TORSION BOUNDS OF LOOP SPACES AND SUSPENSIONS

Hans Joachim Baues

Due to a result of Arkowitz-Curjel [1] we know that a connected finite type Hspace X admits a map g : X → K_{X} with the following properties. The space K_{X} is a product of Eilenberg Mac Lane spaces with homotopy groups π_{n}(K_{X}) = π_{n}(X) for n ≥ 1 and

g_{∗} : π_{n}(X) → π_{n}(K_{X}) = π_{n}(X)

satisfies g_{∗}(a) = N_{n} · a for a ∈ π_{n}(X) where N_{n} is a natural number ≥ 1. The sequence N_{∗} of numbers N_{n} is termed a π-\underline{torsion bound for} X. We are interested in the following questions:

(A) Does any connected H-space admit a π-torsion bound N_{∗} ?

(B) Is there a sequence N_{∗} which is a π-torsion bound for all connected finite

type H-spaces? In fact, if (A) is true then also (B) is true.

Dually to the result of Arkowitz-Curjel a simply connected finite type co-H-space Y admits a map f : M_{Y} → Y with the following properties. The space M_{Y} is a one point union of Moore spaces with homology groups H_{n}(Y) = H_{n}(M_{Y}) for n ≥ 1 and

f_{∗} : H_{n}(Y) = H_{n}(M_{Y}) → H_{n}(Y)

satisfies f_{∗}(a) = M_{n} · a for a ∈ H_{n}(Y) where M_{n} is a natural number ≥ 1. The sequence M_{∗} of numbers M_{n} is termed an H-\underline{torsion bound for} Y. Similarly as in the problems (A) and (B) above we obtain the following questions.

(C) Does any simply connected co-H-space admit an H-torsion bound M_{∗}? (D) Is there a sequence M_{∗} of numbers which is an H-torsion bound for all

simply connected finite type co-H-spaces? Again if (C) holds then also (D) holds.

All problems (A) ... (D) are unsolved. It was, however, observed by Arlettaz-Pointet [AP], [P2], [P1] that there exists a sequence N_{∗} which is a π-torsion bound for a large class of loop spaces. In this paper we recall and sketch proofs of such results and we show that dual results hold. We prove that there exists a sequence M_{∗} which is an H-torsion bound for a large class of suspensions.

These bounds have implications concerning the exponents of the kernel, resp. cokernel of the Hurewicz homomorphism; compare also [A5], [A3]. We are not interested in writing down formulas for such bounds since these depend on the method of proof and tend to be very large; compare [P2], [A4]. The interested reader, however, will be able to deduce from the proofs explicit formulas.

Typeset by AMS-TEX

The results in this paper are nice examples of a duality in homotopy theory which is called Hπ-duality in [B3], [B2] and which is part of Eckmann-Hilton duality. In particular, the boundary invariants introduced in [B2] which are dual to Postnikov invariants play an essential role in the proofs. In fact, the application of boundary invariants was the motivation to consider the problems in this paper. For the convenience of the reader and in order to make the duality explicit we recall from the literature some results on Postnikov-invariants and π-torsion bounds which essentially are, with the exception of (3.14), due to Arlettaz and Pointet. The main new results in this paper are dealing with boundary invariants and H-torsion bounds of co-H-spaces and suspensions.

§ 1 Bounded classes of spaces

Let A be an abelian group. An Eilenberg-Mac Lane space X = K(A, n) is a connected space with homotopy groups π_{n}(X) = A and π_{i}X = 0 for i ≠ n. A Moore space Y = M(A, n) is a 1-connected space with reduced homology groups H_{n}(Y) = A and H_{i}Y = 0 for i ≠ n. We only consider spaces with base point of the homotopy type of a CW-complex and base point preserving maps. Let [X, Y] be the set of homotopy classes of such maps X → Y. Then

(1.1)  \{ \begin{matrix}π_{n}(A, X) = [M(A, n), X] \\ H^{n}(X, A) = [X, K(A, n)] \end{matrix}

are the homotopy groups, resp. cohomology groups of X with coefficients in A. These are groups since ΩK(A, n + 1) = K(A, n) and ΣM(A, n) = M(A, n + 1) where Ω and Σ denote the loop space and the suspension respectively.

In the literature homotopy groups with coefficients are often denoted by π_{n}(X, A). The comparison with the Hom-functor, however, shows that the notation π_{n}(A, X) as above is more natural.Let

(1)  H^{n}(X, A)_{π} ⊂ H^{n}(X, A) = [X, K(A, n)]

be the subgroup of all f ∈ [X, K(A, n)] which induce the trivial map 0 = π_{n}f : π_{n}X → A. Similarly let

(2)  π_{n}(A, X)_{H} ⊂ π_{n}(A, X) = [M(A, n), X]

be the subgroup of all f ∈ [M(A, n), X] which induce the trivial map 0 = H_{n}f : A → H_{n}X.

Recall that a space X is \underline{simple} if X is connected and if π_{1}X acts trivially on π_{n}X for n ≥ 1. This implies π_{1}X is abelian. Clearly simply connected spaces are simple. A simple space X has a \underline{Postnikov decomposition} given by maps (n ≥ 2)

X ⟶^{q} X_{(n)} ⟶^{p_{n}} X_{(n−1)}

where q induces isomorphisms π_{i}(q) for i ≤ n and where π_{i}X_{(n)} = 0 for i > n. The map p_{n} is the fiber of a \underline{k-invariant}

(1.2)  k_{n}(X) ∈ H^{n+1}(X_{(n−1)}, π_{n}(X))_{π}

If all k_{n}(X) are trivial then X is a product of Eilenberg-Mac Lane spaces. The Postnikov tower shows that all spaces X are ‘twisted’ products of Eilenberg-Mac Lane spaces. In the following definition we consider spaces for which all k-invariants are torsion elements. Such spaces are ‘close’ to products of Eilenberg-Mac Lane spaces.

For an abelian group A and a natural number N ∈ ℕ = {1, 2, … } let N : A → A be the homomorphism, which carries a ∈ A to N · a.

(1.3) Definition. A class X of simple spaces is \underline{π-bounded} if there exists a sequence N_{1}, N_{2}, … of natural numbers such that for all n ≥ 1 and X ∈ X one of the following equivalent conditions (1), (2), (3) is satisfied.

(1) The Hurewicz homomorphism h is embedded in a commutative diagram of homomorphisms between abelian groups

π_{n}X ⟶^{N_{n}} π_{n}X

↘ h ↗ r

H_{n}X

(2) There exists a map

g_{n} : X → K(π_{n}X, n)

inducing (g_{n})_{∗} = N_{n} : π_{n}X → π_{n}X.

(3) the n − th k-invariant k_{n}(X) of X is an N_{n} -torsion element, that is

N_{n} · k_{n}(X) = 0. Clearly a space X is π -bounded if the class {X} consisting only of the space X is π -bounded.

There is an obvious way to dualize the definition of a π -bounded class as follows. For this one has to consider the dual of the k-invariant k_{n}(X) which is the boundary invariant

(1.4)  β_{n}(X) ∈ Γ_{n−1}(H_{n}X, X)/Δ im (b_{n+1}X)_{∗}

of a simply connected space X defined in 2.6.4 [B2]. The new concept of boundary invariant in [B2] is not yet common knowledge as is the concept of Postnikov invariant appearing in each reasonable text book of homotopy theory. There are various different ways to define the boundary invariant β_{n}X, see [B2], one of which is recalled below for the convenience of the reader. The striking duality between k-invariants and boundary invariants discussed in [B2], [B3] and the results of this paper should convince the reader that, in fact, boundary invariants are as useful as the classical Postnikov invariants. Boundary invariants are related to homology groups similarly as Postnikov invariants are related to homotopy groups. If all boundary invariants β_{n}(X) are trivial then X is a one point union of Moore spaces.

The \underline{homology decomposition} of a simply connected space X is given by maps (n ≥ 2)

(1)  X_{n−1} ⟶^{i_{n}} X_{n} ⟶^{j} X

where j induce a homology isomorphism H_{r}(j) for r ≤ n. Moreover i_{n} is the cofiber of a map

(2)  k′_{n} ∈ π_{n−1}(H_{n}X, X_{n−1})_{H}

Here k′_{n}, however, is not an invariant of the homotopy type of X. The ”invariant part” of k′_{n} is described by the boundary invariant β_{n}(X) in (1.4). In fact, there is a canonical homomorphism

(3)  λ_{A} : π_{n−1}(A, X_{n−1})_{H} → Γ_{n−1}(A, X)/Δ im(b_{n+1}X)_{∗}

and β_{n}(X) is the image of k′_{n} under this homomorphism, i.e. β_{n}(X) = λ_{A}(k′_{n}) for A = H_{n}X. See section 2.7 [B2] and (1.9) (c) below.

Let SP_{∞}X be the infinite symmetric product of X and let ΓX be the homotopy fiber of X → SP_{∞}X. Then we define the group in (1.4) (3) by

Γ_{n−1}(A, X) = π_{n−1}(A, ΓX)

and Δ : Ext(A, π_{n}ΓX) → π_{n−1}(A, ΓX) is defined by the universal coefficient sequence. Moreover

(b_{n+1}X)_{∗} : Ext(A, π_{n+1}SP_{∞}X) → Ext(A, π_{n}ΓX)

is obtained by the boundary homomorphism b_{n+1}X : π_{n+1}SP_{∞}X → π_{n}ΓX associated to the fibration ΓX → X → SP_{∞}X. Of course b_{n+1}X coincides with the secondary boundary homomorphism b_{n+1} : H_{n+1}X → Γ_{n}X of J.H.C. Whitehead [WH].

There is the following alternative way to define Γ_{n−1}(A, X) by use of the skeleta of X; see 2.2.3 [B2] and 7.5 [B3]. It is well known that

(4)  π_{n}ΓX = Γ_{n}X = image {π_{n}X^{n−1} → π_{n}X^{n}}

is the Γ-group of J.H.C. Whitehead [WH]. Now it is proved in [B2] that Γ_{n}(A, X) can be obtained from π_{n}(A; X^{n}) by the following diagram consisting of a pull back and a push out of abelian groups. Here i : Γ_{n}X → π_{n}X^{n} is the inclusion and p : π_{n+1}X^{n} → Γ_{n+1}X is the projection given by (4).

(5)

Ext(A, Γ_{n+1}X) ⟵^{p_{∗}} Ext(A, π_{n+1}X^{n}) ⟶^{1} Ext(A, π_{n+1}X^{n})
Δ↓  ^{push}  ↓  ↓^{Δ}
Γ_{n}(A, X) ⟵  P  ⟶ π_{n}(A, X^{n})
↓  ^{pull}  ↓^{μ}
Hom(A, Γ_{n}X^{n}) ⟶^{i_{∗}} Hom (A, π_{n}X^{n})

The column at the right hand side is the short exact sequence given by the universal coefficient formula.

The advantage of the boundary invariant β_{n}X compared to the element k′_{n} in (2) is the naturality formula: Let f : X → Y be a map between simply connected spaces and let \overline{H_{n}f} : M(H_{n}X, n − 1) → M(H_{n}Y, n − 1) be any map between Moore spaces representing the homomorphism H_{n}f : H_{n}X → H_{n}Y induced by f in homology. Then the formula

(6)  (\overline{H_{n}f})^{∗}β_{n}(Y) = f_{∗}β_{n}(X)

holds. The elements k′_{n} in (2) do not satisfy such a naturality formula.

The homology decomposition shows that all simply connected spaces X are ‘twisted’ one point unions of Moore spaces. In the following definition we consider spaces for which all boundary invariants are torsion elements. Such spaces are ‘close’ to one point unions of Moore spaces.

(1.5) Definition. A class X of simply connected spaces is \underline{H-bounded} if there exists a sequence N_{1}, N_{2}, … of natural numbers such that for all n ≥ 1 and X ∈ X one of the following equivalent conditions (1), (2), (3) is satisfied.

(1) The Hurewicz homomorphism h is embedded in a commutative diagram of homomorphisms between abelian groups

π_{n}X

s ↗ ↘ h

H_{n}X ⟶^{N_{n}} H_{n}X

(2) There exists a map

g_{n} : M(H_{n}X, n) → X

inducing (g_{n})_{∗} = N_{n} : H_{n}X → H_{n}X

(3) The n − th boundary invariant β_{n}(X) is an N_{n} -torsion element, that is

N_{n} · β_{n}(X) = 0 Using the Hurewicz homomorphism for spectra we can replace in (1.5) the class X of spaces by a class X of connected spectra and we define this way an H-bounded class of spectra.

A simple space X is of \underline{finite type} if all homotopy groups π_{i}X are finitely generated abelian groups or equivalently if all homotopy groups H_{i}X are finitely generated, i ≥ 1. A \underline{rational sphere} S_{ℚ}^{n} is the same as the Moore space M(ℚ, n) and K(ℚ, n) is termed a rational Eilenberg-Mac Lane space. We have K(ℚ, n) = S_{ℚ}^{n} if n is odd and K(ℚ, n) = ΩS_{ℚ}^{n+1} if n is even. Let X_{ℚ} be the rationalization of the simple space X.

(1.6) Proposition. Let X be a simple space

(A) If X is π -bounded then X_{ℚ} ≃ K_{ℚ} where K_{ℚ} is a product of rational

Eilenberg-Mac Lane spaces. Conversely, (B) if X is of finite type and X_{ℚ} ≃ K_{ℚ} then X is π -bounded.

Proof. (A) is clear since g : X → K_{X} is obviously a rational equivalence. Moreover the assumption X_{ℚ} ≃ K_{ℚ} in (B) shows that k_{n}(X) ⊗ ℚ is trivial. Since X is of finite type we obtain therefore a natural number N_{n} with N_{n}k_{n}(X) = 0. q.e.d.

Hence a finite type simple space X is π -bounded if and only if X_{ℚ} ≃ K_{ℚ}. This, however, does not hold for spaces which are not of finite type as the following example of Arlettaz [AP] shows.

Example. Let n be even. Then K = K(ℚ, n) satisfies H_{2n+1}K = 0 and H_{2n}K = ℚ so that by the universal coefficient theorem

(*)  H^{2n+1}(K; ℤ) = Ext (H_{2n}K, ℤ) = Ext (ℚ, ℤ) = ℝ

Let X be the space with exactly two non trivial homotopy groups π_{n}X = ℚ, π_{2n}X = ℤ and k-invariant k_{2n}X = α a non trivial element in (*). Then α is not a torsion element but one readily checks X_{ℚ} ≃ K(ℚ, n) × K(ℚ, 2n) since Ext(ℚ, ℚ) = 0.

We now consider the Hπ -dual of (1.6).

(1.7) Proposition. Let X be a simply connected space.

(A) If X is H-bounded then X_{ℚ} ≃ S_{ℚ} where S_{ℚ} is a one point union of rational

spheres. Conversely, (B) if X is of finite type and X_{ℚ} ≃ S_{ℚ} then X is H-bounded.

Proof. The proof is dual to the proof of (1.6). Again (A) is clear since g : M_{X} → X is obviously a rational equivalence and clearly (M_{X})_{ℚ} = S_{ℚ}. Moreover the assumption X_{ℚ} ≃ S_{ℚ} implies that β_{n}(X) ⊗ ℚ is trivial. Since X is of finite type we obtain therefore a natural number M_{n} with M_{n}β_{n}(X) = 0. q.e.d.

Hence a finite type simply connected space X is H-bounded if and only if X_{ℚ} ≃ S_{ℚ}. Again this does not hold for spaces which are not of finite type as the following example shows.

Example. Let X be the simply connected space with exactly two non trivial homology groups H_{2}X = ℤ and H_{3}X = ℚ and with boundary invariant β_{3}X = α where α is a non trivial element in

Γ_{2}(ℚ, X)/Δ im (b_{n+1}X)_{∗} = Γ_{2}(ℚ; X)

= Ext (ℚ, Γ_{3}X) = Ext (ℚ, Γπ_{2}X) = Ext (ℚ, ℤ) = ℝ.

Compare (2.2.3) [HH]. Then X has a homology decomposition given by

k′_{3} = α ∈ π_{2}(ℚ, S^{2}) = Ext (ℚ, ℤ) = ℝ

Hence X_{ℚ} ≃ S_{ℚ}^{2} ∨ S_{ℚ}^{3} since Ext (ℚ, ℚ) = 0. But the boundary invariant α of X is of infinite order.

The proposition above shows that for finite type spaces X the condition of X being π -bounded or H-bounded is equivalent to a condition on the rationalization X_{ℚ} of X. This clearly is not at all true for classes of spaces X.

(1.8) Lemma. See [AP]. The conditions (1), (2) and (3) defining a π -bounded class in (1.3) are equivalent.

Proof. One obtains (2) ⇒ (1) by the naturality of h applied to the map g_{n} and (1) ⇒ (2) follows from the surjectivity of

H^{n}(X, π_{n}X) → Hom(H_{n}X, π_{n}X)

Moreover (2) ⇒ (3) follows from the naturality of the k-invariant applied to g_{n}. Finally (3) ⇒ (2) is a consequence of the fact that X_{(n)} is the fiber of k_{n}(X). q.e.d.

(1.9) Lemma. The conditions (1), (2) and (3) defining an H-bounded class in (1.5) are equivalent.

Proof. We again obtain (2) ⇒ (1) by the naturality of the Hurewicz map h applied to g_{n}. Next we prove (1) ⇒ (2) as follows. Let 0 → B_{n} → Z_{n} → H_{n}X → 0 be a short free resolution of H_{n}X. Then d : B_{n} → Z_{n} defines a map M(B_{n}, n) → M(Z_{n}, n) with cofiber M(H_{n}X, n). Now s in (1) yields α : Z_{n} → H_{n}X → π_{n}X with αd = 0. Hence the map ᾱ : M(Z_{n}, n) → X given by α has an extension g_{n} over M(H_{n}X, n).

Next we see that (2) ⇒ (3) follows from the naturality of β_{n}(X) applied to g_{n}; compare (2.6.9) (a) in [B2]. The proof (3) ⇒ (2) is more complicated. Let

(a)  β ∈ Γ_{n−1}(H_{n}X, X)

be an element representing β_{n}X in (1.4). Then we have N · β_{n}(X) = 0 for some N ≥ 1 if and only if there exists

(b)  \{ \begin{matrix}α ∈ Ext (H_{n}X, H_{n+1}X) with \\ Nβ = Δ(b_{n+1})_{∗}α \end{matrix}

Here b_{n+1} : H_{n+1}X → Γ_{n}X is part of the exact sequence of J.H.C. Whitehead. We can choose a homomorphism α′ such that the composite

B_{n} ⟶^{α′} Z_{n+1} ↠ H_{n+1}X

represents α. By (2.7.5) [B2] we can choose a map β′ associated to k′_{n} for which the following diagram commutes

(c)  \begin{aligned}& & X^{n} \\ & & ∪ \\ & & X_{n−1} \\ k′_{n} ↗ & & ∪ \\ M(H_{n}, n − 1) & ⟶^{β′} & X^{n−1} \\ ∪ & & ∪ \\ M(Z_{n}, n − 1) & ⟶ & X^{n−2} \end{aligned}

Here k′_{n} is the map of the homology decomposition in (1.4) (2).

In fact, this diagram yields for A = H_{n} the homomorphism λ_{A} in (1.4) (3) which carries k′_{n} to the class λ_{A}(k′_{n}) represented by β′. That is, with the operators in diagram (1.4) (5) we have for a choice of β′ as above that μ(β′) ∈ image i_{∗} so that β′ represents an element β′ ∈ P in the pull back of (1.4) (5). The projection map q : P → Γ_{n−1}(A, X) yields λ_{A}(k′_{n}) = q(β′) and this defines λ_{A}.

Let C_{n+1} = H_{n+1}(X^{n+1}, X^{n}) and let f : M(C_{n+1}, n) → X^{n} be the attaching map of (n + 1) -cells in X. Then f induces the operator b_{n+1} in (b) so that the equation in (b) is equivalent to the condition that the following diagram homotopy commutes with H_{n} = H_{n}(X).

(d)

M(H_{n}, n − 1) ⟶ M(B_{n}, n) ⟶^{α′} M(Z_{n+1}, n) ⟶ M(C_{n+1}, n) ⟶ ∗
N↓  ↓^{f}  ↓
M(H_{n}, n − 1)  ⟶^{k′_{n}}  X_{n−1} ⟶  X^{n}  ⟶ X^{n+1}
↓  ↓  ↓  ↓
∗  ⟶  X_{n}  ⟶ X^{n+1} ⟶ X

Adding the homotopies in this diagram yields the required map g_{n} : M(H_{n}, n) = ΣM(H_{n}, n − 1) → X with the property in (2). q.e.d.

§ 2 Bounded classes of H-spaces and co-H-spaces

An \underline{H-space} is a space X together with a multiplication μ : X × X → X for which the composite

(2.1)  X ∨ X → X × X ⟶^{μ} X

is homotopic to the folding map (1, 1). For example a loop space ΩX is an H-space. A \underline{co-H-space} is a connected space together with a comultiplication μ : X → X ∨X for which the composite

(2.2)  X ⟶^{μ} X ∨ X → X × X

is homotopic to the diagonal map (1, 1). For example a suspension ΣX is a co-H-space. The map μ induces in both cases an operation + on the set of homotopy classes [X, X], that is α + β = μ_{∗}(α, β) and α + β = μ^{∗}(α, β) respectively. For a natural number N let

(2.3)  N · 1_{X} : X → X

be a map given by the N-fold sum N · 1_{X} = (… ((1_{X} + 1_{X}) + 1_{X}) + … + 1_{X}) in [X, X]. The induced maps

(N · 1_{X})_{∗} = N : π_{n}X → π_{n}N for an H-space X
(N · 1_{X})_{∗} = N : H_{n}X → H_{n}X for a co-H-space X

are multiplication by N.

It is well known that the rationalization of a connected H-space is a product of rational Eilenberg-Mac Lane spaces. Therefore (1.6) implies the following result of Arkowitz-Curjel [AC].

(2.4) Proposition. A finite type connected H-space is π -bounded.

Also the Hπ -dual of this result holds by (1.7) since the rationalization of a simply connected co-H-space is known to be a one point union of rational spheres.

(2.5) Proposition. A finite type simply connected co-H-space is H-bounded.

Is it true that (2.4) and (2.5) still hold if one omits the finite type condition? Here the examples in (1.6) and (1.7) cannot be used.

Let X be a class of spaces and let F be a functor which carries spaces to spaces then F X denotes the class of all spaces F X with X ∈ X.

(2.6) Proposition. Let X be a class of connected H-spaces and assume ΩΣX is π -bounded. Then X is π -bounded.

Proof. For an H-space X the suspension map i : X → ΩΣX admits a homotopy retraction r. Compare Rutter [R2]. Now the result follows from the naturality of the k-invariants since

i_{∗}k_{n}(X) = i^{∗}k_{n}ΩΣX

where i_{∗} is injective since i admits a retraction r. Compare also 4.9 and 4.10 in [P1] and 3.3 in [AP]. q.e.d.

(2.7) Proposition. Let X be a class of simply connected co-H-spaces and assume ΣΩX is H-bounded. Then X is H-bounded.

Proof. For a co-H-space the evaluation map ε : ΣΩX → X admits a homotopy section s. Compare Rutter [R1]. Now the naturality of the boundary invariants shows

ε_{∗}β_{n}(ΣΩX) = ε^{∗}β_{n}X

where ε^{∗} is injective since ε admits the section s. q.e.d.

The propositions (2.6) and (2.7) show that it suffices to prove (2.4) and (2.5) for loop spaces and suspensions respectively.

(2.8) Proposition. Let X be a class of connected H-spaces. Then X is π -bounded if and only if there exists a sequence R_{1}, R_{2}, … in ℕ such that for all n ≥ 1 and X ∈ X there is a homotopy commutative diagram

X ⟶^{R̄_{n}} X
↓  ↓
K_{n} ⟶ X_{(n)}

Here R̄_{n} is a map which induces multiplication by R_{n} on homotopy groups, that is (R̄_{n})_{∗} = R_{n} : π_{∗}X → π_{∗}X. Moreover K_{n} is a product of Eilenberg-Mac Lane spaces and X → X_{(n)} is the Postnikov section of X. The space K_{n} can be chosen to be the product of all K(π_{i}, i) with i ≤ n and π_{i} = π_{i}(X).

Proof. If the sequence R_{i}, i ≥ 1, exists then the naturality of k-invariants shows that X is π -bounded. Now assume that X is π -bounded by N_{i}, i ≥ 1. Then we construct R_{i} with the property in (2.8) as follows. Consider the homotopy commutative diagram with N = N_{n}.

X ⟶^{N·1} X
↓  ↓
X_{(n)} ⟶^{N·1} X_{(n)}
p_{n}↓  ↓^{p_{n}}
X_{(n−1)} ⟶^{N·1} X_{(n−1)}

We use the fact that X_{(n)} is an H-space and that the vertical arrows in the diagram are H-maps; see 9.4 [St]. Since p_{n} is a fibration the bottom square can be chosen to be commutative; see 9.4 [St]. Hence we obtain the factorization N · 1 : X_{(n)} → (N · 1)^{∗}p_{n} → X_{(n)}. Here we have

(N · 1)^{∗}p_{n} ≃ X_{(n−1)} × K(π_{n}, n)

since N · k_{n}(X) = 0. Now we apply this inductively and we get for R_{n} = N_{1} · N_{2} · . . . · N_{n} the result. Compare also 1.2 in [AP]. q.e.d.

We do not know whether the dual of (2.8) holds for a class of simply connected co-H-spaces; but we are able to prove the following result.

(2.9) Proposition. Let X be a class of suspensions X = ΣY of simply connected spaces Y. Then X is H-bounded if and only if there exist R_{1}, R_{2}, … in ℕ such that for all n ≥ 1 and X ∈ X there is a homotopy commutative diagram

X ⟵^{R_{n}·1_{X}} X
↑  ↑
M_{n} ⟵ X_{n}

where M_{n} is a one point union of Moore spaces and X_{n} → X is given by a homology decomposition of Y; that is X_{n} = ΣY_{n−1} → ΣY = X. The space M_{n} can be chosen to be the one point union of all Moore spaces M(H_{i}, i) with i ≤ n and H_{i} = H_{i}X.

Proof. If the sequence R_{i}, i ≥ 1, exists the naturality of boundary invariants shows that X is H-bounded. Now assume that X is H-bounded by N_{i}, i ≥ 1. Then we have to construct R_{i} and diagrams as in (2.8). Since X = ΣY ∈ X is a suspension we have

X^{n} = ΣY^{n−1} and X_{n} = ΣY_{n−1}

and all inclusion maps X^{n} ⊂ X_{n} ⊂ X are suspensions. For X ∈ X we know N_{n} · β_{n}(X) = 0. This implies that also the homomorphism

(1)  b_{n}(X) : H_{n}X → Γ_{n−1}X

satisfies N_{n}b_{n}(X) = 0. The homomorphism is represented by the map b_{n} in the commutative diagram

(2)  \begin{matrix}M(C_{n}, n − 1) & ⟶^{f_{n}} & X^{n−1} & ⊂ X^{n} = C(f_{n}) \\ ∪ & & ∪ \\ M(Z_{n}, n − 1) & ⟶^{b_{n}} & X^{n−2} \end{matrix}

Here C_{n} = H_{n}(X^{n}, X^{n−1}) and f_{n} is the attaching map of n-cells. Moreover B_{n} ⊂ Z_{n} ⊂ C_{n} denote the subgroups of boundaries and cycles in C_{n}. We have C_{n} = Z_{n} ⊕ B_{n−1} and the homology decomposition X_{n} of X is obtained by the cofiber sequence

(3)  M(B_{n−1}, n − 1) ⟶^{f̄_{n}} X^{n−1} → X_{n−1} = C(f̄_{n})

where f̄_{n} is the restriction of f_{n} so that X_{n−1} ⊂ X^{n}. We obtain the diagrams in (2.9) inductively as follows. Let

M_{n} = ⋁_{i=1}^{n} M(H_{i}X, i).

The inductive assumption is the existence of a homotopy commutative diagram

(4)  \begin{matrix}X & ⟵^{R_{n−1}} & X \\ ↑ & & ↑ \\ M_{n−1} & ⟵^{S_{n−1}} & X_{n−1} \end{matrix}

with the property that the composites

(5)  \{ \begin{matrix}M(Z_{n+1}, n) ⟶^{b_{n+1}} X^{n−1} ⟶^{I^{n−1}} X_{n−1} ⟶^{S_{n−1}} M_{n−1} \\ M(Z_{n}, n − 1) ⟶^{b_{n}} X^{n−2} ⟶^{I^{n−1}i^{n−2}} X_{n−1} ⟶^{S_{n−1}} M_{n−1} \end{matrix}

are null homotopic. This holds for n = 3 since M_{2} = X_{2} is a point. Clearly X is 2-connected since Y is simply connected. We now consider the following diagram

(6)  X ⟵^{R_{n−1}}  X  ⟵^{N_{n}}  X ; ↑ ; ↑  ^{⊛}  X^{n}  ↑ ; ↑^{i} ; M_{n−1} ⟵^{f_{n−1}} X_{n−1}  ⟵^{N_{n}}  X_{n−1} ; ↑^{b_{n+1}} ; M(Z_{n+1}, n)  ↑^{k′_{n}} ; ↑_{α′} ; M(B_{n}, n) ⟵^{q} M(H_{n}, n − 1)

Here we know by (1) and diagram (1.9) (d) that

α′′ = N_{n}k′_{n} − b_{n+1}α′q

satisfies i_{∗}α′′ = 0. Since

M(Z_{n}, n − 1) ⟶^{b_{n}} X_{n−1} ⟶^{i} X^{n}

is a cofiber sequence this implies that

α′′ = b_{n}α′′′ for some α′′′ : M(H_{n}, n − 1) → M(Z_{n}, n − 1)

Here we use the fact that X is 2-connected. Hence we see by (5) that

(7)  f_{n+1}N_{n}k′_{n} ≃ 0

This shows that we have the following commutative diagram where P denotes a push out

(8)  \begin{matrix}X & ⟵^{R_{n−1}N_{n}} & X \\ ↑ & & & ↑ \\ P & ⟵^{S̄_{n}} & X_{n} \\ ↑ī & & & ↑^{i_{n−1}} \end{matrix}

M_{n−1} ⟵^{S_{n−1}} X_{n−1} ⟵^{N_{n}} X_{n−1}

This diagram is obtained by diagram ⊛ in (6) which we may choose to be commutative since X_{n−1} → X is a cofibration. By (7) we know that there is a homotopy equivalence

(9)  μ : P ≃ M_{n−1} ∨ M(H_{n}, n) = M_{n}

Now we have to obtain condition (5) for n replaced by (n + 1). For this we consider the following composite with N = N_{n+1} · N_{n+2}.

M(Z_{n+2}, n + 1) ⟶^{b_{n+2}} X^{n} ⟶^{N} X^{n} ⟶^{i^{n}} X^{n+1}

which is null homotopic since Nb_{n+2}(X) = 0. Hence the cofiber sequence

M(C_{n+1}, n) ⟶^{f_{n}} X^{n} ⟶^{i^{n}} X^{n+1}

shows that there exists λ : M(Z_{n+2}, n + 1) → M(C_{n+1}, n) such that

(10)  f_{n}λ = Nb_{n+2}

Moreover the diagram

(11)  \begin{matrix}M(C_{n+1}, n) & & = M(B_{n}, n) ∨ M(Z_{n+1}, n) \\ ↓^{f_{n}} & & ↓^{(0,i_{n−1}I^{n−1}b_{n+1})} \\ X^{n} & ⟶^{I^{n}} & X_{n} \end{matrix}

homotopy commutes with i_{n−1}I^{n−1} : X^{n−1} ⊂ X_{n−1} ⊂ X_{n}. Now let

(12)  R_{n} = R_{n−1} · N_{n} · N_{n+1} · N_{n+2}

Then (8), (9) show that we have a homotopy commutative diagram

(13)  \begin{matrix}X & ⟵^{R_{n}} & X \\ ↑ & & ↑ \\ M_{n} & ⟵^{S_{n}} & X_{n} \end{matrix}

with S_{n} = (μS̄_{n})N. Moreover we get the following equations.

S_{n}I^{n}b_{n+2} = (μS̄_{n})NI^{n}b_{n+2},  (13)
= μS̄_{n}I^{n}Nb_{n+2}
= μS̄_{n}I^{n}f_{n}λ,  (10)
= μS̄_{n}(0, i_{n−1}I^{n−1}b_{n+1})λ,  (11)
μS̄_{n}i_{n−1}I^{n−1}b_{n+1} = μīS_{n−1}N_{n}I^{n−1}b_{n+1},  (8)
= μīS_{n−1}I^{n−1}b_{n+1}N_{n},
= 0, see (5)

Here we use the fact that b_{n+1} is a suspension since f_{n+1} is a suspension. This completes the proof that S_{n}I^{n}b_{n+2} = 0. Moreover we get

S_{n}I^{n}i^{n−1}b_{n+1} = μS̄_{n}NI^{n}i^{n−1}b_{n+1},  (13)

= μS̄_{n}I^{n}Ni^{n−1}b_{n+1} = 0

since N_{n+1}i^{n−1}b_{n+1} = 0 by N_{n+1}b_{n+1}(X) = 0. This completes the proof of the inductive step; see (5). q.e.d.

§ 3 Examples of bounded classes of loop spaces and suspensions

We shall prove in this section the following two results (3.1) and (3.2) which are Hπ -dual and we dsecribe various further classes of loop spaces and suspensions which are π -bounded and H-bounded respectively.

(3.1) Theorem. The class of all connected infinite loop spaces is π -bounded.

(3.2) Theorem. The class of all connected spectra is H-bounded.

For the proof of (3.1) and (3.2) compare (3.13) below. See also [A1], [A2], [A4], [AP], [A7]. On the other hand we show:

(3.3) Theorem. See [P1].

(A) The class of all connected finite type loop spaces ΩX with at most two non

trivial homotopy groups is π -bounded. (B) The class of all connected double loop spaces ΩΩX with at most two non

trivial homotopy groups is π bounded.

This result is due to Pointet-Tischler [P1] where part (A) is also stated for nonfinite type X. The proof of this non-finite type case of (A), however, has a gap in 3.26 [P1]. Also the following Hπ -dual of (3.3) holds.

(3.4) Theorem.

(A) The class of all suspensions ΣX of simply connected finite type spaces X

with at most two non trivial homology groups is H-bounded. (B) The class of all double suspensions ΣΣX of simply connected spaces X with

at most two non trivial homology groups is H-bounded.

In (3.3) (B) and (3.4) (B) we do not need the assumption that spaces are of finite type.

(3.5) Definition. Let Ab_{∗} be the category of graded abelian groups {A_{n}; n ≥ 1} and homomorphisms h_{∗} = {h_{n} : A_{n} → B_{n}; n ≥ 1}. We say that a class H of morphisms in Ab_{∗} is \underline{bounded torsion} if there exists a sequence N_{1}, N_{2}, … in ℕ such that for all h_{∗} ∈ H the image of h_{n} : A_{n} → B_{n} is N_{n} -torsion (that is N_{n}h_{n} = 0).

The results above rely on the following two basic facts (3.6) and (3.7) where we use the evaluation map ΣΩK → K and the suspension map M → ΩΣM. For any space X and n ≥ 1 let π′_{n}(X), resp. H′_{n}(X), be the kernel, resp. cokernel of the Hurewicz map h : π_{n}(X) → H_{n}(X). Hence we have the exact sequence

0 → π′_{n}(X) → π_{n}(X) → H_{n}(X) → H′_{n}(X) → 0

(3.6) Lemma. See [P1].

(A) Consider the class of graded homomorphisms

Ω_{∗} = {H_{n}(ΣΩK) ⟶^{Ω_{n}} H_{n}(K) → H′_{n}(K)}_{n≥1}

induced by the evaluation map in homology where K is a product of Eilenberg-Mac Lane spaces. This class is bounded torsion.

(B) Consider the class of graded homomorphisms

Ω_{∗}^{A} = {Ω_{n}^{A} : H^{n}(K, A)_{π} → H^{n}(ΣΩK, A)_{π}}_{n≥1}

induced by the evaluation map in cohomology where K is a finite type product of Eilenberg-Mac Lane spaces and A ∈ Ab. This class is bounded torsion.

This crucial lemma is due to Pointet-Tischler in [P1]; compare also the proof below where we consider also the following lemma which is dual to (3.6).

(3.7) Lemma.

(A) Consider the class of graded homomorphisms

Σ_{∗} = {π′_{n}(M) → π_{n}(M) ⟶^{Σ_{n}} π_{n}(ΩΣM)}_{n≥1}

induced by the suspension map on homotopy groups where M is a one point union of Moore spaces. This class is bounded torsion.

(B) Consider the class of graded homomorphisms

Σ_{∗}^{A} = {Σ_{n}^{A} : π_{n}(A, M)_{H} → π_{n}(A, ΩΣM)_{H}}_{n≥1}

induced by the suspension map for homotopy groups with coefficients where M is a finite type one point union of Moore spaces and A ∈ Ab. This class is bounded torsion.

Proof of (3.6) and (3.7). Since the functors Σ, Ω, π_{n} and H_{n} are compatible with filtered direct limits it suffices to prove (3.6) (A) and (3.7) (A) for spaces K and M respectively which are of finite type. Moreover since the maps Ω_{n} and Σ_{n} carry cross effects to zero it in addition suffices to prove (3.6) (A) and (3.7) (A) for K = K(C, M) and M = M(C, m) respectively where C = ℤ or C = ℤ/p^{r} with p a prime and r ≥ 1. In this case (3.6) (A) is a consequence of the work of Cartan [C]; compare 3.34 Pointet-Tischler [P1]. Moreover for M = M(ℤ, m) lemma (3.7) (A) is a well known property of homotopy groups of spheres. For M = M(ℤ/p^{r}, m) lemma (3.7) (A) is a consequence of the fact that the suspension map

(*)  M(ℤ/p^{r}, m) → ΩΣM(ℤ/p^{r}, m)

factors through the space S^{m+1}{p^{r}} which is the fiber of the map p^{r} · 1_{S^{m+1}} : S^{m+1} → S^{m+1}. This is a result of Cohen-Moore-Neisendorfer in 11.6 [CMN].

For a proof of (3.6) (B) we refer to the proof of Pointet-Tischler where, however, we restrict to the finite type case so that we do not need 3.26 [P1]. For the proof of (3.7) (B) we again use (*) above and the fact that Σ_{n}^{A} carries cross effects to zero. Hence again it suffices to prove (3.7) (B) for M = M(C, n) and this is a consequence of (*). q.e.d.

Proof of (3.3) and (3.4). One obtains (3.3) (A) as a consequence of (3.6) (B) and one obtains (3.4) (A) as a consequence of (3.7) (B). Next we prove (3.3) (B) and (3.4) (B). For this we only use the universal coefficient formulas and (3.6) (A) and (3.7) (A) respectively. In fact let π, G ∈ Ab and consider the following commutative diagram where Ω is the loop map and K = K(π, m), K′ = ΩK, K′′ = ΩΩK.

Ext(H_{n−2}K′, G) ⟶^{(σΩ_{n−2})^{∗}} Ext(H_{n−3}K′′, G)
↓  ↓
H^{n}(K, G)_{π}  ⟶^{Ω} H^{n−1}(K′, G)_{π}  ⟶^{Ω} H^{n−2}(K′′, G)_{π}
↓  ↓
Hom(H_{n}K, G) ⟶_{(σΩ_{n})^{∗}} Hom(H_{n−1}K′, G)

Here σ denotes the suspension isomorphism σ : H_{n−1}X ≅ H_{n}ΣX and Ω_{n}, Ω_{n−2} are given by (3.6) (A). The column in the middle of the diagram is short exact. If NΩ_{n} = 0 and MΩ_{n−2} = 0 then clearly the diagram shows N · M(ΩΩ) = 0. Moreover since Ωk_{n}(X) = k_{n−1}(ΩX) we see that (3.6) (A) implies (3.3) (B). The proof of (3.4) (B) is strictly dual. For this we consider the suspension map Σ in the following commutative diagram where M = M(π, m), M′ = ΣM(π, m) and M′′ = ΣΣM(π, m).

(2) \begin{aligned}& & Ext(G, π_{n+2}M′) & ⟶^{(ωΣ_{n+2})^{∗}} & Ext(G, π_{n+3}M′′) \\ & & ↓ & & ↓ \\ π_{n}(G, M)_{H} & ⟶^{Σ} & π_{n+1}(G, M′)_{H} & ⟶^{Σ} & π_{n+2}(G, M′′)_{H} \\ ↓ & & ↓ \\ Hom(G, π_{n}M) & ⟶^{(ωΣ_{n})^{∗}} & Hom(G, π_{n+1}M′) \end{aligned}

Here ω denotes the canonical isomorphism ω : π_{n}ΩX = π_{n−1}X and Σ_{n} and Σ_{n+2} are given by (3.7) (A). The column in the middle is short exact. Hence NΣ_{n} = 0 and MΣ_{n+2} = 0 implies N · M(ΣΣ) = 0 and this yields (3.4) (B) since we can use the homology decomposition of X. q.e.d.

(3.9) Theorem. Let X be a π -bounded class of connected H-spaces. Then (A) and (B) hold.

(A) The class of all double loop maps

Ω_{∗}^{2} = {Ω^{2} : H^{n}(X, A)_{π} → H^{n−2}(Ω^{2}X, A)_{π}}_{n≥1}

with X ∈ X and A ∈ Ab is bounded torsion. (B) The class of all loop maps

Ω_{∗} = {Ω : H^{n}(X, A)_{π} → H^{n−1}(ΩX, A)_{π}}_{n≥1}

with X ∈ X and X of finite type and A ∈ Ab is bounded torsion.

Proof. This follows from (2.8), (3.8) (1) and (3.6) (B). q.e.d.

(3.10) Theorem. Let X be an H-bounded class of suspensions X = ΣY of simply connected spaces Y. Then (A) and (B) hold.

(A) The class of all double suspension maps

Σ_{∗}^{2} = {Σ^{2} : π_{n}(A, X)_{H} → π_{n+1}(A, Σ^{2}X)_{H}}_{n≥1}

with X ∈ X and A ∈ Ab is bounded torsion. (B) The class of all suspension maps

Σ_{∗} = {Σ : π_{n}(A, X)_{H} → π_{n+1}(A, ΣX)_{H}}_{n≥1}

with X ∈ X and X of finite type and A ∈ Ab is bounded torsion.

Proof. This follows from (2.9), (3.8) (2) and (3.7) (B). q.e.d.

Let X be a class of spaces. Then X_{F} denotes the \underline{class of fibers} F (f) of maps f : X → Y with the following properties:

• X, Y ∈ X and X is an H-space,

• π_{i}X ≠ 0 implies π_{i+1}Y = 0,
• f_{∗} : π_{∗}X → π_{∗}Y is trivial, f_{∗} = 0.

Moreover X_{F}^{ft} is the subclass of X_{F} consisting of all F (f) for which X is of finite type.

Dually let X_{C} be the \underline{class of cofibers} C(f) of maps f : X → Y with the following properties:

• X, Y ∈ X and Y is a suspension Y = ΣY′ of a simply connected space Y′,

• H_{i}X ≠ 0 implies H_{i+1}Y = 0,

• f_{∗} : H_{∗}X → H_{∗}Y is trivial, f_{∗} = 0.

Here H_{∗} denotes the reduced homology. Let X_{C}^{ft} be the subclass of X_{C} consisting of all C(f) for which Y is of finite type.

Using this notation we get the following lemma which is needed for the induction in the proofs of theorem (3.12), (3.14), (3.15) below.

(3.11) Lemma.

(A) If X is a π -bounded class of connected spaces then ΩX_{F}^{ft} and Ω^{2}X_{F} are π

-bounded classes. (B) If X is an H-bounded class of simply connected spaces then ΣX_{C}^{ft} and Σ^{2}X_{C}

are H-bounded classes.

Proof. The proof of (A) is strictly dual to the proof of (B). We therefore prove only (B) as follows. For C(f) ∈ X_{C} we have the short exact sequence

0 → H_{i}Y ⟶^{i} H_{i}C(f) ⟶^{j} H_{i−1}X → 0

where either i or j is an isomorphism; compare the definition of X_{C}. If H_{i}Y ≠ 0 we can use (1.5) (2) for Y and hence for C(f). Now assume H_{i}Y = 0 and H_{i−1}X ≠ 0 and let M = M(H_{i−1}X, i − 1). Then we can use (1.5) (2) for X and we get a map M → X inducing multiplication by N ∈ ℕ on homology H_{i−1}. For the composite α : M → X → Y we can use (3.10) (A), resp. (3.10) (B). Hence we obtain N′ ∈ ℕ with N′Σ^{2}α = 0, resp. N′Σα = 0. Moreover we have the canonical map

C(N′α) → C(f)

which induces multiplication by N′ · N on homology H_{i}. Hence we get also maps

Σ^{3}M → Σ^{2}C(N′α) → Σ^{2}C(f), resp.
Σ^{2}M → ΣC(N′α) → ΣC(f)

inducing multiplication by N′ ·N on homology H_{i+1}, resp. H_{i+1}. Using again (1.5) we obtain the proposition in (3.11) (B). q.e.d.

(3.12) Definition. We say that a space X is Ω \underline{-stable} if X is an Eilenberg-Mac Lane space or if there exists r ≥ 2 such that X is (r − 1) -connected and π_{i}X = 0 for i ≥ 2r − 1. Moreover X is Σ \underline{-stable} if there exists r ≥ 2 such that X is (r − 1) -connected and dim(X) ≤ 2r − 1. In particular Moore spaces are Σ -stable.

(3.13) Theorem.

(A) The class of products of Ω -stable spaces is π -bounded.

(B) The class of one point unions of Σ -stable spaces is H-bounded.

Here a product of CW-complexes is defined by use of the CW-topology of the product set.

Proof of (3.13). Let X be a Ω -stable space. Then there exists for all n ≥ 1 a space Y with X = Ω^{n}Y. Hence we can use the Postnikov decomposition of Y and (3.11) (A) to see that (3.13) (A) holds. Now let X be a Σ -stable space. Then for H_{i}X ≠ 0 the Hurewicz homomorphism h : π_{i}X → H_{i}X is isomorphic to h : π_{i+n}Σ^{n}X → H_{i+n}Σ^{n}X for all n ≥ 0. Hence it suffices to show that the class of all Σ^{n}X with n sufficiently large is H-bounded. But this follows from the homology decomposition of X and (3.11) (B) inductively. q.e.d.

(3.14) Theorem. Let n ≥ 0 and r ≥ 0 and consider the following classes:

(A) the class of all n-connected 2r-fold loop spaces X = Ω^{2r}Y with π_{i}X = 0 for

i > (r + 2)(n + 2r) − 2r, and (B) the class of all n-connected finite type r-fold loop spaces X = Ω^{r}Y with

π_{i}X = 0 for i > (r + 2)(n + r) − r. Then these classes (A) and (B) are π -bounded.

This is an improvement of the corresponding results in [P1]. We also give a more direct proof of the result relying on the ”principal reduction theorem”.

Proof. In case (B) we see that Y is m = n + r connected. Hence by principal reduction the Postnikov tower Y_{(i)} of Y admits fiber sequences

Y_{(jm)} → Y_{(jm−m)} → K_{j}

for j ≥ 2 with Y_{(2m)} = ΩK_{2}; compare 2.3.11 [B1]. All spaces K_{j}, j ≥ 2, are Ω -stable. Hence using (3.13) (A) we can apply (3.11) (A) inductively to see that the class of all Ω^{r}Y_{((r+2)m)} is π -bounded if Y is of finite type. In case (A) have to use Ω^{2r}Y_{((r+2)m)} with m = n + 2r. q.e.d.

(3.15) Theorem. Let n ≥ 2 and r ≥ 0 and consider the following classes:

(A) the class of all 2r-fold suspensions X = Σ^{2r}Y of (n − 1) -connected spaces

Y with dim(Y) ≤ (r + 2)n − 1, and (B) the class of all r-fold suspensions X = Σ^{r}Y of (n − 1) -connected finite type

spaces Y with dim(Y) ≤ (r + 2)n − 1. Then these classes (A) and (B) are H-bounded.

Proof. Since Y is (n − 1) -connected the skeleta Y^{i} satisfy principal reduction in the sense of 2.3.13 [B1]. Hence for j ≥ 2 there exist cofibration sequences

K_{j} → Y^{(j−1)n−1} → Y^{jn−1}

where K_{j} is Σ -stable with X^{2n−1} = ΣK_{2}. Now we can apply (3.11) (B) inductively and we see that (A) and (B) hold. q.e.d.

Literature

[AC] Arkowitz, M. and Curjel, C., The Hurewicz homomorphism and finite homotopy invariants, Trans. Amer. Math. Soc. 110 (1964), 538-551.

[AP] Arlettaz, D. and Pointet-Tischler, N., Postnikov invariants of H-spaces, Preprint, University of Lausanne (1997).

[A1] Arlettaz, D., On the homology of the special linear group over a number field, Comment. Math. Helv. 61 (1986), 556-564.

[A2] Arlettaz, D., On the k-invariants of iterated loop spaces, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 343-350.

[A3] Arlettaz, D., The first k-invariant of a double loop space is trivial, Arch. Math. 54 (1990), 84-92.

[A4] Arlettaz, D., Universal bounds for the exponent of stable homotopy groups, Topology Appl. 38 (1991), 255-261.

[A5] Arlettaz, D., Exponents for extraordinary homology groups, Comment. Math. Helv. 68 (1993), 653-672.

[A6] Arlettaz, D., The Hurewicz homomorphism in algebraic K-theory 71 (1991), Journal of Pure and Applied Algebra, 1-12.

[A7] Arlettaz, D., The order of the differentials in the Atiyah-Hirzebruch spectral sequence, K-theory 6 (1992), 347-361.

[A8] Arlettaz, D., The exponent of the homotopy groups of Moore spectra and the stable Hurewicz homomorphism (1996), Canadian J. Math. 48, 483-495.

[B1] Baues, H.J., Obstruction theory, Springer Lecture Note in Math. 628 (1977).

[B2] Baues, H.J., Homotopy type and homology, Oxford Math. Monographs (1996), Oxford University Press, 496 pages.

[B3] Baues, H.J., Homotopy types, Handbook of algebraic topology, chapter I, (1995) 1-71, Edited by I.M. James, Elsevier Science.

[C] Cartan, H., Algèbres d’Eilenberg-MacLane et homotopie, exposé 11, Sminaire H. Cartan Ecole Norm. Sup. (1954/1955).

[CMN] Cohen, F.R. and Moore, J.C. and Neisendorfer, J.A., Torsion in homotopy theory, Ann. of Math. 109 (1979) 121-168.

[J] James, I.M., Reduced product spaces, Ann. of Math. (2) 62 (1955), 170-197.

[P1] Pointet-Tischler, N., Invariants de Postnikov des espaces de lacets, Thèse Université de Lausanne (1996).

[P2] Pointet-Tischler, N., La suspension cohomologique des espaces d’Eilenberg-MacLane, C.R. Acad. Sci. Paris Sr. I, 325 (1997) 1113-1116.

[R1] Rutter, J.W., The suspension of the loop on a space with comultiplication, Math. Ann. 209 (1974) 69-82.

[R2] Rutter, J.W., The loops on the suspension of a space with multiplication, Math. Ann. 214 (1975) 81-94.

[St] Stasheff, J., H-spaces from a homotopy point of view, Springer Lecture Note in Math. 161 (1970).

[WG2] Whitehead, G., Elements of homotopy theory, Grad. Texts Math. 61, (Springer 1978).

[WH] Whitehead, J.H.C., A certain exact sequence, Ann. of Math. (2) 52 (1950), 51-110.