generally, groups for which the lower central series stabilizes), we have isomorphisms L_{g}G ≅ G/ΓG ≅ L_{E(f)}G ≅ G_{Hℤ} (cf. [3]).

We now want to realize localization with respect to g: D ↠ D/ΓD topologically, by exhibiting a localization functor of spaces that induces g-localization on the fundamental group and which is intermediate between Quillen’s plus construction and Bousfield’s Hℤlocalization (and, hence, does not change the integral homology of a space).

We will need the following proposition, which is similar to results obtained in [5] and [6].

Proposition 2.9 Let ψ: A → B be any map which induces an epimorphism ψ_{∗} = π_{1}(ψ): π_{1}(A) ↠ π_{1}(B) and suppose that A is a CW-complex of dimension at most two. Then ψ-localization of spaces is π_{1}-compatible, i.e.

π_{1}(L_{ψ}X) ≅ L_{ψ_{∗}}(π_{1}X),

for all spaces X.

Proof. For any space X, the map X → L_{ψ}X is a ψ-equivalence. Hence, by [5, Proposition 3.3], the induced homomorphism π_{1}(X) → π_{1}(L_{ψ}X) is a ψ_{∗}-equivalence. Moreover, we claim that π_{1}(L_{ψ}X) is ψ_{∗}-local. To see this, it suffices to prove that the restriction of any homomorphism ℓ: π_{1}A → π_{1}(L_{ψ}X) to ker ψ_{∗} is trivial. However, since the dimension of A is at most two, there exists a map ξ: A → L_{ψ}X inducing ℓ on the fundamental group. Since L_{ψ}X is ψ-local, we infer that there exists a map χ: B → L_{ψ}X such that χ ∘ ψ ≃ ξ, which implies that the restriction ℓ| ker ψ_{∗} = (π_{1}(χ) ∘ ψ_{∗})| ker ψ_{∗} is trivial. □

Now choose a two-dimensional CW-complex MD such that π_{1}MD = D. Attach 2cells to MD, thereby obtaining a map i: MD ↪ C which induces g: D ↠ D/ΓD on the fundamental group. We then obtain a diagram

π_{2}MD → π_{2}C
↓  ↓
0 → H_{2}MD → H_{2}C → H_{2}(C, MD) → 0
↓  ↓
H_{2}(π_{1}MD) → H_{2}(π_{1}C) →  0
↓  ↓
0  0

Moreover, since H_{2}(π_{1}C) = H_{2}(D/ΓD) = 0, we infer that H_{2}C is an epimorphic image of π_{2}C. This means that we can kill H_{2}C by attaching 3-cells to C, through a map j: C → C′. It is now easily verified that the composition φ of

MD ⟶^{i} C ⟶^{j} C′  (2.2)

is an integral homology equivalence and that φ induces the homomorphism g: D ↠ D/ΓD on the fundamental group (cf. [2, Lemma 6.1]).

Theorem 2.10 Let φ: MD → C′ be the composition given in (2.2). Then φ-localization of spaces induces g-localization on the fundamental group. Moreover, for any space X, there are natural maps

X → X^{+} → L_{φ}X → X_{Hℤ}.

Proof. Since MD is a two-dimensional CW-complex and since φ induces an epimorphism on the fundamental group, we know by Proposition 2.9 that φ-localization is π_{1}- compatible, so that it induces g-localization on the fundamental group. The second claim is obvious, since φ is clearly an integral homology equivalence, and the perfect radical of π_{1}(L_{φ}X) being trivial (cf. [1]). □

To see that the natural maps given in Theorem 2.10 are not equivalences in general, and thus that we have constructed a functor which is really different from Quillen’s plus construction and from Hℤ-localization, observe the following. On one hand, L_{φ}(S^{1}∨S^{1}) ≃ S^{1} ∨ S^{1} ≃ (S^{1} ∨ S^{1})^{+}. Indeed, the fact that ℤ ∗ ℤ is g-local implies that S^{1} ∨ S^{1} is φ-local. On the other hand, π_{1}(L_{φ}BΣ_{3}) ≅ L_{g}(π_{1}BΣ_{3}) ≅ ℤ/2, while π_{1}BΣ_{3}^{+} ≅ P_{A(f)}(π_{1}BΣ_{3}) ≅ Σ_{3}.

3 Homology equivalences inducing an epimorphism

on the fundamental group

In this section we want to explore some immediate consequences of our results. In particular, we want to show that there are some restrictions on (integral) homology equivalences that induce an epimorphism on the fundamental group.

Proposition 3.1 Let ψ : X → Y be an integral homology equivalence of spaces such that the induced homomorphism f = π_{1}ψ is an epimorphism. Then R_{f}G ⊂ ΓG for all groups G, where R_{f} denotes the radical associated to the epireflection class ⟨f⟩.

Proof. By hypothesis we know that f is an epimorphic Hℤ-map. Hence, ⟨f⟩ ≤ ⟨g⟩, where g denotes the universal epimorphic Hℤ-map of Theorem 2.6. Hence, there are natural homomorphisms G ↠ L_{f}G ≅ G/R_{f}G ↠ L_{g}G ≅ G/ΓG, for any group G. □

In particular, we have the following result.

Proposition 3.2 Let ψ: X → Y be an integral homology equivalence of spaces such that the induced homomorphism f = π_{1}ψ is an epimorphism. Then ker f ⊂ Γπ_{1}X.

Proof. Since f is an epimorphic Hℤ-map, we know that ker f is π_{1}X-perfect. □

In other words, for any space X, the maximal subgroup that can be factored out of π_{1}X without altering the integral homology of X, is precisely Γπ_{1}X. In particular, this implies that there is a restriction on the possibility of realizing topologically a universal epimorphic Hℤ-equivalence of groups (i.e. of finding an integral homology equivalence of spaces which induces localization with respect to a universal epimorphic Hℤ-equivalence of groups on the fundamental group).

Proposition 3.3 Let E(f) denote a universal epimorphic Hℤ-equivalence of groups. Then the following assertions are equivalent:

(i) There exists an integral homology equivalence ψ: X → Y of spaces such that π_{1}ψ = E(f) and L_{ψ} is π_{1}-compatible;

(ii) ker(l_{Hℤ}: G → G_{Hℤ}) = ΓG for all groups G.

Proof. To see that (i) implies (ii), it suffices to show that ker l_{Hℤ} ⊂ ΓG. However, this is an immediate consequence of Proposition 3.2, since L_{π_{1}ψ}G ≅ L_{E(f)}G ≅ G/ ker l_{Hℤ} for all groups G. Finally (ii) implies (i), as is shown by our construction of φ in (2.2). □

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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