GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 1, 1995, 55-77

ON SOME PROPERTIES OF COHOMOTOPY-TYPE FUNCTORS

S. KHAZHOMIA

Abstract. It is shown that if Π^{n}, n > 2, are Chogoshvili’s cohomotopy functors [1, 2, 3], then 1) the isomorphism

Π^{n}(X_{1} × X_{2}) ≈ Π^{n}(X_{1}) ⊕ Π^{n}(X_{2})

holds for topological spaces X_{1} and X_{2}; 2) the relations

rank Π^{n}(X) = rank π_{n}(X),
Tor Π^{n+1}(X) ≈ Tor π_{n}(X)

hold under certain restrictions for the space X.

In [2–5] we investigated Chogoshvili’s cohomotopy functors Π^{n}(−; −; G; H) (see [1]) from the category of pairs of topological spaces with base points into the category of abelian groups. The aim of this paper is to give full proofs of the results announced in [4, 5]. Like everywhere in our discussion below, we considered the case when H is the integral singular theory of cohomologies (G = ℤ). In this paper all definitions, notions, and notation used in [3] are understood to be known and hence are not explained in the sequel. Note only that the auxiliary subcategories needed for defining functors Π^{n} will be denoted here by L_{n} (as distinct from [3] where they are routinely denoted by K_{n}). The base points of all spaces will be denoted by ∗ and, as a rule, will not be indicated if this does not cause any confusion. Moreover, as follows from [2], the base points may not be indicated at all for simply connected spaces which are mainly considered here.

We shall treat here mainly the absolute groups which were defined for n > 2 (see [3]). Therefore an arbitrary connected and simply connected space X with a finite-type module of homologies and H^{i}(X) = 0, 0 < i < n,

55 can be regarded as the object of L_{n}, n > 2. We shall also use the fact that for X ∈ L_{n} the analogues of Gurevich homomorphisms in the Π theory

1991 Mathematics Subject Classification. 55Q55.

Key words and phrases. Extensions of functors, cohomology, cohomotopy, Moore-Postnikov decompositions.

1072-947X/95/0100-0055$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

d_{i} : H^{i}(X) → Π^{i}(X)

are isomorphisms for 2 < i ≤ n (see [2]).

The paper consists of two sections. In Section 1 we state our results and prove some of them. In Section 2 we prove the statements that have been given in Section 1 without proof.

1. In this section we present our results and develop some of their proofs.

Proposition 1.1. If a space X consists only of a base point ∗, then Π^{n}(X) = 0 for all n > 2.

Proof. Let p ∈ Π^{n}(X) and

α = (X_{n}; f) ∈ ω(X; n)

be an arbitrary index. Then f ∼ 0 and from Corollary 1.3 of [3] it follows that P_{α} = 0. Therefore p = 0 and the proposition is proved.

By K(ℤ, m) denote any Eilenberg–MacLane space of the type (ℤ, m), having the homotopy type of a countable CW -complex.

Proposition 1.2. For n > 2 and m = 1, 2, ⋯ we have

Π^{n}(K(ℤ, m)) = \begin{cases}0, if n ≠ m; \\ ℤ, if n = m. \end{cases}

Proof. For m ≥ n we have K(ℤ, m) ∈ L_{n} and the proposition follows from the isomorphism

d_{n} : H^{n}(K(ℤ, m)) → Π^{n}(K(ℤ, m)).

Let now n > m, p ∈ Π^{n}(K(ℤ, m)), and

α = (X; f) ∈ ω(K(ℤ, m); n), X ∈ L_{n},

be an arbitrary index. Then since H^{m}(X) = 0, the mapping f : X → K(ℤ, m) is null-homotopic. Hence p_{α} = 0. Therefore p = 0 and the proposition is proved.

In Section 2 we shall prove

Theorem 1.3. For n > 2 the isomorphism

Π^{n}(X_{1} × X_{2}) ≈ Π^{n}(X_{1}) ⊕ Π^{n}(X_{2}).

holds for arbitrary spaces X_{1} and X_{2}.

By K_{m}^{t} denote an arbitrary space of the homotopy type ∏_{α=1}^{t} K_{α} where K_{α} = K(ℤ, m), α = 1, 2, …, t. By K_{m}^{0} denote a space containing only one point ∗.

Proposition 1.2 and Theorem 1.3 imply

Corollary 1.4. For n > 2, m = 1, 2, …, t > 0, we have

Π^{n}(K_{m}^{t}) = \begin{cases}0, & if m ≠ n; \\ \underbrace{ℤ ⊕ ℤ ⊕ ⋯ ⊕ ℤ}_{t}, & if m = n. \end{cases}

Let B_{n} ∈ L_{n}, n > 4. Choose some system h_{1}, h_{2}, …, h_{s} of generators of the group H^{n}(B_{n}) and consider the corresponding mappings

h_{i} : B_{n} → K(ℤ, n), i = 1, 2, …, s.

Then we can construct the mapping

f = ∏_{i=1}^{s} h_{i} : B_{n} → K_{n}^{s}.

The homomorphism f^{∗} is obviously an epimorphism in the dimension n. Let further

K_{n−1}^{s} → EK_{n}^{s} → K_{n}^{s}

be the standard Serre path-space fibering (K_{n−1}^{s} = ΩK_{n}^{s}) and

K_{n−1}^{s} ⟶^{i} E_{n−1} ⟶^{p_{n}} B_{n}

be the principal fibering induced by the mapping f (see [6]). Consider the diagram

K_{n−2}^{t} ⟶^{i′} E_{n+1}

↓p_{n−1}

K_{n−1}^{s} ⟶^{i} E_{n−1} ⟶^{f′} K_{n−1}^{t}  (1)

↓p_{n}

B_{n} ⟶^{f} K_{n}^{s} ,

where the number t, the mapping f′, and the fibering p_{n−1} are defined similarly to s, f, and p_{n}, respectively.

In Section 2 we shall prove Lemma 1.5. For n > 4 we have

1) E_{n−1} ∈ L_{n−1} and Π^{n−1}(E_{n−1}) ≈ H^{n−1}(E_{n−1}) are free groups;

2) Π^{n}(E_{n−1}) = H^{n}(E_{n−1}) = 0;

3) The homomorphism p_{n}^{∗} : H^{n+1}(B_{n}) → H^{n+1}(E_{n−1}) is an isomorphism;

4) Homomorphisms p_{n}^{#} : Π^{n+k}(B_{n}) → Π^{n+k}(E_{n−1}) are isomorphisms for k > 0;

5) E_{n+1} ∈ L_{n+1}, t ≤ s, and if H^{n}(B_{n}) is a free group, then t = 0 and E_{n−1} = E_{n+1}.

Let F →^{i} E →^{p} B be a fibering in the Serre sense. In [2] we obtained the semiexact sequence

⋯ ⟶ Π^{n−1}(B) ⟶^{p^{#}} Π^{n−1}(E) ⟶^{i^{#}} Π^{n−1}(F) ⟶^{δ̄^{#}} Π^{n}(B) ⟶ · · · . (2)

Here the coboundary homomorphism δ̄^{#} is defined from the semiexact sequence of the pair (E, F) by means of the isomorphism (see [2])

p^{#} : Π^{n}(B, ∗) → Π^{n}(E, F), n > 3.

Namely: δ̄^{#} = p^{#−1}δ^{#}, where δ^{#} is the coboundary homomorphism of the pair (E, F). We shall now give another definition of the homomorphism δ̄^{#}. For simplicity consideration will be given to fiberings in the Gurevich sense, which is sufficient for our purposes. These are, for example, all principal fiberings (see [6]). Let f : (X_{n}, ∗) → (B, ∗), where X_{n} ∈ L_{n} is an arbitrary mapping. Consider the mapping

f_{X} : (CΩX_{n}, ΩX_{n}) → (B, ∗)

and the diagram

[figure: (CΩX_{n}, ΩX_{n}); (SΩX_{n}, ∗); (X_{n}, ∗); (B, ∗); (E, F); f̄; p; e_{X}; π_{X}; f]

where CX and SX denote respectively the cone and the suspension over the space X in the category of spaces with base points, e_{X} and π_{X} are standard mappings, and f̄ is the lifting of the mapping f_{X} = fπ_{X}e_{X}. As is known, the mappings f̄ and f_{X} define each other uniquely up to homotopy. We introduce the notation π_{X}e_{X} = ψ_{X} and, moreover, consider the mapping

ḡ = f̄|ΩX_{n} : ΩX_{n} → F

and the indices

α = (X_{n}; f) ∈ ω(B; n),
ᾱ = (ΩX_{n}; ḡ) ∈ ω(F; n − 1).

Note that X_{n} ∈ L_{n} implies ΩX_{n} ∈ L_{n−1}, n > 3, and SX_{n} ∈ L_{n+1} (see [2]). Besides, as is known (see, for example, [2]), for the spaces X_{n} ∈ L_{n}, n > 3, the induced homomorphism π_{X}^{∗} will be an isomorphism in cohomologies of dimension n. In the sequel we may not specially indicate this fact. Let q ∈ Π^{n−1}(F). We assume

q̄_{α} = ψ_{X}^{∗−1}(δ(qᾱ)),

where δ is a cohomological coboundary operator of the pair (CΩX_{n}, ΩX_{n}). In the sequel we may not indicate the corresponding pair for the operator δ.

We shall now show that the set {q̄_{α}} defines an element of the group Π^{n}(B). Consider the diagram

[figure: f_{2}; ψ_{Y}; (Y, ∗); (B, ∗); (CΩY_{n}, ΩY_{n}); (CΩX_{n}, ΩX_{n}); (X, ∗); (B, ∗); f̄_{1}; p; f_{1}; ψ_{X}; φ_{1}; φ; p; f̄_{2}; (E, F); (E, F)]

where f_{2}φ ∼ f_{1} and the mapping φ_{1} = CΩφ is defined in the standard manner. Then

pf̄_{2}φ_{1} = f_{2}ψ_{Y} φ_{1} = f_{2}φψ_{X} ∼ f_{1}ψ_{X} = pf̄_{1}.

Accordingly, f̄_{2} ∼ f̄_{1}. Let

ḡ_{1} = f̄_{1}|ΩX_{n}, ḡ_{2} = f̄_{2}|ΩY_{n}, φ_{0} = φ_{1}|ΩX_{n}.

Then ḡ_{2}φ_{0} ∼ ḡ_{1}. Consider the indices

ᾱ_{1} = (ΩX_{n}; ḡ_{1}) ∈ ω(F; n − 1),
ᾱ_{2} = (ΩY_{n}; ḡ_{2}) ∈ ω(F; n − 1),
α_{1} = (X_{n}; f_{1}) ∈ ω(B; n),
α_{2} = (Y_{n}; f_{2}) ∈ ω(B; n).

We have α_{1} < α_{2} and ᾱ_{1} < ᾱ_{2}. Then

φ^{∗}(q̄_{α_{2}}) = φ^{∗}(ψ_{Y}^{∗−1}(δ(qᾱ_{2}))) = ψ_{X}^{∗−1}(φ_{1}^{∗}(δ(qᾱ_{2}))) =
= ψ_{X}^{∗−1}(δ(φ_{0}^{∗}(qᾱ_{2}))) = ψ_{X}^{∗−1}(δ(qᾱ_{1})) = q̄_{α_{1}}.

This means (see [3]) that we have defined the element q̄ ∈ Π^{n}(B, ∗) and thereby the mapping

δ̃^{#} : Π^{n−1}(F, ∗) → Π^{n}(B, ∗).

Consider the diagram

[figure: Π^{n−1}(F, ∗); Π^{n}(E, F); δ^{#}; Π^{n}(B, ∗); p^{#}; δ̃^{#}]  (3)

Proposition 1.6. Diagram (3) is commutative and the mapping δ̃^{#} = p^{#−1}δ^{#} is a homomorphism.

Proof. Let q ∈ Π^{n−1}(F). It is sufficient to check the equality p^{#}(δ̃^{#}(q)) = δ^{#}(q) for indices of the form (see [2])

α = (CX_{n−1}, X_{n−1}; f) ∈ ω(E, F; n), X_{n−1} ∈ L_{n−1}.

Consider the commutative diagram

\begin{matrix}(CΩSX_{n−1}, ΩSX_{n−1}) & ⟵^{k_{1}} & (CX_{n−1}, X_{n−1}) & ⟶^{f} & (E, F) \\ ↓e_{1} & & ↓e & & ^{↓p} \\ (SΩSX_{n−1}, ∗) & ⟶^{π} & (SX_{n−1}, ∗) & ⟶^{φ} & (B, ∗) \end{matrix} ,

where the mapping φ is induced by the mapping f, the mappings e_{1}, e and π are standard mappings, and the mapping k_{1} = ck is obtained from the standard mapping

k : X_{n−1} → ΩSX_{n−1}.

We introduce the notation πe_{1} = ψ_{1}. Let

Φ : (CΩSX_{n−1}, ΩSX_{n−1}) → (E, F)

be the lifting of the mapping φψ_{1}. Consider the indices

α_{1} =(X_{n−1}; f|X_{n−1}) ∈ ω(F; n − 1),
P (α) =(CX_{n−1}, X_{n−1}; pf) ∈ ω(B, ∗; n),
β =(SX_{n−1}; φ) ∈ ω(B; n),
β_{1} =(ΩSX_{n−1}; Φ|ΩSX_{n−1}) ∈ ω(F; n − 1).

Since pΦk_{1} = φψ_{1}k_{1} = φe = pf, we have Φk_{1} ∼ f. Therefore

(Φ|ΩSX_{n−1})k = (Φk_{1})|X_{n−1} ∼ f|X_{n−1}.

Thus α_{1} < β_{1}. Then, on the one hand, we have (see [1,2])

[δ^{#}(q)]_{α} = δ(q_{α_{1}})

and, on the other hand,

[p^{#}(δ̃^{#}(q))]_{α} = [δ̃^{#}(q)]_{P (α)} = e^{∗}([δ̃^{#}(q)]_{β}) =
= e^{∗}(ψ_{1}^{∗−1}(δ(q_{β_{1}}))) = (e^{∗}ψ_{1}^{∗−1})(δ(k^{∗−1}(q_{α_{1}}))) =
= (e^{∗}ψ_{1}^{∗−1})(k_{1}^{∗−1}(δ(q_{α_{1}}))) =
= (e^{∗}ψ_{1}^{∗−1}k_{1}^{∗−1})(δ(q_{α_{1}})) = δ(q_{α_{1}}).

Therefore p^{#}δ̃ = δ^{#}.

In the sequel the homomorphisms δ̃^{#} and δ̄^{#} will both be denoted by δ^{#}.

Remark. The boundary homomorphism δ^{#} has been defined here only for fiberings in the Gurevich sense because we wanted arbitrary indices to participate in the definition. For a fibering in the sense of Serre the definition will be more sophisticated, since we shall have to use CW -approximations of spaces. On the other hand, it should be noted that Theorem 2 from [3] could make it possible for the indices to limit our consideration to the suspended CW -complexes X_{n} = SX_{n−1}, X_{n−1} ∈ L_{n−1}. Then a simpler diagram

[figure: (CX_{n−1}, X_{n−1}); (SX_{n−1}, ∗); (B, ∗); (E, F); e; f; f̄; p]

than ours would allow us to define the homomorphism δ^{#} for all fiberings. As is known, using this diagram we can easily define the boundary homomorphism ∂_{#} in fiberings for homotopy groups.

Consider now the diagram

\begin{aligned}0 & → & H^{n−1}(E_{n−1}) & →^{i^{∗}} & H^{n−1}(K_{n−1}^{s}) & →^{τ} & H^{n}(B_{n}) & → & 0 \\ & & d_{n−1}↓ ≈ & & d_{n−1}↓ ≈ & & d_{n}↓ ≈ \\ 0 & → & Π^{n−1}(E_{n−1}) & →^{i^{#}} & Π^{n−1}(K_{n−1}^{s}) & →^{δ^{#}} & Π^{n}(B_{n}) & → & 0, \end{aligned} (4)

where the upper row is part of the cohomological exact Serre sequence for the fibering p_{n}.

In Section 2 we shall prove

Proposition 1.7. Diagram (4) is commutative and its lower row is also an exact sequence.

Corollary 1.4, Lemma 1.5, and Proposition 1.7 imply that the following proposition is valid.

Proposition 1.8. The semiexact sequence (2) for the fibering p_{n} splits into short exact sequences

0 ⟶ Π^{n−1}(E_{n−1}) ⟶^{i^{#}} Π^{n−1}(K_{n−1}^{s}) ⟶^{δ^{#}} Π^{n}(B_{n}) ⟶ 0,

0 ⟶ Π^{n+k}(B_{n}) ⟶^{p^{#}} Π^{n+k}(E_{n−1}) ⟶ 0, k > 0.

Now consider segments of the exact sequences for the fiberings p_{n} and p_{n−1} from diagram (1) for the functors Π^{n} and π_{n}:

a) By Proposition 1.8, for cohomotopy functors Π^{n} we have

0 ⟶ Π^{n−1}(E_{n−1}) ⟶^{i^{#}} Π^{n−1}(K_{n−1}^{s}) ⟶^{δ^{#}} Π^{n}(B_{n}) ⟶ 0,

0 ⟶ Π^{n−2}(K_{n−2}^{t}) ⟶^{δ^{#}} Π^{n−1}(E_{n−1}) ⟶ 0.

After replacing the group Π^{n−1}(E_{n−1}) by the group Π^{n−2}(K_{n−2}^{t}), we shall have the exact sequence

0 ⟶ Π^{n−2}(K_{n−2}^{t}) ⟶^{δ_{n−1}} Π^{n−1}(K_{n−1}^{s}) ⟶ Π^{n}(B_{n}) ⟶ 0,

where the imbedding homomorphism δ_{n−1} is defined as the composition δ_{n−1} = i^{#}δ^{#}. We shall repeat our construction procedure this time for the space E_{n+1} (in the role of B_{n}). We have

Π^{n+1}(B_{n}) ≈ Π^{n+1}(E_{n−1}) ≈ Π^{n+1}(E_{n+1}).

After replacing the group Π^{n+1}(E_{n+1}) by the group Π^{n+1}(B_{n}), we obtain an exact sequence

0 ⟶ Π^{n−1}(K_{n−1}^{t′}) ⟶^{δ_{n}} Π^{n}(K_{n}^{s′}) ⟶ Π^{n+1}(B_{n}) ⟶ 0

and so forth.

b) For homotopy functors π_{n} we have

0 ⟶ π_{n}(E_{n−1}) ⟶^{p_{n#}} π_{n}(B_{n}) ⟶^{∂_{#}} π_{n−1}(K_{n−1}^{s}) ⟶^{i_{#}}

⟶^{i_{#}} π_{n−1}(E_{n−1}) ⟶^{p_{n#}} π_{n−1}(B_{n}) ⟶ 0,

0 ⟶ π_{n−1}(E_{n−1}) ⟶^{∂_{#}} π_{n−2}(K_{n−2}^{t}) ⟶ 0,

π_{n}(E_{n−1}) ≈ π_{n}(E_{n+1}).

After replacing the groups π_{n}(E_{n−1}) and π_{n−1}(E_{n−1}) respectively by the groups π_{n}(E_{n+1}) and π_{n−2}(K_{n−2}^{t}), we obtain an exact sequence

\begin{matrix}0 ⟶ π_{n}(E_{n+1}) ⟶ π_{n}(B_{n}) ⟶ π_{n−1}(K_{n−1}^{s}) ⟶^{∂_{n−1}} \\ ⟶^{∂_{n−1}} π_{n−2}(K_{n−2}^{t}) ⟶ π_{n−1}(B_{n}) ⟶ 0, \end{matrix}  (5)

where the homeomorphism ∂_{n−1} is defined as the composition ∂_{n−1} = ∂_{#}i_{#}. Note, besides, that the isomorphisms

(p_{n}p_{n−1})_{#} : π_{i}(E_{n+1}) ⟶ π_{i}(B_{n})

holds for i ≥ n + 1. On repeating our construction procedures this time for the space E_{n+1} (in the role of B_{n}), we obtain an exact sequence

\begin{matrix}0 ⟶ π_{n+1}(E_{n+2}) ⟶ π_{n+1}(E_{n+1}) ⟶ π_{n}(K_{n}^{s′}) ⟶^{∂_{n}} \\ ⟶^{∂_{n}} π_{n−1}(K_{n−1}^{t′}) ⟶ π_{n}(E_{n+1}) ⟶ 0. \end{matrix}  (6)

Moreover, we have

π_{n+1}(E_{n+1}) ≈ π_{n+1}(E_{n−1}) ≈ π_{n+1}(B_{n}).

After replacing the group π_{n+1}(E_{n+1}) in (6) by the group π_{n+1}(B_{n}), combining sequences (5) and (6) in the term π_{n}(E_{n+1}), and continuing our constructions, we obtain a gradually long exact sequence

⋯ ⟶ π_{n+1}(B_{n}) ⟶ π_{n}(K_{n}^{s′}) ⟶^{∂_{n}} π_{n−1}(K_{n−1}^{t′}) ⟶ π_{n}(B_{n}) ⟶

⟶ π_{n−1}(K_{n−1}^{s}) ⟶^{∂_{n−1}} π_{n−2}(K_{n−2}^{t}) ⟶ π_{n−1}(B_{n}) ⟶ 0.

Note that because of the fact that the groups π_{n−1}(B_{n}), π_{n}(E_{n+1}), and so forth are the finite ones the image of homomorphisms ∂_{i}, i ≥ n − 1, for an arbitrary element of the corresponding free abelian group contains some multiple of this element.

Let now X be an arbitrary space, n > 2, p ∈ Π^{n}(X), and f : S^{n} → X be an arbitrary mapping where S^{n} is the standard sphere. Consider the index

α = (S^{n}; f) ∈ ω(X; n)

and assume

ε(p)([f]) = p_{α}.

As follows from [3], we have correctly defined the mapping

ε(p) : π_{n}(X) → H^{n}(S^{n}).

In Section 2 we shall prove

Proposition 1.9. ε(p) is a group homomorphism.

For p = 0 we evidently have ε(p)([f]) = p_{α} = 0. Besides, let p, q ∈ Π^{n}(x). Then

ε(p + q)([f]) = [p + q]_{α} = p_{α} + q_{α} = ε(p)([f]) + ε(q)([f]).

Thus we have defined the homomorphism

ε : Π^{n}(X) → Hom(π_{n}(X), H^{n}(S^{n})), n > 2.

Proposition 1.10. If X = K_{n}^{t}, then ε is an isomorphism of dimension n.

Proof. Consider the space S^{(t)} = V_{k=1}^{t}S_{k}, S_{k} = S^{n}, k = 1, 2, …, t. Let i_{k} : S^{n} → S^{(t)} be the standard imbeddings. Fix the system of generators of the group π_{n}(K_{n}^{t}):

f_{k} : S^{n} → K_{n}^{t}, k = 1, 2, …, t.

Consider the commutative diagram

[figure: S^{n}; S^{(t)}; K_{n}^{t}; f_{k}; f; i_{k}]

where f = V f_{k}. It is understood that the homomorphism f^{∗} induced in cohomologies of dimension n will be an isomorphism. Let φ be an arbitrary homomorphism from

Hom(π_{n}(K_{n}^{t}), H^{n}(S^{n})).

Define an element h ∈ H^{n}(S^{(t)}) by the condition i_{k}^{∗}(h) = φ([f_{k}]), k = 1, 2, …, t. Let h_{0} = f^{∗−1}(h) ∈ H^{n}(K_{n}^{t}). Assume that

p = d_{n}(h) ∈ Π^{n}(K_{n}^{t}),

where d_{n} is the analogue of the Gurevich homomorphism in the Π theory. Then

ε(p)([f_{k}]) = p_{α_{k}} = f_{k}^{∗}(h_{0}) = (i_{k}^{∗}f^{∗})(f^{∗−1}(h)) = i_{k}^{∗}(h) = φ([f_{k}]).

Therefore ε(p) = φ and ε is an epimorphism. By applying the dimensional reasoning (Corollary 1.4) we now have Ker ε = 0.

Let

Σ = e^{∗−1}δ : H^{i}(X) → H^{i+1}(SX)

be the suspension isomorphism in cohomologies where δ is the coboundary homomorphism and

e : (CX, X) → (SX, ∗)

is the standard mapping. Consider the m-sphere S^{m} as the space S^{m}(S^{0}) where S^{0} is the 0-dimensional sphere. Fix the isomorphism

ξ_{1} : H^{n−1}(S^{n−1}) → Z

and set ξ_{2} = ξ_{1}Σ, where

Σ : H^{n−2}(S^{n−2}) → H^{n−1}(S^{n−1})

is the suspension isomorphism.

Let [f] ∈ π_{n−1}(K_{n−1}^{s}) and p ∈ Π^{n−1}(K_{n−1}^{s}). Define the isomorphism

ε_{1} : Π^{n−1}(K_{n−1}^{s}) → Hom(π_{n−1}(K_{n−1}^{s}), ℤ)

by the equality

ε_{1}(p)([f]) = ξ_{1}(ε(p)([f])).

Let now n > 4. Consider the diagram

Hom(π_{n−2}(K_{n−2}^{t}), ℤ) \underrightarrow{_{∂_{n−1}}} Hom(π_{n−1}(K_{n−1}^{s}), ℤ)
ε_{2}↓^{≈}  ^{ε_{1}}↓^{≈}
Π^{n−2}(K_{n−2}^{t})  ⟶^{δ_{n−1}}  Π^{n−1}(K_{n−1}^{s}),

(7)

where ∂_{n−1} and δ_{n−1} are the above-defined homomorphisms, ˜∂_{n−1} is the homomorphism dual to ∂_{n−1}, and the isomorphism ε_{2} is defined analogously to ε_{1}.

In Section 2 we shall prove

Proposition 1.11. Diagram (7) is commutative.

Theorem 1.12. Let X be a linearly and simply connected space for which homology groups H_{i}(X, ℤ) are finitely generated for all i and cohomology groups H^{i}(X, ℤ) = 0 for 1 ≤ i ≤ 4. Then for n > 2 we have

rank Π^{n}(X) = rank π_{n}(X),
Tor Π^{n+1}(X) ≈ Tor π_{n}(X).

Proof. Consider a sequence of fiberings constructed analogously to diagram (1) (by the conditions of the theorem X ∈ L_{5})

X = X_{5} ⟵^{k_{4}^{s_{5}}} X′_{6} ⟵^{k_{3}^{t_{5}}} X_{6} ⟵^{k_{5}^{s_{6}}} X′_{7} ⟵^{k_{4}^{t_{6}}} X_{7} ⟵ ⋯

⋯ ⟵ X_{n} ⟵^{k_{n−1}^{sn}} X′_{n+1} ⟵^{k_{n−2}^{tn}} X_{n+1} ⟵ ⋯,

where instead of the mappings we give the fibers of the corresponding fiberings. Note that this sequence is a certain Moore–Postnikov decomposition of the mapping ∗ → X (see [6]). Then, as follows from the above discussion (items a) and b)), we obtain short exact sequences

0 ⟶ Π^{n−2}(K_{n−2}^{t_{n}}) ⟶^{δ_{n−1}} Π^{n−1}(K_{n−1}^{s_{n}}) ⟶ Π^{n}(X) ⟶ 0,

where n > 4 and a long exact sequence

⋯ ⟶ π_{n}(X) ⟶ π_{n−1}(K_{n−1}^{s_{n}}) ⟶^{∂_{n−1}} π_{n−2}(K_{n−2}^{t_{n}}) ⟶ π_{n−1}(X) ⟶ ⋯

⋯ ⟶ π_{5}(X) ⟶ π_{4}(K_{4}^{s_{5}}) ⟶^{∂_{4}} π_{3}(K_{3}^{t_{5}}) ⟶ π_{4}(X) ⟶ 0.

Proposition 1.11 implies that the homomorphisms ∂_{i} and δ_{i}, i ≥ 4, can be regarded as homomorphisms dual to each other. Now, recalling the well-known results of homological algebra and taking into account the last remark of item b), we obtain the statement of the theorem for n > 4. For n = 4 we have Tor Π^{5}(X) ≈ Tor π_{4}(X), while the equality rank Π^{4}(X) = rank H^{4}(X) = rank π_{4}(X) = 0 is obvious. For n = 3 all groups from the statement of the theorem are trivial.

2. In this section we prove the statements that have been given in Section 1 without proof.

1. Proof of Theorem 1.3. It is more convenient for us to assume that the objects of subcategories L_{n} are finite CW -complexes without cells of dimension 1, 2, …, n − 2 (see [3]). Let X ∈ L_{n} and h ∈ H^{n}(X × X). Consider the diagram

X ⟶^{i′_{1}} X × X ⟵^{i′_{2}} X
↑ Δ
X

where i′_{1} and i′_{2} are the standard imbeddings and Δ is the diagonal mapping. Write h in the form

h = i_{1}^{′∗}(h) × 1 + 1 × i_{2}^{∗}(h),

where 1 ∈ H^{0}(X) = ℤ. Then

Δ^{∗}(h) = Δ^{∗}(i_{1}^{′∗}(h) × 1) + Δ^{∗}(1 × i_{2}^{′∗}(h)) =
= i_{1}^{′∗}(h) · 1 + 1 · i_{2}^{′∗}(h) =_{1} i^{′∗}(h) + i_{2}^{′∗}(h).

Thus we have the equality

Δ^{∗}(h) = i_{1}^{′∗}(h) + i_{2}^{′∗}(h)

which we shall use below without indicating that we do so.

Let

j_{t} : X_{1} × X_{2} → X_{t}, t = 1, 2,

be the standard projections and

i_{t} : X_{t} → X_{1} × X_{2}, t = 1, 2,

be the standard imbeddings. Consider the homomorphisms

j_{t}^{#} : Π^{n}(X_{t}) → Π^{n}(X_{1} × X_{2}), t = 1, 2.

Let us show that j_{t}^{#} are monomorphisms. Let p ∈ Π^{n}(X_{1}) and j_{1}^{#}(p) = 0. Consider the index

α = (X; f) ∈ ω(X_{1}; n)

and the commutative diagram

[figure: X; X_{1} × X_{2}; X_{1}; j_{1}; g_{1}; f]

where g_{1} = i_{1}f. Let

β = (X; g_{1}) ∈ ω(X_{1} × X_{2}; n).

Then

0 = [j_{1}^{#}(p)]_{β} = p_{α}.

Therefore p = 0. Applying an analogous procedure, we obtain that j_{2}^{#} is a monomorphism.

Now we shall show that

Jmj_{1}^{#} ∩ Jmj_{2}^{#} = 0.

Let

p = j_{1}^{#}(p_{1}) = j_{2}^{#}(p_{2}) ∈ Π^{n}(X_{1} × X_{2}),

where p_{t} ∈ Π^{n}(X_{t}), t = 1, 2. Consider an arbitrary index

α = (X; f) ∈ ω(X_{1} × X_{2}; n)

and the mappings

f_{1} = j_{1}f : X → X_{1},
f′ = i_{1}j_{1}f : X → X_{1} × X_{2}.

We have the commutative diagram

[figure: X_{1}; X_{1} × X_{2}; X; j_{i}; f_{1}; f; f′; .]

Consider the indices

α_{1} = (X; f_{1}) ∈ ω(X_{1}; n),
α′ = (X; f′) ∈ ω(X_{1} × X_{2}; n).

Then

p_{α} = [j_{1}^{#}(p_{1})]_{α} = [p_{1}]_{α_{1}},
p_{α′} = [j_{1}^{#}(p_{1})]_{α′} = [p_{1}]_{α_{1}}.

Therefore p_{α} = p_{α′}. Consider the commutative diagram

[figure: X; X_{1} × X_{2}; X_{2}; j_{2}; f′; 0]

and the index

α_{0} = (X; 0) ∈ ω(X_{2}; n),

where 0 is the constant mapping. Using Corollary 1.3 from [3], we obtain

P_{α′} = [j_{2}^{#}(p_{2})]_{α′} = [p_{2}]_{α_{0}} = 0.

Therefore p_{α} = p_{α′} = 0 and p = 0.

Let now p be an arbitrary element of the group Π^{n}(X_{1} × X_{2}). We shall show that

j_{1}^{#}(i_{1}^{#}(p)) + j_{2}^{#}(i_{2}^{#}(p)) = p.

Consider an arbitrary index

α = (X; f) ∈ ω(X_{1} × X_{2}; n)

and the commutative diagram

[figure: X × X; X; X_{1} × X_{2}; X; X; Δ; i′_{2}; i′_{1}; f_{2}; f_{1}; f_{1} × f_{2}]

where f_{t} = i_{t}j_{t}f, t = 1, 2. It is clear that (f_{1} × f_{2})Δ = f and X × X ∈ L_{n}. Consider the indices

α_{0} = (X × X; f_{1} × f_{2}),
α_{t} = (X; f_{t}), t = 1, 2,

where α_{0}, α_{1}, α_{2} ∈ ω(X_{1} × X_{2}; n). Now

p_{α_{1}} + p_{α_{2}} = i_{1}^{′∗}(p_{α_{0}}) + i′_{2}(p_{α_{0}}) = Δ^{∗}(p_{α_{0}}) = p_{α}.

Consider also the commutative diagram

[figure: X_{1}; X; X_{1} × X_{2}; X_{1} × X_{2}; j_{1}; i_{1}; f′; f_{1}; f]

where f′ = j_{1}f, and the index

α′ = (X; f′) ∈ ω(X_{1}; n).

Now

[j_{1}^{#}(i_{1}^{#}(p))]_{α} = [i_{1}^{∗}(p)]_{α′} = p_{α_{1}}.

By an analogous procedure we obtain [j_{2}^{#}(i_{2}^{#}(p))]_{α} = p_{α_{2}}. Now

[j_{1}^{#}(i_{1}^{#}(p)) + j_{2}^{#}(i_{2}^{#}(p))]_{α} = [j_{1}^{#}(i_{1}^{#}(p))]_{α} +
+[j_{2}^{#}(i_{2}^{#}(p))]_{α} = p_{α_{1}} + p_{α_{2}} = p_{α}.

2. Proof of Lemma 1.5. Consider the homotopy sequence of the fibering p_{n}

⋯ → π_{n−1}(K_{n−1}^{s}) → π_{n−1}(E_{n−1}) → π_{n−1}(B_{n}) → 0.

Hence it follows that we have π_{i}(E_{n−1}) = 0, i < n − 1, for the linear connected space E_{n−1}. The spaces B_{n} and K_{n−1}^{s} have finite-type modules of homologies. Therefore E_{n−1} also has a finite-type module of homologies. Thus E_{n−1} ∈ L_{n−1}. Since π_{n−2}(E_{n−1}) = 0, H^{n−1}(E_{n−1}) is a free group. Besides, the homomorphism

d_{n−1} : H^{n−1}(E_{n−1}) → Π^{n−1}(E_{n−1})

is an isomorphism. We have thereby proved the first item.

Now let us consider the diagram of the term E_{2} of a cohomological spectral sequence of the fibering p_{n}

[figure: ℤ; 0; 0; H^{n−1}(K_{n−1}^{s}); n − 1; n + 1; n; H^{n+1}(B_{n}); H^{n}(B_{n}); n; n + 1; τ; 0]

In our case the transgression τ is an epimorphism. Therefore H^{n}(E_{n−1}) = 0. Besides, it follows from the diagram that

p_{n}^{∗} : H^{n+1}(B) → H^{n+1}(E_{n−1})

is an isomorphism. We have thereby proved the third item and half of the second item.

The number t is defined by the quantity of generators of the free group

H^{n−1}(E_{n−1}) ≈ Ker τ ⊂ H^{n−1}(K_{n−1}^{s}).

If H^{n}(B_{n}) is a free group, then Ker τ = 0 and hence t = 0. Therefore in that case E_{n−1} ∈ L_{n+1}. Applying now our above reasoning to the fibering p_{n−1}, we obtain that E_{n+1} ∈ L_{n} and, moreover,

H^{n}(E_{n+1}) ≈ H^{n}(E_{n−1}) = 0.

Therefore E_{n+1} ∈ L_{n+1} ⊂ L_{n}. We have proved the fifth item. Let

α = (X_{n}; φ) ∈ ω(E_{n−1}; n)

be an arbitrary index (here and in the sequel, when an analogous situation occurs, it will be assumed that the objects of auxiliary subcategories L_{n} are the finite CW -complexes (see [3])). The mapping

φ : X_{n} → E_{n−1}

can be lifted to the mapping in the space of the fibering p_{n−1}

φ̃: X_{n} → E_{n+1}.

This follows from the fact that the obstruction to this lifting is zero (see [6])

c(φ) ∈ H^{n−1}(X_{n}, ⊕ℤ) = 0.

Consider the index

β = (E_{n+1}, p_{n−1}) ∈ ω(E_{n−1}; n).

Since p_{n−1}φ̃ = φ, we have α < β. Let p ∈ Π^{n}(E_{n−1}). Then p_{β} ∈ H^{n}(E_{n+1}) = 0. Therefore

p_{α} = φ^{∗}(p_{β}) = φ^{∗}(0) = 0.

Hence p = 0. We have thereby completely proved the second item. It remains to prove the fourth item.

Let p ∈ Π^{n+k}(E_{n−1}), k > 0, and let

φ : X_{n+1} → B_{n}

be an arbitrary mapping, X_{n+k} ∈ L_{n+k}. Obstructions to the lifting of the mapping φ and to the homotopy between two such liftings belong, respectively, to the groups (see [6])

H^{n}(X_{n+k}, ⊕ℤ) = 0,
H^{n−1}(X_{n+k}, ⊕ℤ) = 0.

Therefore we can choose up to homotopy a unique lifting φ̃ of the mapping φ

φ̃: X_{n+k} → E_{n+1},

p_{n}φ̃ = φ. Consider the indices

α = (X_{n+k}; φ) ∈ ω(B_{n}, n + k),
α̃ = (X_{n+k}; φ̃) ∈ ω(E_{n−1}, n + k).

Assume that

[ε(p)]_{α} = p_{α}.

As follows from [3], our definition is correct. Let us show that the set [ε(p)]_{α} defines an element ε(p) of the group Π^{n+k}(B_{n}). Consider the diagram

[figure: g; Y_{n+k}; X_{n+k}; B_{n}; E_{n−1}; ψ; ψ̃; φ̃; ,; p_{n}; φ]

where ψg ∼ φ, p_{n}ψ̃ = ψ, φ̃ = ψ̃g, Y_{n+k} ∈ L_{n+k}. Then

p_{n}φ̃ = p_{n}ψ̃g = ψg ∼ φ.

Also consider the indices

β = (Y_{n+k}; ψ) ∈ ω(B_{n}, n + k),

β̃ = (Y_{n+k}; ψ̃) ∈ ω(E_{n−1}, n + k).

Then α̃ < β̃ and we have

g^{∗}([ε(p)]_{β}) = g^{∗}(p_{β}) = p_{α} = [ε(p)]_{α}.

Thus we have defined the mapping

ε : Π^{n+k}(E_{n−1}) → Π^{n+k}(B_{n}).

Let now p ∈ Π^{n+k}(B_{n}) and

φ : X_{n+K} → B_{n}

be an arbitrary mapping. Consider the indices

α = (X_{n+k}; φ) ∈ ω(B_{n}; n + k),
α̃ = (X_{n+k}; φ̃) ∈ ω(E_{n−1}; n + k),
p_{n}(α̃) = (X_{n+k}; p_{n}φ̃) = (X_{n+k}; φ) = α ∈ ω(B_{n}; n + k).

Then

[ε(p_{n}^{#}(p))]_{α} = [p_{n}^{#}(p)]_{α} = p_{p_{n}(_{α})} = p_{α}.

Therefore εp_{n}^{#} = id.

Let further p ∈ Π^{n+k}(E_{n−1}) and

φ̃: X_{n+k} → E_{n−1}

be an arbitrary mapping. Consider the indices

α̃ = (X_{n+k}; φ̃) ∈ ω(E_{n−1}; n + k), α = p_{n}(α̃) = (X_{n+k}; p_{n}φ̃) ∈ ω(B_{n}; n + k).

We can use the mapping φ̃ as the lifting of p_{n}φ̃. Then

[p_{n}^{#}(ε(p))]_{α} = [ε(p)]_{p_{n}(_{α})} = [ε(p)]_{α} = p_{α}.

Therefore p_{n}^{#}ε = id. We have thereby completed the proof of the fourth item and Lemma 1.5.

3. Proof of Proposition 1.7. The commutativity of the left square follows from the fact that homomorphisms di are natural. Consider the right square where K_{n−1}^{s} = ΩK_{n}^{s}. Let h ∈ H^{n−1}(ΩK_{n}^{s}). Since B_{n} ∈ L_{n}, it suffices to check the equality

(δ^{#}d_{n−1})(h) = (d_{n}τ)(h)

only for the index

β = (B_{n}; id) ∈ ω(B_{n}; n).

The fibering

ΩK_{n−1}^{s} ⟶ E_{n−1} ⟶^{p_{n}} B_{n}

is obtained here from the standard fibering

ΩK_{n−1}^{s} ⟶ EK_{n}^{s} ⟶^{p} K_{n}^{s}

by the mapping

f : B_{n} → K_{n}^{s}.

Therefore

τ = f^{∗}τ_{0} = f^{∗}π^{∗−1}Σ,

where the mapping

π : ΣΩK_{n}^{s} → K_{n}^{s}

is the known mapping, the homomorphism

Σ : H^{n−1}(ΩK_{n}^{s}) → H^{n}(SΩK_{n}^{s})

is an isomorphism of the suspension in cohomologies, and τ_{0} = π_{∗}^{−1}Σ is the transgression of the fibering p. Now consider the commutative diagram

[figure: (B_{n}, ∗); π_{B}; (SΩB_{n}, ∗); e_{B}; (CΩB_{n}, ΩB_{n}); (E_{n−1}, ΩK_{n}^{s}); (SΩK_{n}^{s}, ∗); (K_{n}^{s}, ∗); (CΩK_{n}^{s}, ΩK_{n}^{s}); p_{n}; f; SΩf; π; e; CΩf; Φ]

where

E_{n−1} = {(b, e)|b ∈ B_{n}, e ∈ EK_{n}^{s}, f(b) = p(e)}.

Next define the lifting mapping Φ. But first note that when defining the cone CX, we contract the subspace

(X × 0) ∪ (∗ × I) ⊂ X × I

to the base point.

Let

σ ∈ ΩB_{n}, 0 ≤ t ≤ 1, 0 ≤ ξ ≤ 1.

Define the path s_{t} ∈ EK_{n}^{s} by the equality

s_{t}(ξ) = ((Ωf)(σ))(ξt)

and set

Φ([(σ, t)]) = (σ(t), s_{t}).

It is obvious that for t = 1 we have Φ|ΩB_{n} = Ωf. Consider the index

β_{1} = (ΩB_{n}; Ωf) ∈ ω(ΩK_{n}^{s}, n − 1).

Then we have

[δ^{∗}(d_{n−1}(h))]_{β} = ψ_{B}^{∗−1}(δ[d_{n−1}(h)]_{β_{1}}) = ψ_{B}^{∗−1}(δ((Ωf)^{∗}(h))),

where ψ_{B} = π_{B}e_{B}. On the other hand,

[d_{n}(τ(h))]_{β} = τ(h) = f^{∗}(τ_{0}(h)) = f^{∗}(π^{∗−1}(Σ(h))) =
= (f^{∗}π^{∗−1})(e^{∗−1}(δ(h))) = (π_{B}^{∗−1}(SΩf)^{∗}e^{∗−1})(δ(h)) =
= (π_{B}^{∗−1}e_{B}^{∗−1})((CΩf)^{∗}δ)(h) = ψ_{B}^{∗−1}(δ((Ωf)^{∗}(h))).

4. Proof of Proposition 1.9. It is obvious that ε(p)(0) = 0. Let now

f_{i} : S^{n} → X, i = 1, 2,

be arbitrary mappings. Let, moreover,

i_{k} : S^{n} → S^{n} ∨ S^{n}, k = 1, 2,

be standard imbeddings and

j_{k} : S^{n} ∨ S^{n} → S^{n}, k = 1, 2,

be standard projections. Also consider the mapping

ν : S^{n} → S^{n} ∨ S^{n}

which brings the H-cogroup structure onto the sphere. We have the commutative diagram

[figure: X; S^{n} ∨ S^{n}; S^{n}; S^{n}; S^{n}; f_{2}; i_{2}; ν; f_{1} ∨ f_{2}; f_{1}; i_{1}; .]

We introduce the notation (f_{1} ∨ f_{2})ν = f_{3} and consider the indices

α_{i} = (S^{n}; f_{i}), i = 1, 2, 3,
β = (S^{n} ∨ S^{n}; f_{1} ∨ f_{2}).

Then α_{i} < β, i = 1, 2, 3. Let e ∈ H^{n}(S^{n}) be the generator of the group H^{n}(S^{n}). Then the elements j_{1}^{∗}(e) and j_{2}^{∗}(e) are the generators of the group H^{n}(S^{n} ∨ S^{n}). Let

p_{β} = k_{1}j_{1}^{∗}(e) + k_{2}j_{2}^{∗}(e), k_{1}, k_{2} ∈ ℤ.

Then

p_{α_{3}} = ν^{∗}(p_{β}) = k_{1}ν^{∗}(j_{1}^{∗}(e)) + k_{2}ν^{∗}(j_{2}^{∗}(e)) = k_{1}e + k_{2}e,
p_{α_{1}} = i_{1}^{∗}(p_{β}) = k_{1}i_{1}^{∗}(j_{1}^{∗}(e)) + k_{2}i_{1}^{∗}(j_{2}^{∗}(e)) = k_{1}e + k_{2}0 = k_{1}e.

Analogously, p_{α_{2}} = k_{2}e. Therefore p_{α_{3}} = p_{α_{1}} + p_{α_{2}}. Then

ε(p)([f_{1}] + [f_{2}]) = p_{α_{3}} = p_{α_{1}} + p_{α_{2}} = ε(p)([f_{1}]) + ε(p)([f_{2}]).

5. Proof of Proposition 1.11. Let

g : S^{n−1} → K_{n−1}^{s}

be an arbitrary mapping. Consider the commutative diagram

[figure: (E_{n−1}, ∗); ig; (S^{n−1}, ∗); π = id; (CS^{n−2}, S^{n−2}); (S^{n−!}, ∗); (E_{n−1}, ∗); (E_{n+1}, K_{n−2}^{t}); (E_{n+1}, K_{n−2}^{t}); g^{=}; p_{n−1}; π_{s}; ig; p_{n−1}; ḡ; K_{c}; (CΩS^{n−1}, ΩS^{n−1}); e_{s}; e; (SΩS^{n−1}, ∗); (SS^{n−2}, ∗); π_{S}]

where the mapping

k : S^{n−2} → ΩS^{n−1} = ΩSS^{n−2}

and the mappings π_{s}, e_{s}, e are the standard ones, the mapping ḡ is obtained by lifting the mapping (ig)π_{s}e_{s}, and g^{=}= ḡk_{c}. Consider the mappings

g_{0} =g^{=} |S^{n−2} : S^{n−2} → K_{n−2}^{t},

g_{2} = ḡ|ΩS^{n−1} : ΩS^{n−1} → K_{n−2}^{t}.

Then we obviously have

∂_{n−1}([g]) = ∂_{#}([id]) = [g_{0}], g_{0} = g_{2}k.

We introduce the indices

β = (S^{n−1}; g) ∈ ω(K_{n−1}^{s}; n − 1),
β_{0} = (S^{n−2}; g_{0}) ∈ ω(K_{n−2}^{s}; n − 2),
β_{1} = (S^{n−1}; ig) ∈ ω(E_{n−1}; n − 1),
β_{2} = (ΩS^{n−1}; g_{2}) ∈ ω(K_{n−2}^{t}; n − 2).

Then β_{0} < β_{2}. Let p ∈ Π^{n−2}(K_{n−2}^{t}). Then

(˜∂_{n−1}(ε_{2}(p)))([g]) = ε_{2}(p)(∂_{n−1}([g])) = ξ_{2}(ε(p)(∂_{#}([ig]))) =
= ξ_{2}(p_{β_{0}}) = ξ_{1}(Σ(p_{β_{0}})) = ξ_{1}(Σ(K^{∗}(p_{β_{2}}))) = ξ_{1}((ΣK^{∗})(p_{β_{2}})).

But, on the other hand,

(ε_{1}(δ_{n−1}(p)))([g]) = ξ_{1}((ε(δ_{n−1}(p)))([g])) = ξ_{1}([δ_{n−1}(p)]_{β}) =
= ξ_{1}([i^{#}(δ^{#}(p))]_{β}) = ξ_{1}([δ^{#}(p)]_{β_{1}}) = ξ_{1}((ψ_{s}^{∗−1}δ)(p_{β_{2}})),

where ψ_{s} = π_{s}e_{s}. Since ξ_{1} is an isomorphism, it is sufficient for us to check the equality ΣK^{∗} = ψ_{s}^{∗−1}δ. Using the equality, π_{s}e_{s}K_{c} = e we shall have

ΣK^{∗} = (e^{∗−1}δ)(δ^{−1}K_{c}^{∗}δ) = e^{∗−1}K_{c}^{∗}δ =
= e^{∗−1}e^{∗}e_{s}^{∗−1}π_{s}^{∗−1}δ = ψ_{s}^{∗−1}δ.

References

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2. S. M. Khazhomia, On some properties of functors dual to homotopy functors (Russian). Trudy Tbiliss. Mat. Inst. Razmadze 91(1988) , 81-87.

3. S. Khazhomia, On cohomotopy-type functors. Georgian Math. J. 1(1994), No. 2, 151-171.

4. S. M. Khazhomia, On the dual homotopy functors. Bull. Acad. Sci. Georgia 146(1992), No.1, 13-16.

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6. E. H. Spanier, Algebraic topology. N.Y.: McGraw-Hill, 1966.

(Received 19.05.1993)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Scieces 1, Z. Rukhadze St., Tbilisi 380093 Republic of Georgia