Homology, Homotopy and Applications, vol.6(1), 2004, pp.109–135

ON THE HOMOTOPY TYPE OF A CHAIN ALGEBRA

MAHMOUD BENKHALIFA

(communicated by James Stasheff)

Abstract Let R be a P.I.D and let A be a dga over R. It is well-known that the graded homology modules H_{∗}(A) and T or_{∗}^{A}(R, R) alone do not suffice (in general) to determine the homotopy type of the dga A. J.H. Baues had built a more precise invariant, the “certain” exact sequence of Whitehead associated with A. Whitehead had built it for CW-complexes. In this work we explore this sequence to show how it can be used to classify the homotopy types of A.

1. Introduction

The classification of homotopy types is a classical and fundamental task in homotopy theory. There are only a few explicit results in the literature (see [4], [6], [7], [8], [9], [12], [17]) within the context of finite polyhedra and (see [4], [5], [6], [11]) within the context of differential graded algebras and differential graded Lie algebras. Recall that the existence of the models of Adams-Hilton [1], Anick [2], [3] and Quillen [16] justify the geometrical interest of these types of algebras.

Let R be a principal ideal domain P.I.D. We denote by DGA_{∗}(flat) the category of R-flat, differential, graded, associative, augmented and connected algebras (dga). Recall that DGA_{∗}(flat) is a cofibration category [5, 6], so an object cylinder is well defined in this category; hence we can define the notion of homotopy.

In [18] J.H.C. Whitehead has introduced a “certain” exact sequence associated with a simply connected CW-complex and derived the classification of 4-dimensional simply connected CW-complexes. After these results J.H. Baues proved, in [5], that the Whitehead sequence exists also for a dga A and showed that this sequence can establish a classification of the homotopy types of 4-dimensional simply connected dgas.

Let V = (V_{i})_{i⩾0} be a graded module, with V_{0} = 0. The tensor algebra T (V) is the graded algebra whose underlying graded module is the graded sum ⊕_{n⩾0}V^{⊗n}

(here V^{⊗0} = 0). We say that a dga A is free if, forgetting differentials, one has

The author gratefully acknowledges the referee for helpful comments and corrections. The author also would like to thank Professor James Stasheff for his help regarding the publication of this paper. Received July 7, 2003, revised March 3, 2004; published on April 5, 2004. 2000 Mathematics Subject Classification: 55Q15, 55U40. Key words and phrases: Differential graded algebra, Whitehead exact sequence, Detecting functor, Homotopy type. \begin{matrix}c \\ ◯ \end{matrix} 2004, Mahmoud Benkhalifa. Permission to copy for private use granted.

A ≅ T (V) for some V. In this case the differential on A is determined by its restriction V → T (V) and the graded module of indecompsable of A is isomorphic to V. A dga-morphism f : A → B is called a quasi-isomorphism if the induced morphism H_{∗}(f) : H_{∗}(A) → H_{∗}(B) is an isomorphism.

In [5] Baues showed that for each object A of DGA_{∗}(flat) there exists a quasiisomorphism (T (V), ∂) → A. The tensor algebra (T (V), ∂) is called a model of A. It is not unique up to isomorphism but it is unique up to homotopy.

Let (T (V), ∂ be a model of A, then the Whitehead exact sequence associated with A is the following long exact sequence:

⋯ → T or_{n+2}^{A}(R, R) ⟶^{b_{n+2}} Γ_{n} ⟶ H_{n}(A) ⟶ T or_{n+1}^{A}(R, R) ⟶^{b_{n+1}} ⋯

Recall that T or_{∗}^{A}(R, R) = H_{∗}(s^{−1}V, d), where s^{−1} is the desuspension graded homomorphism and where d is the linear part of the differential ∂ (see [4, 5]). Recall also that Γ_{n} is defined by setting:

Γ_{n} = ker(H_{n}(T (V_{⩽n})) ⟶ V_{n}).

In this paper we study the problem of the classification of homotopy types in the category DGA_{∗}(flat). How we can recognize the homotopy type of an object in DGA_{∗}(flat) and how we can compute the number of homotopy types of a certain object in DGA_{∗}(flat) satisfying some given data? Our tool to address this problem is the Whitehead exact sequence. Indeed this sequence can be taken apart to give the following characteristic elements which we call the Γ-system associated with A:

- a graded module T or_{n⩾2}^{A}(R, R),

- a family of homomorphisms (b_{n+2})_{n⩾2}, where b_{n+2} : T or_{n+2}^{A}(R, R) → Γ_{n},

- a family of extensions (i_{∗}(π_{n}) ∈ Ext(T or_{n+1}^{A}(R, R), coker b_{n+2}))_{n⩾2} , where π_{n} ∈ Ext(ker_{n+1}, cokerb_{n+2}) denotes the extension represented by the short exact se-

quence coker b_{n+2} ↣ H_{n}(A) ↠ ker b_{n+1} and where ker b_{n+1} ↪^{i} Tor_{n+1}^{A}(R, R).

We start by defining an algebraic category denoted Γ. Roughly speaking each object of this category is the Γ-system associated with a certain free dga. Then we define a surjective function:

F : ObDGA_{∗}(flat)⧸_{≃} → ObΓ,

by using the Γ-system associated with A. But unfortunately the function F is not a functor since it is not natural with respect to dga morphisms. Therefore, to make this correspondence natural, we introduce the subcategory ΓDGA_{∗}(flat) of DGA_{∗}(flat) whose morphisms are the dga morphisms α satisfying the condition (3.3) and thereby F becomes a functor DGA_{∗}(flat)⧸_{≃} → Γ which satisfies the properties of a “detecting functor”, a notion introduced by H.J. Baues in [5], which implies that the functor F induces the following results:

Theorem 1. Two objects A and B in DGA_{∗}(flat) have the same homotopy type if their Γ-systems are isomorphic.

Theorem 2. Homotopy types in the category ΓDGA_{∗}(flat)⧸_{≃} are in bijection with the classes of isomorphisms of objects in the category Γ.

Moreover, we have:

Theorem 3. Two objects A and B in ΓDGA_{∗}(flat) have the same homotopy type if and only if their Whitehead exact sequences are isomorphic.

If the condition (4.1) is satisfied, we can identify the category DGA_{∗}(flat) with its subcategory ΓDGA_{∗}(flat). Therefore, under this condition, we conclude that theorems 2 and 3 are also true in DGA_{∗}(flat).

The functor F is surjective on objects then for every Γ-system there exists a free dga (T (V), ∂) such that F ((T (V), ∂)) = Γ-system. When R is a field of any characteristic we show that the dga (T (V), ∂) coincides with the minimal model defined in [10]. So the notion of Γ-system is probably the best substitute for the minimal model in the case of differential algebras over a P.I.D R, rather than over a field.

In the last section we treat a particular case where the definition of the category Γ may be simplified. Indeed when the dga A is an object of ΓDGA_{n}^{3n+2}(flat) (the subcategory of ΓDGA_{∗}(flat) of which the objects are those satisfying the relations T or_{i}^{A}(R, R) = 0 for i ⩽ n and i ⩾ 3n + 3) we can denote the graded module Γ_{∗} simply by the graded module T or_{∗}^{A}(R, R). Then we define an algebraic category Γ_{n}^{3n+2} and a functor:

F_{n}^{3n+2} : ΓDGA_{n}^{3n+2}(flat)⧸_{≃} → Γ_{n}^{3n+2}

and we show that F_{n}^{3n+2} is also a “detecting functor”. Hence we derive the following homotopy classification theorem in ΓDGA_{n}^{3n+2}(flat):

Theorem 4. Homotopy types of objects in ΓDGA_{n}^{3n+2}(flat) are in bijection with the proper equivalence classes ( see definition 15) of tuples (b_{3n+2}, π_{3n}, ...., b_{2n+2}, π_{2n}) where b_{k+2} ∈ Hom(H_{k+2}, Γ_{k}) and where π_{k} ∈ Ext(H_{k+1}, coker b_{k+2}) for each k ⩽ 3n.

This article is organized as follows. In section 2, the Whitehead exact sequence associated with a dga is defined and its essential properties are given. Section 3 is devoted to the Γ-homotopy systems of order n, a notion needed to define the category Γ and to introduce the functor F and therefore to announce the main results in section 4. We conclude with some geometric applications and examples in section 5.

2. Whitehead exact sequence associated with a dga

In this section we give the definition and the essential properties of the Whitehead exact sequence associated to an associative differential graded algebra. Recall that Baues has constructed this sequence for dgas in [5] and he proved that it is an exact sequence.

Let A be a dga and let (T (V), ∂) be a model of A. Form the following long exact sequence:

⋯ → H_{n}(T (V_{⩽n})) ⟶^{j_{n}} V_{n} ⟶^{β_{n}} H_{n−1}(T (V_{⩽n−1})) → ⋯

where the connecting β_{n} is defined by:

β_{n}(v_{n}) = \overline{∂(v_{n}),}  (2.1)

where \overline{∂(v_{n})} ∈ H_{n−1}(T (V_{⩽n−1})) is the homology class of the (n − 1)-cycle ∂(v_{n}) ∈ T_{n−1}(V_{⩽n−1}), we define the graded module (Γ_{n})_{n⩾2} by setting:

Γ_{n} = ker(H_{n}(T (V_{⩽n})) ⟶^{j_{n}} V_{n}).  (2.2)

Recall that the linear part d of the differential ∂ is given by:

d_{n+1} = j_{n−1} ∘ β_{n} ∀n ⩾ 2.  (2.3)

The Whitehead exact sequence associated with the dga A is by definition ( see [5]) the following long exact sequence:

⋯ → T or_{n+2}^{A}(R, R) ⟶^{b_{n+2}} Γ_{n} ⟶ H_{n}(A) ⟶ T or_{n+1}^{A}(R, R) ⟶^{b_{n+1}} ⋯

where b_{n+2}(\overline{z}) = β_{n+1}(z).

Remark 1. Since V_{n} is free, for each n ⩾ 2, from the short exact sequence:

Γ_{n} ↣ H_{n}(T (V_{⩽n})) ↠ ker β_{n} ⊂ V_{n},

we deduce that:

H_{n}(T (V_{⩽n})) ≅ Γ_{n} ⊕ ker β_{n},  (2.4)

and in terms of the differential d_{n+2} : V_{n+1} → V_{n} we deduce the following decomposition:

V_{n+1} ≅ (Im d_{n+2})′ ⊕ ker d_{n+2},  (2.5)

where (Im d_{n+2})′ ⊂ V_{n+1} is a copy of Im d_{n+2} ⊂ V_{n}. Therefore the short exact sequence:

(Im d_{n+2})′ ↣^{d_{n+1}} ker d_{n+1} ↠ T or_{n+1}^{A}(R, R),

is a free resolution of the module T or_{n+1}^{A}(R, R).

- Since d_{n+2} ((Imd_{n+2})′) ⊂ ker β_{n} then the short exact sequence:

(Im d_{n+2})′ ↣^{d_{n+2}} ker β_{n} ↠ ker b_{n+1},  (2.6)

is a free resolution of the sub-module ker b_{n+1} ⊂ T or_{n+1}^{A}(R, R).

- According to the relations (2.5) and (2.4), if (z_{n+2,σ})_{σ∈∑} and (l_{n+2,σ′})_{σ′∈∑′} denote respectively the basis of the free modules ker d_{n+2} and (Im d_{n+2})′, the formula (2.1) can be written :

β_{n+1}(z_{n+2,σ} + l_{n+2,σ′}) = b_{n+2}(\overline{z_{n+2,σ}}) + φ_{n}(l_{n+2,σ′}) + d_{n+2}(l_{n+2,σ′}),  (2.7)

where φ_{n} : (Im d_{n+2})′ → Γ_{n} is a homomorphism given by the differential in ∂.

Proposition 1. Let A be a dga, then for each n ⩾ 2, we have:

H_{n}(A) ≅ \frac{coker b_{n+2} ⊕ ker β_{n}}{Im\overline{φ_{n}} ⊕ Im d_{n+2}} ,  (2.8)

where:

\overline{φ_{n}} : (Im d_{n+2})′ →^{φ_{n}} Γ_{n}↠coker b_{n+2}.

Proof. Let (T (V), ∂) be a model of A. From the following exact sequence:

V_{n+1} ⟶^{β_{n+1}} H_{n}(T (V_{⩽n})) → H_{n}(T (V_{⩽n+1})) → 0,

and the relation (2.4) we get:

H_{n}(T (V_{⩽n+1})) ≅ \frac{Γ_{n} ⊕ ker β_{n}}{Im β_{n+1}} .  (2.9)

Substituting the relation (2.7) in the formula (2.9) we get:

H_{n}(T (V_{⩽n+1})) ≅ \frac{Γ_{n} ⊕ ker β_{n}}{Im φ_{n} + Im b_{n+2} + Im d_{n+2}} ,  (2.10)

but T (V) is a model of A, so:

H_{n}(T (V_{⩽n+1})) = H_{n}(T (V)) = H_{n}(A),

thus the relation (2.10) can be written as:

H_{n}(A) ≅ H_{n}(T (V)) ≅ \frac{coker b_{n+2} ⊕ ker β_{n}}{Im \overline{φ_{n}} + Im d_{n+2}} ,

as desired.

It’s well know that the Whitehead exact sequence is natural with respect to dga morphisms; namely, a dga morphism f : A → B induces the following useful commutative diagram:

[figure: … → T orA_{n+2}(R, R) ⟶^{b_{n+2}} Γ_{n}^{A} ⟶ H_{n}(A) ⟶ T orA_{n+1}(R, R) ⟶^{b_{n+1}} ...; … → T orB_{n+2}(R, R) ⟶^{b′_{n+2}} Γ_{n}^{B} ⟶ H_{n}(B) ⟶ T orB_{n+1}(R, R) ⟶^{b′_{n+1}} ...; T or_{n+2}^{f} (R, R); (A); T or_{n+1}^{f} (R, R); γ_{n}^{f}; H_{n}(f)]

where γ_{∗}^{f} : Γ_{∗}^{A} → Γ_{∗}^{B} is the graded homomorphism induced by the dga morphism f. The commutativity of the above diagram induces the formula:

(T or_{n+1}^{f}(R, R))^{∗} ([H_{n}(B)]) = (\overline{γ_{n}^{f}})_{∗}([H_{n}(A)]),  (2.11)

where [H_{n}(A)] ∈ Ext(ker b_{n+1}, coker b_{n+2}), [H_{n}(B)] ∈ Ext(ker b′_{n+1}, coker b′_{n+2}) and where:

(T or_{n+1}^{f}(R, R))^{∗} : Ext(ker b′_{n+1}, coker b′_{n+2}) → Ext(ker b_{n+1}, coker b′_{n+2})
(\overline{γ_{n}^{f}})_{∗} : Ext(ker b_{n+1}, coker b_{n+2}) → Ext(ker b_{n+1}, coker b′_{n+2}).

Remark 2. Formula (2.11) means the following: consider:

(Im d_{n+2})′ →^{d_{n+2}} ker β_{n} ↠ ker b_{n+1}

(Im d′_{n+2})′ →^{d′_{n+2}} ker β′_{n} ↠ ker b′_{n+1},

as two free resolutions of ker b_{n+1} and ker b′_{n+1} respectively. To the given extensions [H_{n}(A)], [H_{n}(B)] and homomorphisms γ_{n}^{f}, T or_{n+1}^{f}(R, R) there correspond the following diagrams:

[figure: (Im d_{n+2})′ ↣^{d_{n+2}} ker β_{n} ↠ ker b_{n+1}; (Im d′_{n+2})′ ↣^{d′_{n+2}} ker β′_{n} ↠ ker b′_{n+1}; (Im d_{n+2})′ ↣^{d_{n+2}} ker β_{n} ↠ ker b_{n+1}; coker b′_{n+1}; coker b_{n+1}; \overline{φ_{n}}; coker b′_{n+1}; \overline{γ_{n}^{f}}; \overline{φ′_{n}}; ξ_{n+2}]

where [φ_{n}] = [H_{n}(A)] , [φ′_{n}] = [H_{n}(B)] and where \overline{γ_{n}^{f}} (respectively ξ_{n+2}) is the homomorphism induced by γ_{n}^{f} (respect. by T or_{n+1}^{f}(R, R)) on the quotient module coker b_{n+2} (respectively on the sub-module (Im d_{n+2})′).

The homomorphisms(\overline{γ_{n}^{f}})_{∗} and (T or_{n+1}^{f}(R, R))^{∗} satisfy the following relations:

(\overline{γ_{n}^{f}})_{∗}([H_{n}(A)]) = [\overline{γ_{n}^{f}} ∘ \overline{φ_{n}}]
(T or_{n+1}^{f}(R, R))^{∗}([H_{n}(B)]) = [\overline{φ′_{n}} ∘ ξ_{n+2}],

so the formula (2.11) is equivalent to the relation:

[\overline{γ_{n}^{f}} ∘ \overline{φ_{n}}] = [\overline{φ′_{n}} ∘ ξ_{n+2}] in Ext(ker b_{n+1}, coker b′_{n+2}),

or that there exists a homomorphism g_{n} : ker β_{n} ⟶ coker b′_{n+2} satisfying the relation:

\overline{γ_{n}^{f}} ∘ \overline{φ_{n}} − \overline{φ′_{n}} ∘ ξ_{n+2} = g_{n} ∘ d_{n+2}.  (2.12)

3. Γ-Homotopy systems of order n and their category

The notion of homotopy systems of order n was defined by Baues in [5] for CWcomplexes. For him a homotopy system of order n is a triple constituting with a CW-complex X^{n} of dimension n, a chain complex (C_{∗}, d) which coincides with the cellular chain complex C_{∗}(X^{n}) in dimension below n + 1 and a homomorphism of abelian groups f_{n+1} : C_{n+1} → π_{n}(X^{n}) which satisfies the cocycle condition f_{n+1} ∘ d_{n+2} = 0.

In this section we introduce the notion of homotopy systems of order n for dgas and their morphisms. Although this definition is completely different from Baues one but they express the same ideas.

Definition 1. Let n ⩾ 2. A Γ-homotopy system of order n is a triple (T (V, ∂^{n}), b_{n+2}, π_{n}) where: 1- T (V_{⩽n}, ∂^{n}) is an R-free graded algebra. 2- (V_{⩾1}, d_{∗}) is a positive chain complex. 3- b_{n+2} is a homomorphism of modules:

b_{n+2} : H_{n+2}(s^{−1}V_{∗}) → Γ_{n} ⊂ H_{n}(T (V_{⩽n})).

4- π_{n} is an extension such that:

π_{n} ∈ Ext(H_{n+1}(s^{−1}V_{∗}), coker b_{n+2}).

Recall that the R-module Γ_{n} is given by the formula (2.2) and recall that s^{−1} is the desuspension graded homomorphism.

Definition 2. A morphism between two Γ-homotopy systems (T (V, ∂^{n}), b_{n+2}, π_{n}) and (T (W, δ^{n}), b′_{n+2}, π′_{n}) of order n is a pair (ξ_{∗}, α^{n}) such that: α^{n} : T ((V_{⩽n}, ∂^{n})) → T (W_{⩽n}, δ^{n})) is a dga-morphism, ξ_{∗} : s^{−1}V_{∗} → s^{−1}W_{∗} is a chain map such that:

H_{i}(ξ_{∗}) = T or_{i}^{α^{n}}(R, R), ∀i ⩽ n + 1,

satisfying the following two conditions:

1 − b′_{n+2} ∘ H_{n+2}(ξ_{∗}) = γ_{n}^{α^{n}} ∘ b_{n+2}
2 − [H_{n+1}(ξ_{∗})]^{∗}(π′_{n}) = (γ_{n}^{α^{n}})_{∗}(π_{n}),

where the homomorphism γ_{n}^{α^{n}} is induced by α^{n} and where:

[H_{n+1}(ξ_{∗})]^{∗} : Ext(H_{n+1}(s^{−1}W_{∗}), coker b′_{n+2}) → Ext(H_{n+1}(s^{−1}V_{∗}), coker b′_{n+2})

(γ_{n}^{α^{n}})_{∗} : Ext(H_{n+1}(s^{−1}V_{∗}), coker b_{n+2}) → Ext(H_{n+1}(s^{−1}V_{∗}), coker b′_{n+2}).

Remark 3. As in the remark (2), the last condition means the following: Put H_{n+1} = H_{n+1}(s^{−1}V_{∗}) and H′_{n+1} = H_{n+1}(s^{−1}W_{∗}) and consider:

(Im d_{n+2})′ →^{d_{n+2}} ker d_{n+1} ↠ H_{n+1}

(Imd ′_{n+2})′ →^{d′_{n+2}} ker d′_{n+1} ↠ H′_{n+1},

as two free resolutions of H_{n+1} and H′_{n+1} respectively. To the given extensions π_{n} and π′_{n} and homomorphisms γ_{n}^{α^{n}}, H_{n+1}(ξ_{∗}) there corresponds the following diagram:

[figure: (Im d_{n+2})′ ↣^{d_{n+2}} ker d_{n+1} ↠ H_{n+1}; (Im d′_{n+2})′ ↣^{d′_{n+2}} ker d′_{n+1} ↠ H′_{n+1}; (Im d_{n+2})′ ↣^{d_{n+2}} ker d_{n+1} ↠ H_{n+1}; coker b′_{n+1}; coker b_{n+1}; \overline{φ_{n}}; coker b′_{n+1}; \overline{γ_{n}^{η}}; \overline{φ′_{n}}; ξ_{n+2}]

where [\overline{φ_{n}}] = π_{n}, [\overline{φ′_{n}}] = π′_{n} and where \overline{γ_{n}^{α^{n}}} is the homomorphism induced by γ_{n}^{α^{n}} on the quotient module coker b_{n+2}. The homomorphisms (\overline{γ_{n}^{α^{n}}})_{∗} and [H_{n+1}(ξ_{∗})]^{∗} satisfy the relation:

(\overline{γ_{n}^{α^{n}}})_{∗}(π_{n}) = [\overline{γ_{n}^{α^{n}}} ∘ \overline{φ_{n}}]
[H_{n+1}(ξ_{∗})]^{∗} (π′_{n}) = [\overline{φ′_{n}} ∘ ξ_{n+2}].

So the second condition is equivalent to the existence of a homomorphism h_{n} : ker d_{n+1} ⟶ coker b′_{n+2} satisfying the relation:

\overline{γ_{n}^{f}} ∘ \overline{φ_{n}} − \overline{φ′_{n}} ∘ ξ_{n+2} = h_{n} ∘ d_{n+2}.  (3.1)

Denote by H_{n} the category of Γ-homotopy systems of order n and their morphisms.

Definition 3 (see [5]). Let (ξ_{∗}, α^{n}) and (ξ′_{∗}, α^{′n}) be two morphisms in the category H_{n}. We say that they are homotopic in H_{n} and we write (ξ_{∗}, α^{n}) ≃ (ξ′_{∗}, α^{′n}), if α^{n} and α^{′n} are homotopic as a dga morphisms and if there exist homomorphisms σ_{k+1} : (s^{−1}V)_{k} → (s^{−1}W)_{k+1}, k ⩾ n, such that:

ξ_{k} − ξ′_{k} = σ_{k} ∘ d_{k} + d′_{k+1} ∘ σ_{k+1}, k ⩾ n + 1.

Denote by H_{n}⧸_{≃} the category whose objects are those of H_{n} and whose morphisms are the homotopy classes {(ξ_{∗}, α^{n})} of morphisms (ξ_{∗}, α^{n}) of H_{n}. We denote also by H_{n}^{n+1} the subcategory of H_{n} whose objects are the Γ-homotopy systems (T (V, ∂^{n}), b_{n+2}, π_{n}) of order n such that V_{i} = 0 for all i ⩾ n + 2.

3.1. Definition of the functor F_{n}^{n+1}

The Whitehead exact sequence introduced in the previous section allows us to define a function:

F_{n} : OBDGA_{∗}(flat)⧸_{≃} → OBH_{n}⧸_{≃},

as follows : let (T (V, ∂) ∈ OBDGA_{∗}(flat)⧸_{≃} and let:

⋯ → Tor_{n+2}^{T (V)}(R, R) ⟶^{b_{n+2}} Γ_{n} → H_{n}(T (V)) → Tor_{n+1}^{T (V)}(R, R) ⟶^{b_{n+1}} ⋯

be the Whitehead exact sequence associated with T (V).

This sequence gives us a homomorphism b_{n+2} : Tor_{n+2}^{T (V)}(R, R) → Γ_{n} and a short exact sequence coker b_{n+1} ↣ H_{n}(T (V)) ↠ ker b_{n+1}. But according to relation (2.8) we have:

H_{n}(A) ≅ \frac{coker b_{n+2} ⊕ ker β_{n}}{Im \overline{φ_{n}} + Im d_{n+2}} ,

where φ_{n}^{−} : (Im d_{n+1})′ ↣^{φ_{n}} Γ_{n} →^{pr} coker b_{n+1}.

Since the relation (2.3) implies ker β_{n} ⊂ ker d_{n+1} we set:

π_{n} = [\frac{coker b_{n+2} ⊕ ker d_{n+1}}{Im \overline{φ_{n}} + Im d_{n+2}} ] .  (3.2)

Therefore, π_{n} ∈ Ext(T or_{n+1}^{T (V)}(R, R), coker b_{n+2}). We define F_{n} by:

F_{n}(T (V)) = (T (V, ∂^{n}), b_{n+2}, π_{n}).

We now wish to define F_{n} on dga morphisms by setting F_{n}(α) = T or_{∗}^{α}(R, R), but unfortunately this definition is not natural with respect to dga morphisms since we have seen that if α is a dga morphism then the induced graded homomorphism T or_{∗}^{α}(R, R) implies the formula (2.12). But as we know, in order to be a morphism in H_{n} a graded homomorphism f_{∗} should satisfy the formula (3.1).

Therefore to make this correspondence natural with respect to dga morphisms, we define ΓDGA_{∗}(flat) as the subcategory of DGA_{∗}(flat) whose morphisms are the dga morphisms α satisfying the following condition: for all n ⩾ 2 :

[\overline{γ_{n}^{α}} ∘ \overline{φ_{n}}] = [\overline{φ′_{n}} ∘ ξ_{n+2}] in Ext(T or_{n+1}^{T (V)}(R, R), coker b′_{n+2}).  (3.3)

Note that this condition is trivial when we work over a field of any characteristic. Thus we define the functor:

F_{n} : ΓDGA_{∗}(flat)⧸_{≃} → H_{n}⧸_{≃},

by setting:

F_{n}(T (V)) = (T (V, ∂^{n}), b_{n+2}, π_{n})
F_{n}({α}) = {(ξ_{∗}, α^{n})},

where α^{n} denote the restriction of the dga-morphism α on T (V_{⩽n}, ∂^{n}) and where the ξ_{∗} is the chain transformation induces by the dga-morphism α on the indecomposables.

Corollary 1. The condition (3.3) implies that the morphism (ξ_{∗}, α^{n}) satisfies the second condition which define the morphisms in H_{n}.

Let DGA_{∗}^{n+2}(flat) be the subcategory of DGA_{∗}(flat) on which the objects A are those satisfying the relation T or_{i}^{A}(R, R) = 0 , ∀i ⩾ n + 3.

Remark 4. If A is an object of the category DGA_{∗}^{n+2}(flat) then a model T (V_{⩽n+1}, ∂) of A can be chosen such that V_{i} = 0 for each i ⩾ n + 2. Therefore T or_{n+2}^{A}(R, R) = ker d_{n+2} ⊂ V_{n+1}. So T or_{n+2}^{A}(R, R) is free.

We can also consider the functor:

F_{n}^{n+1} : ΓDGA_{∗}^{n+2}(flat)⧸_{≃} → H_{n}^{n+1}⧸_{≃},

given by the formula:

F_{n}^{n+1}(T (V_{⩽n+1}, ∂)) = (T (V_{⩽n+1}, ∂^{n}), b_{n+2}, π_{n})
F_{n}^{n+1}({α^{n+1}}) = {(ξ_{∗}, α^{n})},

which is the restriction of the functor F_{n} to the subcategory ΓDGA_{∗}^{n+2}(flat)⧸_{≃} of DGA_{∗}(flat)⧸_{≃}.

3.2. Properties of the functor F_{n}^{n+1} Now we approach the study of the functor F_{n}^{n+1}.

Theorem 5. A morphism α : T (V_{⩽n+1}, ∂) → T (W_{⩽n+1}, δ) in ΓDGA_{∗}^{n+2}(flat) is a quasi-isomorphism if and only if F_{n}^{n+1}(α) is an isomorphism in H_{n}^{n+1}⧸_{≃}.

Proof. Recall that by definition of the functor F_{n}^{n+1} we have the formulas:

F_{n}^{n+1}(T (V_{⩽n+1}, ∂)) = (T (V_{⩽n+1}, ∂^{n}), b_{n+2}, π_{n})
F_{n}^{n+1}(T (W_{⩽n+1}, δ)) = (T (W_{⩽n+1}, δ^{n}), b′_{n+2}, π′_{n}),

and the relation:

T or_{∗}^{T (V_{⩽n+1})}(R, R) = H_{∗}(s^{−1}V_{⩽n+1}, d_{∗})
T or_{∗}^{T (W_{⩽n+1})}(R, R) = H_{∗}(s^{−1}W_{⩽n+1}, d′_{∗}).

So if F_{n}^{n+1}(α) is an isomorphism in H_{n}^{n+1}⧸_{≃} then we deduce that the graded homomorphism:

T or_{∗}^{α}(R, R) : T or_{∗}^{T (V_{⩽n+1})}(R, R) → T or_{∗}^{T (W_{⩽n+1})}(R, R),

is an isomorphism and then we apply the classical theorem of Moore [13, 15] which asserts that a dga morphism f in DGA_{∗}(flat) is a quasi-isomorphism if and only if T or_{∗}^{f}(R, R) is a quasi-isomorphism as a chain morphism.

Theorem 6. For each Γ-homotopy system (T (V_{⩽n+1}, ∂^{n}), b_{n+2}, π_{n}) in H_{n}^{n+1}, there exists a free dga T (V_{⩽n+1}, ∂) in DGA_{∗}^{n+2}(flat) such that:

F_{n}^{n+1}(T (V_{⩽n+1}, ∂)) = (T (V_{⩽n+1}, ∂^{n+1}), b_{n+2}, π_{n})
T or_{n+2}^{T (V_{⩽n+1})}(R, R) = H_{n+2}(s^{−1}V_{⩽n+1}).

Proof. Consider:

(Im d_{n+2})′ ↣^{d_{n+2}} ker β_{n} ↠ ker b_{n+1},

as a free resolution of ker b_{n+1}. The inclusion ker b_{n+1} ↪^{i} H_{n+1}(s^{−1}V_{∗}) implies the homomorphism:

Ext(H_{n+1}(s^{−1}V_{∗}), coker b_{n+2}) ⟶^{i_{∗}} Ext(ker b_{n+1}, coker b_{n+2}).

Therefore for the given extension π_{n} ∈ Ext(H_{n+1}(s^{−1}V_{∗}), coker b_{n+2}) and the homomorphism b_{n+2}, there exist homomorphisms μ_{n+2}, φ_{n} which make the following diagram commutative:

ker d_{n+2}  Z_{n}(T (V_{⩽n})) μ_{n+2}  Γ_{n} ↠ coker b_{n+2} [figure] b_{n+2}  [figure] (Im d_{n+2})′ 0 pr  ker β_{n} [figure] ker b_{n+1} φ_{n} \overline{φ_{n}}

where [\overline{φ_{n}}] = i_{∗}(π_{n}). Using the decomposition of the module V_{n+1} = (Im d_{n+2})′ ⊕ker d_{n+2} in the relation (2.5), we define ∂ on T (V_{⩽n+1}) by the formulas:

∂_{n+1}(z_{n+2,σ} + l_{n+2,σ′}) = d_{n+2}(l_{n+2,σ′}) + φ_{n}(l_{n+2,σ′}) + μ_{n+2}(z_{n+2,σ})
∂_{⩽n} = ∂_{⩽n}^{n},

where (z_{n+2,σ})_{σ∈∑} and (l_{n+2,σ′})_{σ′∈∑′} denote respectively bases of the free submodules ker d_{n+2} and (Im d_{n+2})′. Note that by the above diagram the element:

φ_{n}(l_{n+2,σ′}) + μ_{n+2}(z_{n+2,σ}) ∈ Z_{n}(T (V_{⩽n})),

hence ∂_{⩽n}^{n}(φ_{n}(l_{n+2,σ′}) + μ_{n+2}(z_{n+2,σ})) = 0. Since ∂_{⩽n}^{n}(d_{n+2}(l_{n+2,σ′})) = d_{n+1} (d_{n+2}(l_{n+2,σ′})) = 0 we deduce that ∂ is a differential on T (V_{⩽n+1}). It’s easy to see that the Whitehead exact sequence associated with the dga A = (T (V_{⩽n+1}), ∂) can be written:

ker d_{n+2} = T or_{n+2}^{A}(R, R) ⟶^{b_{n+2}} Γ_{n} → H_{n}(A) → T or_{n+1}^{A}(R, R) → ⋯

then according to the definition of the functor F_{n}^{n+1}, it easy to check that:

F_{n}^{n+1}(A) = (T (V_{⩽n+1}, ∂^{n}), b_{n+2}, π_{n})
T or_{⩽n+2}^{A}(R, R) = H_{⩽n+2}(s^{−1}V_{⩽n+1}),

and the proof is completed.

Theorem 6 allows us to describe an action of the group Hom(T or_{n+2}^{T (V)}(R, R), Γ_{n}) on the set OBDGA_{∗}(flat) as follows:

Corollary 2. Given a free dga (T (V), ∂), let b_{n+2} ∈ Hom(T or_{n+2}^{T (V)}(R, R), Γ_{n}). Perturb the differential ∂^{n+1} in the dga (T (V_{⩽n+1}, ∂^{n+1}) by setting:

∂^{′n+1}(z_{n+2,σ} + l_{n+2,σ′}) = ∂^{n+1}(z_{n+2,σ} + l_{n+2,σ′}) + μ_{n+2}(z_{n+2,σ}).  (3.4)

For a given extension π_{n} ∈ Ext(T or_{n+1}^{T (V)}(R, R), coker b′_{n+2}), also perturb ∂^{′n+1} to obtain a new differential δ^{n+1} of T (V_{⩽n+1}) by setting:

δ^{n+1}(z_{n+2,σ} + l_{n+2,σ′}) = ∂^{′n+1}(z_{n+2,σ} + l_{n+2,σ′}) + d_{n+2}(l_{n+2,σ′})  (3.5)
+φ_{n}(l_{n+2,σ′}),  (3.6)

so that the differential δ is defined by the formulas:

δ^{i} = ∂^{′i} = ∂^{i} ∀i ≠ n + 1
δ^{n+1} = ∂_{n+1}^{′n+1} + d_{n+2} + φ_{n} = ∂^{n+1} + d_{n+2} + μ_{n+2}.

Definition 4. The dga (T (V_{⩽n+1}), ∂) constructed below is called the dga associated with the Γ-homotopy system (T (V_{⩽n+1}, ∂^{n}), b_{n+2}, π_{n}) of order n.

Definition 5. We call the pair (b_{n+2}, π_{n}) an adapted couple for the dga (T (V_{⩽n}), ∂^{n}).

Definition 6. A dga morphism α^{n} : (T (V_{⩽n}), ∂^{n}) → (T (W_{⩽n}), δ^{n}) is said to be n-diagonal if α^{n} maps T (V_{⩽n−1}) onto T (W_{⩽n−1}) and the direct factor ker d_{n+1} of V_{n} onto the module W_{n}.

Lemma 1. If α^{n} : (T (V_{⩽n}), ∂^{n}) → (T (W_{⩽n}), δ^{n}) is n-diagonal then according to the splitting H_{n}(T (V_{⩽n})) ≅ Γ_{n} ⊕ ker b_{n+1} (respectively. H_{n}(T (W_{⩽n})) ≅ Γ′_{n} ⊕ ker b′_{n+1}), the homomorphism:

H_{n}(α^{n}) : H_{n}(T (V_{⩽n})) → H_{n}(T (W_{⩽n})),

splits into:

H_{n}(α^{n}) = γ_{n}^{α^{n}} ⊕ ξ_{n+1},  (3.7)

where γ_{n}^{α^{n}} is the homomorphism induced by α^{n} on the sub-module Γ_{n}, and where ξ_{n+1} : ker β_{n} → ker β′_{n} is the chain transformation induced by α^{n} on the indecomposables.

Theorem 7. Let (T (V_{⩽n+1}, ∂) and (T (W_{⩽n+1}, δ) be two free dgas in DGA_{∗}^{n+2}(flat) and let:

(ξ_{∗}, α^{n}) : F_{n}^{n+1} ((T (V_{⩽n+1}, ∂)) → F_{n}^{n+1}((T (W_{⩽n+1}, δ)),

be a morphism in H_{n}^{n+1} such that α^{n} is n-diagonal. Then there exists a dgamorphism Λ : (T (V_{⩽n+1}), ∂) → (T (W_{⩽n+1}), δ) such that Λ is (n + 1)-diagonal and satisfies:

F_{n}^{n+1}(Λ) = H_{n+2}(ξ_{∗}).  (3.8)

In order to prove the theorem we consider the following diagram:

V_{n+1}  W_{n+1} [figure] [figure]  [figure] [figure] ξ_{n+2} β_{n+1}  β′_{n+1} j_{n}  j′_{n} d′_{n+2} d_{n+2} V_{n}  W_{n} ⋮  ⋮ ξ_{n+1} H_{n}(α^{n}) H_{n}(T (V_{⩽n}))  H_{n}(T (W_{⩽n}))

- where the left and right triangles commute by the definition of the linear differentials,

- where the lower trapezoid commutes by the definition of the given morphism (ξ_{∗}, α^{n}),

- where the central square commutes by the definition of ξ_{∗}.

Since j′_{n}(H_{k}(α^{n}) ∘ β_{n+1} − β′_{n+1} ∘ ξ_{n+2}) = 0, then:

Im (H_{n}(α^{n}) ∘ β_{n+1} − β′_{n+1} ∘ ξ_{n+2}) ⊂ Γ′_{n}.

We begin by giving the following lemma:

Lemma 2. There exists a homomorphism g_{n} : V_{n} → Γ′_{n} such that

H_{n}(α^{n}) ∘ β_{n+1} − β′_{n+1} ∘ ξ_{n+2} = g_{n} ∘ d_{n+1} (mod Im b′_{n+2}).

Proof. Since the dga morphism α^{n} is n-diagonal, the relations (2.7) and (3.7) give us an explicit expression of the homomorphism H_{n}(α^{n}) ∘ β_{n+1} − β′_{n+1} ∘ ξ_{n+2} so it’s easy to show that:

H_{n}(α^{n}) ∘ β_{n+1} − β′_{n+1} ∘ ξ_{n+2} = γ_{n}^{α^{n}} ∘ φ_{n} − φ′_{n} ∘ ξ_{n+2}.

Since pr ∘ (γ_{n}^{α^{n}} ∘ φ_{n} − φ′_{n} ∘ ξ_{n+2}) = \overline{γ_{n}^{α^{n}}} ∘ \overline{φ_{n}} − \overline{φ′_{n}} ∘ ξ_{n+2} and according to the condition (2) defining a morphism in the category H_{n} and remark 3, there exists a homomorphism g_{n} which makes the following diagram commutative after composition with the projection pr:

Γ′_{n}  coker b′_{n+2} γ_{n}^{α^{n}}φ_{n} − φ′_{n}ξ_{n+2}  g_{n} 0  (Im d_{n+2})′  ker d_{n+1} [figure] pr d_{n+2}

Since V_{n} = (Im d′_{n+1}) ⊕ ker d_{n+1} then we can extend g_{n} to V_{n} and the proof is completed.

Proof. (of theorem (7)) According to lemma 2 we have:

H_{n}(α) ∘ β_{n+1} − β′_{n+1} ∘ ξ_{n+2} − g_{n} ∘ d_{n+2} : (Im d′_{n+2}) → Im b′_{n+2},

or:

(H_{n}(α) − g_{n} ∘ j_{n}) ∘ β_{n+1} − β′_{n+1} ∘ ξ_{n+2} : (Im d′_{n+2}) → Im b′_{n+2}.

As a result there exists a homomorphism λ_{n+1} which makes the upper triangle in the following diagram commutative:

ker d′_{n+2} [figure] [figure] Z_{n}(T (W_{⩽n})) V_{n+1}  Im b′_{n+2} ⊂ Γ′_{n}^{(H_{n}(α^{n}) − g_{n}j_{n})β_{n+1} − β′_{n+1}ξ_{n+2}}  pr λ_{n+1} ^{(α^{n} − h_{n})∂ − δξ_{n+2}}

where the homomorphism h_{n} : V_{n} → Z_{n}(T (W_{⩽n})) satisfies the relation pr ∘h_{n} = g_{n} and where Z_{n}(T (W_{⩽n})) is the module of n-cycles of the dga T (W). Choose (z_{n+2,σ})_{σ∈∑} and (l_{n+2,σ′})_{σ′∈∑′} respectively as bases of the free sub-modules (Im d_{n+2})′ and ker d_{n+2}. Recall that:

V_{n+1} ≅ (Im d_{n+2})′ ⊕ (ker d_{n+2}).

By the commutativity of the above diagram, for each σ ∈ Σ and σ′ ∈ Σ′, there exists an element t_{n+2,σ,σ′} ∈ T_{n+1}(W_{⩽n}) such that:

((α_{n} − h_{n})∂ − δξ_{n+2})(z_{n+2,σ} + l_{n+2,σ′}) = δ ∘ λ_{n+1}(l_{n+2,σ′})  (3.9)
+δ(t_{n+2,σ,σ′}).

If z_{n+2,σ} + l_{n+2,σ′} ∈ V_{n+1}, then we define Λ : (T (V_{⩽n+1}), ∂) ⟶ (T (W_{⩽n+1}), δ) by setting:

Λ_{n+1}(z_{n+2,σ} + l_{n+2,σ′}) = λ_{n+1}(l_{n+2,σ,σ′}) + t_{n+2,σ,σ′} + ξ_{n+2}(z_{n+2,σ} + l_{n+2,σ′})
Λ_{n} = α_{n} − h_{n}
Λ_{i} = α_{i}  i ≠ n, n + 1.

Λ is a dga morphism i.e. Λ∘∂ = δ ∘Λ. Indeed, since for each z_{n+2,σ} +l_{n+2,σ′} ∈ V_{n+1} we have ∂(z_{n+2,σ} + l_{n+2,σ′}) ∈ Z_{n} (T (W_{⩽n})) , then according to the definition of Λ we have

Λ ∘ ∂(z_{n+2,σ} + l_{n+2,σ′}) = (α_{n} − h_{n}) ∘ ∂(z_{n+2,σ} + l_{n+2,σ′}),

and, on the other hand, according to the relation (3.9) and the definition of Λ, we have

δ ∘ Λ_{n+1}(z_{n+2,σ} + l_{n+2,σ′}) = δ ∘ λ_{n+1}(l_{n+2,σ′}) + δ ∘ ξ_{n+2}(z_{n+2,σ} + l_{n+2,σ′})
+ δ(t_{n+2,σ′})
= ((α_{n} − h_{n}) ∘ ∂ − δ ∘ ξ_{n+2})((z_{n+2,σ} + l_{n+2,σ′}))
+ δ ∘ ξ_{n+2}(z_{n+2,σ} + l_{n+2,σ′})
= (α_{n} − h_{n}) ∘ ∂((z_{n+2,σ} + l_{n+2,σ′})).

Therefore Λ is a dga morphism. Finally by the definition, it’s clear that α is (n+1)- diagonal and satisfies the relation (3.8).

Following H.J. Baues in [5], let A and B be two categories and let F : A → B be a functor. We say that F is a “detecting functor ” if the following conditions are satisfied: 1 -A morphism α : A_{1} → A_{2} in the category A is an isomorphism if and only if F (α) : F (A_{1}) → F (A_{2}) is an isomorphism. 2 - For each object B in the category B, there exists an object A in A such that F (A) = B. 3 - Let A_{1} and A_{2} be two objects in A and let β : F (A_{1}) → F (A_{2}) be a morphism in B, there exists a morphism α : A_{1} → A_{2} verifying F (α) = β. We summarize the properties of the functor F_{n}^{n+1} in the following result:

Theorem 8. The functor:

F_{n}^{n+1} : ΓDGA_{∗}^{n+2}(flat)⧸_{≃} → H_{n}^{n+1}⧸_{≃},

is a detecting functor.

4. The category Γ

4.1. Notion of Γ-system and Γ-morphism Roughly speaking, each object of the category Γ gives a long exact sequence which is the Whitehead exact sequence associated with a certain object in DGA_{∗}(flat). Given a graded R-module H_{∗}, choose a free chain complex (s^{−1}V_{∗}, d) such that H_{∗} = H_{∗}(s^{−1}V_{∗}, d) .

Definition 7. A Γ-system is a triple (H_{n}, b_{n+2,}π_{n})_{n⩾2} where, for each n, (b_{n+2,}π_{n}) is an adapted couple (see definition 5) for the dga (T (V_{⩽n}), ∂^{n}) which is constructed from the dga (T (V_{⩽2}), d) and the pairs (b_{k+2,}π_{k})_{2⩽k⩽n−1} according to theorem 6.

Thus if we iterate this process we find a free dga (T (V), ∂) having the following property: for each n ⩾ 2 we have:

F_{n}((T (V), ∂)) = (T (V, ∂^{n}), b_{n+2}, π_{n}).

Definition 8. The dga (T (V), ∂) constructed below is called the dga associated with the Γ-system (H_{n}, b_{n+2}, π_{n})_{⩾2}.

Remark 5. The (T (V), ∂) is not unique but we shall prove later that (T (V), ∂) is unique up to homotopy.

Let (H_{n}, b_{n+2}, π_{n})_{n⩾2} and (H′_{n}, b′_{n+2}, π′_{n})_{n⩾2} be two Γ-systems and let f_{∗} : H_{∗} → H′_{∗} be a graded homomorphism of degree 0. We say that f_{∗} is a Γ-morphism if f_{∗} satisfies the following inductive construction: let (T (V), ∂) and (T (W), δ) be two free dgas associated respectively with the given Γ-systems. By virtue of the homotopy extension theorem [14], there exists a chain transformation ξ_{∗} : (s^{−1}V, d) → (s^{−1}W, d′) such that H_{∗}(ξ_{∗}) = f_{∗}. We begin by considering the dga morphism α^{2} : (T (V_{⩽2}), d) → (T (W_{⩽2}), d′) given by:

α^{2} = ξ_{3} on V_{2}
α^{2} = ξ_{2} on V_{1}.

Since ker d_{3} = H_{3} we deduce that (ξ_{∗}, α^{2}) is a morphism in the category H_{2}^{3} and it’s clear that α^{2} is 2-diagonal by the definition, so theorem 7 gives a dga morphism α^{3} : (T (V_{⩽3}), ∂^{3}) → (T (W_{⩽3}), δ^{3}) which is 3-diagonal and such that T or_{i}^{α^{3}}(R, R) = f_{i} for all i ⩽ 4. Suppose now that we have built a dga morphism:

α^{n} : (T (V_{⩽n}), ∂^{n}) → (T (W_{⩽n}), δ^{n}),

by using the above process, such that α^{n} is n-diagonal and such that:

T or_{i}^{α^{n}}(R, R) = f_{i} for all i ⩽ n + 1.

If we assume that (ξ_{∗}, α^{n}) is a morphism in H_{n}^{n+1}, then theorem 7 allows us to construct a dga morphism α^{n+1} : (T (V_{⩽n+1}), ∂^{n+1}) → (T (W_{⩽n+1}), δ^{n+1}) which is (n + 1)-diagonal and such that:

T or_{i}^{α^{n+1}}(R, R) = f_{i} for all i ⩽ n + 2.

Thus if we iterate this process for all n ⩾ 2 we find a dga morphism α : (T (V), ∂) → (T (W), δ) satisfying the following properties: 1- T or_{∗}^{α}(R, R) = f_{∗}. 2- For each n ⩾ 2, (ξ_{∗}, α^{n}) is a morphism in the category H_{n}, where α^{n} : T (V_{⩽n}), ∂) → (T (W_{⩽n}), δ) is the restriction of α.

Definition 9. A graded homomorphism f_{∗} : H_{∗} → H′_{∗} satisfying the above inductive condition for each n ⩾ 2 is called a Γ-morphism.

Definition 10. The dga morphism α : (T (V), ∂) → (T (W), δ) constructed by the above process is called the dga morphism associated with the Γ-morphism f_{∗}.

Remark 6. Since T or_{∗}^{α}(R, R) = f_{∗} we deduce that the dga morphism α constructed below satisfies condition (3.3). So it is a morphism in the category ΓDGA_{∗}(flat).

4.2. Definition of the category Γ and the functor F Now we are able to give the following definition:

Definition 11. The category Γ is defined as follows: Objects: the Γ-systems defined in the definition (7) Morphisms: the Γ-morphisms defined in the definition (9)

Thus we define the functor F : ΓDGA_{∗}(flat)⧸_{≃} → Γ by setting:

F ((T (V), ∂)) = (T or_{k}^{T (V)}(R, R), b_{k+2}, π_{k})_{k⩾2}
F ({α}) = T or_{∗}^{α}(R, R).

We now give some properties of the functor F and we show that the functor satisfies the properties of a “detecting functor”. We begin with the following propositions:

Proposition 2. Let (T (V), ∂) and (T (W), δ) be two free dgas. If the Γ-systems (T or_{k}^{T (V)}(R, R), b_{k+2}, π_{k})_{k⩾2} and (T or_{k}^{T (W)}(R, R), b′_{k+2}, π′_{k})_{k⩾2} are isomorphic in the category Γ, then the given dgas have the same homotopy type.

Proof. Let f_{∗} : (T or_{k}^{T (V)}(R, R), b_{k+2}, π_{k})_{k⩾2} → (T or_{k}^{T (W)}(R, R), b′_{k+2}, π′_{k})_{k⩾2} be the isomorphism between the two Γ-systems and let α : (T (V), ∂) → (T (W), δ) be the dga morphism associated with the Γ-morphism f_{∗} (see definition 10). Since T or_{∗}^{α}(R, R) = f_{∗} and again we apply the classical Moore’s theorem to get the result.

Proposition 3. A dga-morphism α is a quasi-isomorphism if and only if F (α) is an isomorphism.

Proof. Obvious since F (α) = T or_{∗}^{α}(R, R).

Remark 7. ¿From the proposition 3, we conclude that the dga morphism α associated to the Γ-morphism f_{∗} is unique up to homotopy.

Now we are able to announce the main theorem in this paper:

Theorem 9. The functor:

F : ΓDGA_{∗}(flat)⧸_{≃} → Γ,

is a detecting functor.

Proof. Definition 7 implies that to each Γ-system is associated a dga (T (V), ∂) such that F (T (V)) is the given Γ-system. Definition 10 implies that to each Γ-morphism is associated a morphism in ΓDGA_{∗}(flat) which has for image, by the functor F, the given Γ-morphism. We conclude the proof by proposition 3.

As a consequence of theorem 9 we deduce the following result:

Theorem 10. Homotopy types in the category ΓDGA_{∗}(flat)⧸_{≃} are in bijection with the classes of isomorphisms of objects in the category Γ.

Moreover, we derive the following theorem:

Theorem 11. Two objects of the category ΓDGA_{∗}(flat) have the same homotopy type if and only if their Whitehead exact sequences are isomorphic.

Now we give a relation identifying the subcategory ΓDGA_{∗}(flat) with DGA_{∗}(flat).

Proposition 4. Let:

⋯ → T or_{n+2}^{T (V)}(R, R) ⟶^{b_{n+2}} Γ_{n} ⟶ H_{n}(T (V)) ⟶ T or_{n+1}^{T (V)}(R, R) ⟶^{b_{n+1}} ⋯

⋯ → T or_{n+2}^{T (W)}(R, R) ⟶^{b′_{n+2}} Γ′_{n} ⟶ H_{n}(T (W)) ⟶ T or_{n+1}^{T (W)}(R, R) ⟶^{b′_{n+1}} ⋯

be the Whitehead exact sequences associated respectively with two free dgas (T (V), ∂) and (T (W), δ). If

Ext (\frac{T or_{n+1}^{T (V)}(R, R)}{ker b_{n+1}}, coker b′_{n+1}) = 0 for all n ⩾ 2,  (4.1)

then the condition (3.3) is satisfied.

Proof. ¿From the following short exact sequence:

ker β_{n} ↪^{i} ker d_{n+1} ↠ \frac{ker d_{n+1}}{ker β_{n}} ,

where i is the inclusion, and by the second isomorphism theorem which asserts that:

\frac{ker d_{n+1}}{ker β_{n}} ≅ \frac{T or_{n+1}^{T (V)}(R, R)}{ker b_{n+1}},

we deduce that:

Ext (\frac{T or_{n+1}^{T (V)}(R, R)}{ker b_{n+1}}, coker b′_{n+1}) =\frac{Hom(ker β_{n}, cokerb′_{n+1})}{i^{∗}(Hom(ker d_{n+1}, coker b′_{n+1}))} = 0. (4.2)

¿From the relation (2.12) we know that the homomorphism \overline{γ_{n}^{f}} ∘ \overline{φ_{n}} − \overline{φ′_{n}} ∘ ξ_{n+1} satisfies the following relation:

\overline{γ_{n}^{f}} ∘ \overline{φ_{n}} − \overline{φ′_{n}} ∘ ξ_{n+1} = g_{n} ∘ d_{n+1}

where g_{n} : ker β_{n} ⟶ coker b′_{n+2}. So the relation (4.2) implies that g_{n} ∈ i^{∗}(Hom(ker d_{n+1}, coker b′_{n+1})). Hence there exists a homomorphism h_{n} : ker d_{n+1} → coker b′_{n+1} such that g_{n} = h_{n} ∘ i. Therefore the relation (4.1) implies the condition (3.3).

As a consequence of proposition 4, we derive the following result, which is the main result in this paper

Theorem 12. If the relation (4.1) is satisfied, then two free dgas (T (V), ∂) and (T (W), δ) have the same homotopy type if and only if their Whitehead exact sequences are isomorphic.

Corollary 3. If the relation (4.1) is satisfied for all dga (T (V), ∂), then we have:

DGA_{∗}(flat) = ΓDGA_{∗}(flat).

Let k be a field of any characteristic. In [[10], thm 2.3] Baues and Lemaire establish the following result: for every dga A over k there exists a quasi-isomorphism α : (T (V), ∂) → A, where the differential ∂ is decomposable, i.e: ∂ : V → T_{⩾2}(V). This tensor algebra is called the minimal model of A. It is unique up to isomorphism. Over a field k, following the construction in definition (7), the tensor algebra (T s^{−1}(T or_{∗}^{A}(R, R)), ∂) can be taken as the dga associated with the Γ-system Γ(A). Therefore by theorem 9 there exists a quasi-isomorphism α : (T (s^{−1}T or_{∗}^{A}(R, R)), ∂) → A, where the differential ∂ is decomposable. The dga (T (s^{−1}T or_{∗}^{A}(R, R)), ∂) is not unique, but the Γ-system Γ(A) = (T or_{n}^{A}(R, R)), b_{n+2})_{n⩾2} is unique.

5. The category Γ_{n}^{3n+2}

In this section we treat a partial special case where the notion of the Γ-system may be simplified, even over a P.I.D. Indeed we will show when the dga A is an object of ΓDGA_{n}^{3n+2}(flat) (the subcategory of ΓDGA_{∗}(flat) of which the objects are those satisfying the relations T or_{i}^{A}(R, R) = 0 for i ⩽ n and i ⩾ 3n + 3, we can denote the graded module Γ_{∗} simply by the graded module T or_{∗}^{A}(R, R) and the homomorphism b_{2n+2}. This section is motivated by the following theorem which gives us explicitly the graded module Γ_{∗}^{A} which appear in the Whitehead sequence associated with A.

Theorem 13. Let A be an object in DGA_{n}^{3n+2}(flat). If we put H_{k} = T or_{k}^{A}(R, R), then we have:

Γ_{k}^{A} = ⊕_{i=n+1}^{k−n+1}H_{i} ⊗ H_{k−i+2} ⊕ ⊕_{i=n+1}^{k−n}T or_{R}(H_{i}, H_{k+1−i}) , 2n ⩽ k ⩽ 3n − 1
Γ_{3n}^{A} = ⊕_{i=n+1}^{2n+1}H_{i} ⊗ H_{3n−i+2} ⊕ \frac{H_{n+1} ⊗ H_{n+1} ⊗ H_{n+1}}{(Im b_{2n+2}) ⊗ H_{n+1} + H_{n+1} ⊗ (Im b_{2n+2})}
⊕ ⊕_{i=n+1}^{2n}T or_{R}(H_{i}, H_{3n+1−i})
Γ_{k}^{A} = 0, k ⩽ 2n − 1.

Proof. Recall that Γ_{∗} is defined, for each i ⩾ 2, by the formula:

Γ_{i} = ker(H_{i}(T (V_{⩽i})) → V_{i}),  (5.1)

where (T (V), δ) is a free model of A. Since A is an object of DGA_{n}^{3n+2}(flat), we can choose (T (V), δ) such that:

V_{i} = 0 if i ⩽ n − 1 and i ⩾ 3n + 1.  (5.2)

Now we filter the dga (T (V), δ) by setting :

F^{p} = ⊕_{i⩾p}V^{⊗i},

which induces a spectral sequence (E_{∗,∗}^{−r}, d_{∗,∗}^{−r}) of the first quadrant converging to H_{∗}(T (V), δ). According to the relation (5.1), we have

Γ_{k}^{A} = ⊕_{i⩾2}E_{i,k−i}^{∞} , k ⩽ 3n,

from (5.2), we have

E_{i,k−i}^{∞} = E_{i,k−i}^{−1} = H_{k}(V^{⊗i}, d) , k ⩽ 3n − 1,

so for k ⩽ 3n − 1 we deduce

E_{i,k−i}^{∞} = 0 for i ⩾ 2,

and we get

Γ_{k} = E_{2,k−2}^{∞} = H_{k}(V^{⊗2}, d).

Finally by the Künneth formula we get

Γ_{k} = ⊕^{k}_{i=0}H_{i} ⊗ H_{k−i+2} ⊕ ⊕^{k−n}_{i=n+1}T or_{k}(H_{i}, H_{k+1−i}), k ⩽ 3n − 1.

Since H_{i} = 0, i ⩽ n then we deduce

Γ_{k} = 0 , i ⩽ 2n − 1.

For k = 3n, we remark that E_{i,3n−i}^{∞} = 0 , i ⩾ 4 because E_{i,3n−i}^{−1} = H_{3n}(V^{⊗i}, d), so we get

Γ_{3n} = E_{2,3n−2}^{∞} ⊕ E_{3,3n−3}^{∞}.

Observe that E_{2,3n−2}^{∞} = E_{2,3n−2}^{−1} = H_{3n}(V^{⊗2}) so by the Künneth formula we get

E_{2,3n−2}^{∞} = ⊕^{2n+1}_{i=n+1}H_{i} ⊗ H_{3n−i+2} ⊕ ⊕^{2n}_{i=n+1}T or_{R}(H_{i}, H_{3n−i+1}).

To compute E_{3,3n−3}^{∞}, note that we have

E_{2,2n−1}^{−1} ⟶^{d_{2,3n−1}^{−1}} E_{3,3n−3}^{−1} → E_{4,3n−5}^{−1} = 0.  (5.3)

Applying the Künneth formula to (5.3), we get

⊕^{2n+2}_{i=n+1}H_{i} ⊗ H_{3n−i+2} ⊕ ⊕^{2n+1}_{i=n+1}T or_{R}(H_{i}, H_{3n−i+1}) ⟶^{d_{i,3n−1}^{−1}} H_{n+1} ⊗ H_{n+1} ⊗ H_{n+1} → 0

and it’s easy to see that the differential d_{i,3n−1}^{−1} is identified with the homomorphism:

d_{2,3n−1}^{−1} = b_{2n+2} ⊗ id_{H_{n+1}} + (−1)^{n}id_{H_{n+1}} ⊗ b_{2n+2},

so

Imd_{2,3n−1}^{−1} = Im b_{2n+2} ⊗ H_{n+1} + H_{n+1} ⊗ Im b_{2n+2}.

Finally we get

E_{3,3n−3}^{∞} = \frac{H_{n+1} ⊗ H_{n+1} ⊗ H_{n+1}}{Im b_{2n+2} ⊗ H_{n+1} + H_{n+1} ⊗ Im b_{2n+2}} ,

and the theorem is proved.

As a consequence of this result, we derive the following proposition, which is the version of the Hurewicz theorem in the category DGA_{n}^{3n+2}(flat).

Proposition 5. Let R be a P.I.D and A an object of DGA_{n}^{3n+2}(flat). The Hurewicz homomorphism:

h_{k} : H_{k}(A) ⟶ T or_{k+1}^{A}(R, R),

is an isomorphism for k ⩽ 2n − 1 and surjective for k = 2n.

Proof. One only need apply theorem 13 to the Whitehead exact sequence associated with the dga A.

That’s the same result obtained in ([10] ) where R is a field of any characteristic. The next corollary is a known geometrical version of proposition 5.

Corollary 4. Let X be a (n − 1)-connected (3n + 1)-dimensional CW-complex. Then the homomorphism:

h_{k} : H_{k}(ΩX, R) ⟶ H_{k+1}(X, R),

is an isomorphism for k ⩽ 2n − 1 and surjective for k = 2n.

Proof. It suffices to apply proposition 5 to the free dga (T_{X}(V), ∂^{X}) which is the Adams-Hilton model of X in [1]. Note that since X is (n − 1)-connected (3n + 1)- dimensional then the dga (T_{X}(V), ∂^{X}) is an object in DGA_{n}^{3n+2}(flat).

5.1. Definition of the category Γ_{n}^{3n+2}

We recall that H_{∗} is said to be n-connected, (3n + 2)-dimensional graded module if H_{i} = 0 for i ⩽ n and i ⩾ 3n + 3.

Definition 12. For each n-connected, (3n + 2)-dimensional graded module H_{∗} and for each b_{2n+2} ∈ Hom(H_{2n+2}, H_{n+1} ⊗ H_{n+1}), we define the graded module Γ_{⩽3n} by setting:

Γ_{k} = 0  k ⩽ 2n − 1
Γ_{k} = ⊕_{i=n+1}^{k−n+1}H_{i} ⊗ H_{k−i+2} ⊕ ⊕_{i=n+1}^{k−n}T or_{R}(H_{i}, H_{k+1−i})  2n ⩽ k ⩽ 3n − 1
Γ_{3n} = ⊕_{i=n+1}^{2n+1}H_{i} ⊗ H_{3n−i+2} ⊕ \frac{H_{n+1} ⊗ H_{n+1} ⊗ H_{n+1}}{Im b_{2n+2} ⊗ H_{n+1} + H_{n+1} ⊗ Im b_{2n+2}} ⊕

⊕_{i=n+1}^{2n}T or_{R}(H_{i}, H_{3n+1−i}).

Γ_{⩽3n} is called the graded module associated with H_{∗} and the homomorphism b_{2n+2}.

Note that by theorem 13, Γ_{⩽3n}^{A} is the graded module associated with T or_{∗}^{A}(R, R) and b_{2n+2} given by the Whitehead sequence associated with A.

Definition 13. For each graded homomorphism f_{∗} : H_{∗} → H′_{∗} which makes the following diagram commutative:

H′_{2n+2}  H′_{n+1} ⊗ H′_{n+1} f_{2n+2}  f_{n+1} ⊗ f_{n+1} H_{2n+2}  H_{n+1} ⊗ H_{n+1} b_{2n+2} b′_{2n+2}

we define the homomorphisms γ_{⩽3n} by setting:

γ_{k} = 0  k ⩽ 2n − 1
γ_{k} = ⊕_{i=n+1}^{k−n+1}f_{i} ⊗ f_{k−i+2} ⊕ ⊕_{i=n+1}^{k−n}T or_{R}(f_{i}, f_{k+1−i})  2n ⩽ k ⩽ 3n − 1
γ_{3n} = ⊕_{i=n+1}^{2n+1}f_{i} ⊗ f_{3n−i+2} ⊕ \overline{f_{n+1}}^{⊗3} ⊕ ⊕_{i=n+1}^{2n}T or_{R}(f_{i}, f_{3n+1−i}),

where:

\overline{f_{n+1}}^{⊗3} : \frac{H_{n+1} ⊗ H_{n+1} ⊗ H_{n+1}}{Imb_{2n+2} ⊗ H_{n+1} + H_{n+1} ⊗ Im b_{2n+2}} →

\frac{H′_{n+1} ⊗ H′_{n+1} ⊗ H′_{n+1}}{Im b′_{2n+2} ⊗ H′_{n+1} + H′_{n+1} ⊗ Im b′_{2n+2}} .

γ_{∗} is called the graded homomorphism associated with f_{∗}.

Proposition 6. Let α : A → B be a morphism in DGA_{n}^{3n+2}(flat), then γ_{∗}^{α} : Γ_{∗}^{A} → Γ_{∗}^{B} is the graded homomorphism associated with T or_{∗}^{α}(R, R).

Proof. The proof is very simple but it’s very long. It is just a simple computation and verification (see [11])

Definition 14. The category Γ_{n}^{3n+2} is defined as follows: Object: a collection (H_{∗}, (b_{i+2}, π_{i})_{2n⩽i⩽3n}) such that: H_{∗} is an n-connected, (3n + 2)-dimensional graded module. For each 2n ⩽ i ⩽ 3n, b_{i+2} is a homomorphism of modules H_{i+2} → Γ_{i} where Γ_{⩽3n} is the graded module associated with H_{∗}. For each 2n ⩽ i ⩽ 3n, π_{i} ∈ Ext(H_{i+1}, coker b_{i+2}).

Morphism: a morphism between two objects (H_{∗}, (b_{i+2}, π_{i}))_{2n⩽i⩽3n} and (H′_{∗}, (b′_{i+2}, π′_{i}))_{2n⩽i⩽3n} is a graded homomorphism f_{∗} : H_{∗} → H′_{∗} satisfying the conditions:

f_{i+2} ∘ b′_{i+2} = γ_{i} ∘ b_{i+2},  (f_{i+1})^{∗}(π′_{i}) = (\overline{γ_{i}})_{∗}(π_{i})  i ⩽ 3n,

where γ_{⩽3n} is the homomorphism associated with f_{∗} and where the homomorphisms (f_{i+1})^{∗}, (\overline{γ_{i}})_{∗} are given by:

(\overline{γ_{i}})_{∗} : Ext(H_{i+1}, coker b_{i+2}) → Ext(H′_{i+1}, coker b′_{i+2})
(f_{i+1})^{∗} : Ext(H′_{i+1}, coker b′_{i+2}) → Ext(H_{i+1}, coker b′_{i+2})

and where \overline{γ_{i}} is induced by γ_{i} on coker b_{i+2}.

5.2. The functor F_{n}^{3n+2} As in the previous section we define the functor:

F_{n}^{3n+2} : ΓDGA_{n}^{3n+2}(flat)⧸_{≃} → Γ_{n}^{3n+2},

by setting:

F_{n}^{3n+2}(A) = (T or_{∗}^{A}(R, R), (b_{i+2}, π_{i})_{2n⩽i⩽3n})
F_{n}^{3n+2}({α}) = T or_{∗}^{α}(R, R).

The main result in this section is the following theorem:

Corollary 5. Under the condition (4.1), the functor

F_{n}^{3n+2} : DGA_{n}^{3n+2}(flat)⧸_{≃} → Γ_{n}^{3n+2},

is a “detecting functor ”

Proof. The proof is the same as in theorem 9.

As a corollary we have the result which we can use to compute the number of the homotopy types in the category DGA_{n}^{3n+2}(flat).

Theorem 14 (Homotopy classification theorem ). Homotopy types of objects in ΓDGA_{n}^{3n+2}(flat) are in bijection with the proper equivalence classes of tuples (b_{3n+2}, π_{3n}, ...., b_{2n+2}, π_{2n}) where b_{i+2} ∈ Hom(H_{i+2}, Γ_{i}), π_{i} ∈ Ext(H_{i+1}, coker b′_{i+2}), 2n ⩽ i ⩽ 3n and where H_{∗} is n-connected, (3n + 2)-dimensional graded module.

Recall that (Γ_{i})_{i⩽3n} is the graded module associated with H_{∗} and b_{2n+2}.

Definition 15. Two tuples (b_{3n+2}, π_{3n}, ...., b_{2n+2}, π_{2n}) and (b′_{3n+2}, π′_{3n}, ...., b′_{2n+2}, π′_{2n}) are called proper equivalent if there exists a graded automorphism f_{⩽3n} : H_{⩽3n} ≅ H_{⩽3n} such that for every i ⩽ 3n:

(f_{i+1})^{∗}(π′_{i}) = (\overline{γ_{i}})_{∗}(π_{i}), and γ_{i} ∘ b_{i+2} = b′_{i+2} ∘ f_{i+2}.

5.3. Examples

We conclude this work by giving some geometric applications to the above theorems.

Proposition 7. Let A an object of DGA_{n}^{3n+2}(flat). Denote by S_{A} the set consisting of the dgas B satisfying:

T or_{i}^{B}(R, R) = T or_{i}^{A}(R, R) for every n ⩽ i ⩽ 3n.

Assume that T or_{i}^{A}(R, R) = 0 for each i satisfying:

i ⩽ \frac{3n + 2}{2}  if n is even  (5.4)
i ⩽ \frac{3n + 1}{2}  if n is odd

then all objects of the set S_{A} have the same homotopy type.

Proof. If B is an element of S_{A} then from the hypothesis (5.4) and theorem 13 we deduce that the module Γ_{i} is trivial for all i ⩽ 3n. Hence we get:

F_{n}^{3n+2}(B) = (T or_{∗}^{B}(R, R), 0, 0) = (T or_{∗}^{A}(R, R), 0, 0) = F_{n}^{3n+2}(A).

The relation (4.1) is trivially satisfied . So by corollary 5, F_{n}^{3n+2} is a detecting functor, thus B and A have the same homotopy type.

Proposition 8. Let n ∈ N and X be a CW-complex such that H_{k}(X, R) = 0 for each k satisfying:

k ⩽ \frac{3n + 2}{2} if n is even
k ⩽ \frac{3n + 1}{2} if n is odd
k ⩾ 3n + 3

If we put H_{k} = H_{k}(X, R) then we have :

H_{i}(ΩX, R) = ⊕_{k}H_{k}^{⊗q}  if i = q(k − 1)
H_{i}(ΩX, R) = ⊕_{k}( ⊕_{s+t=q}T or_{R}(H_{k}^{⊗s}, H_{k}^{⊗t}))  if i = q(k − 1) + 1
H_{i}(ΩX, R) = 0  otherwise.

Proof. Let M(H_{k}, k) be a Moore space and let :

Y = ∨_{k}M(H_{k}, k)

be the wedge of the spaces M(H_{k}, k) and let A(X) and A(Y) be the Adams-Hilton models associated respectively to X and Y. We recall that we have:

T or_{∗}^{A(X)}(R, R) = H_{∗}(X, R) and T or_{∗}^{A(Y)}(R, R) = H_{∗}(Y, R).

Since H_{∗}(X, R) = H_{∗}(Y, R), the relation implies that A(Y) is an object of the set Σ_{A(X)} in proposition 7, so that A(X) and A(Y) have the same homotopy type. We deduce that:

H_{∗}(A(X)) ≅ H_{∗}(A(Y)),  (5.5)

since A(X) and A(Y) are respectively the Adams-Hilton models of X and Y. Then:

H_{∗}(A(X)) = H_{∗}(ΩX, R) and H_{∗}(A(Y)) = H_{∗}(ΩY, R),

so according to (5.5) we get:

H_{∗}(ΩX, R) ≅ H_{∗}(Ω ∨_{k} M(H_{k}, k), R) ≅ ⊕_{k}H_{∗}(M(H_{k}, k), R).

Now the next lemma allows us to achieve the proof.

Lemma 3. The graded algebra H_{∗}(ΩM(H_{k}, k), R) is given by:

H_{i}(ΩM(H_{k}, k), R) = H_{k}^{⊗q}  if i = q(k − 1)
H_{i}(ΩM(H_{k}, k), R) = ⊕_{s+t=q}T or_{R}(H_{k}^{⊗s}, H_{k}^{⊗t}) if i = q(k − 1) + 1
H_{i}(ΩM(H_{k}, k), R) = 0  otherwise.

Proof. Let A(M(H_{k}, k)) be an Adams-Hilton model of the Moore space M(H_{k}, k). The Whitehead sequence associated with A(M(H_{k}, k)) is:

→ H_{i+2}(M(H_{k}, k), R) → Γ_{i} → H_{i}(ΩM(H_{k}, k), R) → H_{i+1}(M(H_{k}, k), R) →

Since we have:

H_{i+2}(M(H_{k}, k), R) = 0 unless i = k − 2

it follows that

H_{i}(ΩM(H_{k}, k), R) ≅ Γ_{i}.

According to section (3), we can realize the object Γ ({A(M(H_{k}, k)}) by the free

dga T (V_{k}, V_{k−1}, d) where V_{k} ↣^{d} V_{k−1} ↠ H_{k} is a free resolution of H_{k}. Since the module Γ_{i} = ker (H_{i}(T (V_{k}, V_{k−1}, d)) ⟶ V_{i}) and since the differential d is linear, the Künneth formula allows us to write:

Γ_{i} = H_{k}^{⊗q}  if i = q(k − 1)
Γ_{i} = ⊕_{s+t=q}T or_{R}(H_{k}^{⊗s}, H_{k}^{⊗t})  if i = q(k − 1) + 1
Γ_{i} = 0  otherwise

and the proof is achieved.

Example 1. Let X be a CW-complex 1-connected having the following homology groups:

H_{2}(X, ℤ) = ℤ_{2}, H_{3}(X, ℤ) = ℤ_{3}, H_{4}(X, ℤ) = ℤ_{3},
H_{5}(X, ℤ) = ℤ, H_{i}(X, ℤ) = 0 otherwise.

How many homotopy type of the dga C_{∗}(ΩX) exist?

According to theorem 13 we have:

Γ_{2} = H_{2} ⊗ H_{2} = ℤ_{2}
Γ_{3} = H_{2} ⊗ H_{3} ⊕ H_{3} ⊗ H_{2} ⊕ \frac{H_{2} ⊗ H_{2} ⊗ H_{2}}{Im b_{4} ⊗ H_{2} + H_{2} ⊗ Im b_{4}} = ℤ_{2}.

We begin by computing the homomorphisms b_{4}. We have:

Hom(H_{4}(X), Γ_{2}) = Hom(ℤ_{3}, ℤ_{2}) = 0.

So we deduce that b_{4} = 0 and:

Ext(H_{3}, coker b_{4}) = Ext(ℤ_{3}, ℤ_{2}) = 0.

Then we only find the trivial extension π_{2} = 0. Now we have:

b_{5} ∈ Hom(H_{5}, Γ_{3}) = Hom(ℤ, ℤ_{2}) = ℤ_{2},

so we find two homomorphisms b_{5}^{(0)} = 0 and b_{5}^{(1)} = 1. For b_{5}^{(0)} = 0 we get Ext(H_{4}, coker b_{5}^{(0)}) = Ext(ℤ_{3},ℤ_{2}) = 0 so in this case we also find the trivial extension π_{3}^{(0)} = 0. For b_{5}^{(1)} = 1 then b_{5}^{(1)} is onto and we deduce that Ext(H_{4}, cokerb_{5}^{(1)}) = 0 and also we get the trivial extension π_{3}^{(1)} = 0.

Then we find two Γ-systems (0, 0, 0, 0) , (1, 0, 0, 0) which are obviously not isomorphic in their category.

Now since by computation we have:

Ext (\frac{H_{4}}{ker b_{4}} , coker b_{5}^{(0)}) = Ext(ℤ_{3}, ℤ_{2}) = 0

Ext (\frac{H_{4}}{ker b_{4}} , coker b_{5}^{(1)}) = Ext(ℤ_{3}, 0) = 0

Ext (\frac{H_{5}}{ker b_{5}^{(0)}} , coker b_{6}) = Ext(0, ℤ_{2}) = 0

Ext (\frac{H_{5}}{ker b_{5}^{(1)}} , coker b_{6}) = Ext(ℤ, ℤ_{2}) = 0

Ext (\frac{H_{n+1}}{ker b_{n+1}} , coker b_{n+2}) = Ext(0, ℤ_{2}) = 0 for n ⩾ 5,

then condition (4.1) is satisfied. Therefore, according to the theorem 14 of classification of homotopy types, we have two homotopy types having the above homology groups.

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2004/n1a8/v6n1a8.(dvi,ps,pdf)

Mahmoud Benkhalifa makhalefah@kku.edu.sa,  hasy dz@yahoo.fr

Department of Mathematics, Faculty of Sciences, King Khalid University, P.O Box 9004, Abha, SAUDI ARABIA