PRIMARY DECOMPOSITION OF A 0-CONNECTED NILPOTENT SPACE

It is well known that the different stages of the Cartan-Whitehead decomposition of a 0-connected space can be obtained as the Adams cocompletion of the space with respect to suitable sets of morphisms. In thispaper Cartan-Whitehead decomposition is obtained for a nilpotent space, in terms of Adams cocompletion, using the primary homotopy theory developed by Neisendonfer. Indexing terms/


INTRODUCTION
Deleanu, Frei and Hilton have developed the notion of generalized Adams completion in a categorical context; they have also studied the dual notion, namely the Adams cocompletion of an object in a category [9]. It is to be emphasized that many algebraic and geometrical constructions in Algebraic Topology, Differential Topology, Differentiable Manifolds, Algebra, Analysis, Topology, etc., can be viewed as Adams completions or cocompletions of objects in suitable categories, with respect to carefully chosen sets of morphisms. Behera and Nanda [3] have shown that the different stages of the Cartan-Whitehead decomposition of a 0-connected space are the Adams cocompletion of a space with respect to suitable sets of morphisms. In [12], Neisendonfer has studied the primary homotopic theory in an exhaustive manner.. The central idea of this note is to study how Cartan-Whitehead decomposition of a 0-connected nilpotent space is characterized in terms of its Adams cocompletions; it is done using the primary homotopy theory developed by Neisendonfer.
Let be an arbitrary category and S a set of morphisms of . Let [S −1 ] denote the category of fractions of with respectS andF: → [S −1 ]be the canonical functor. Let denote the category of sets and functions. Then for a given object Y of , defines a covariant functor. If this functor is representable by an object Y S of , that is, if then Y S is called the (generalized) Adams cocompletion of Y with respect to the set of morphisms S or simply the Scocompletion ofY. We shall often refer to Y S as the cocompletion of Y [9].
Given aset S of morphisms of , the saturation of S, denoted as S is the set of all morphisms u in such that F(u) is an isomorphism in S −1 . Furthermore,S is said to be saturated if S = S [4,9].
Deleanu, Frei and Hilton have shown that if the set of morphisms Sis saturated then the Adams cocompletion of a space is characterized by a certain couniversal property ( [9],dual of Theorem 1.2). In most of the applications, however, the set of morphisms S is not saturated. There is a stronger version of Deleanu, Frei and Hilton's characterization of Adams cocompletion in terms of couniversal property as described below.
Theorem1.1. ([4], dual of Theorem 1.2) Let S be a set of morphisms of admitting a calculus of right fractions. Then an object Y S of is the S-cocompletion of the object Y with respect to S if and only if there exists a morphism e ∶ Y S → Y in S which is couniversal with respect to morphisms inS : given a morphism s ∶ Z → Y in S there exists a unique morphism t ∶ Y S → Z in S such that st = e. In other words, the following diagram is commutative : Also the above theorem turns out to be essentially the dual of Theorem 1.2 [9] if we assume S to be saturated; hence the proposition can be proved by recasting the dual of the proof of the Theorem 1.2 [9] with minor changes. The details are omitted.
The following Theorem (dual of Theorem 1.3, [4]) shows that under certain conditions the morphisms e: Y S → Y always belongs to S. (ii) fg ∈ S 1 implies thatg ∈ S 1 .
Then e ∈ S.

2.The category .
Let S m denote the m-dimensional sphere. Suppose m ≥ 2, and let k ∶ S m → S m denote a map of degree k. The space S m −1 e m k is denoted by P m k or P m (ℤ kℤ ). If m ≥ 2, the m-th modkhomotopy group of X is [P m k ; X], denoted by π m (X; ℤ kℤ ) [12]. If m ≥ 3, π m (X; ℤ kℤ ) is a group and m ≥ 4, π m (X; ℤ kℤ ) is an abelian group [12]. N o v e m b e r 1 5 , 2 0 1 5 If f ∶ X ⟶ Y is a map, then there are induced maps f * : π m X; ℤ kℤ → π m (Y; ℤ kℤ )defined by f * g = [ fg ]. If m ≥ 3, f * is a homomorphism and, if m = 2, f * is compatible with the action of π 2 [12].
For a group G, the lower central series G is said to be nilpotent if Γ = {1} for sufficiently large [10].
A connected -complex is said to nilpotent if 1 ( ) is nilpotent and operates nilpotently on ( ) for every ≥ 2 [10]. It remains to be shown that ∈ . We assume that the map ∶ → is a fibration with fibre . We note that is also the fibre of ∶ → . We have the following commutative diagram.

Existence of Adams cocompletion in .
Since the category 0 as stated above is neither complete nor small, the dual of Theorem 2.6 [7] nan not be used to show the existence of Adams cocompletion of an object in the category 0 with respect to the set of morphisms .The following theorem shows that under certain conditions the Adams cocompletion of an object in the category 0 always exists; the theorem is essentially the dual of Theorem 4.7 [1] and dual of Theorem 3.8 [2] (it is also a generalization of the dual of the Theorem in [7]). Assume that the family and the object of satisfy the condition : ( * )There exists a subset of the set ∶ ′ → ∈ }such that is an element of the universe and for each ∶ ′ → , ∈ , there exist an ′ ∶ ′′ → in and a morphism ∶ ′′ → ′ of rendering the following diagram is commutative : Then the Adams cocompletion of does exist.
As remarked by Adams of page 34 of [2] this result remains valid if is the homotopy category of 0-connected based nilpotent spaces (whose underlying sets belong to ). It is to be emphasized that condition ( * ) is essential in order to be able to apply E.H. Brown's representability theorem to prove this result.  Proof. The proof follows from the dual of Theorem 1.3 [4]. We take 1 = ∶ → 0 * ∶ ; ℤ ℤ → ; ℤ ℤ is a monomorphism for ≥ } and 2 = { ∶ → 0 | * ∶ ; ℤ ℤ → ; ℤ ℤ is an epimorphism for ≥ + 1}. Clearly (a) = 1 ∩ 2 and (b) 1 and 2 satisfy all conditions of Theorem 1.2; hence ∈ . This completes the proof of the Proposition 3.3.

4.A primary decomposition of a 0-connected based nilpotent space.
Now we obtain the primary decomposition of a 0-connected based nilpotent space with the help of the set of morphisms as described below. In fact the different stages of the Cartan-Whitehead decomposition of a 0-connected nilpotent space are the Adams cocompletions of the space with respect to the sets of morphisms . In the process, starting from a 0-connected based nilpotent space we get a tower of spaces, ⋯ → +1 +1 → ⋯ → 1 1 → 0 and the direct limit of this tower gives us a space which in some sense is the Cartan-Whitehead decomposition of . First we prove the following proposition. (a) * ∶ ; ℤ kℤ → π m (X; ℤ kℤ )is an isomorphism form > andπ m X n ; ℤ kℤ = 0form ≤ n, (b) e n+1 = e n ∘ θ n+1 .
Proof. For each integer n ≥ 3, let X n be the S n -completion of X and e n ∶ X n → X be the canonical map as stated in Theorem 3.1. Since e n ∈ S n ⊂ S n+1 , it follows from the couniversal property of e n+1 that there exists a unique morphism θ n+1 ∶ X n+1 → X n such that e n+1 = e n ∘ θ n+1 , i.e., the following diagram is commutative: θ n+1 X n+1 e n +1 X ↓ ↗ e n X n The maps θ n can of course be replaced by fibrations in the usual manner. Since e n ∈ S n , e n * ∶ π m X n ; ℤ kℤ → π m (X; ℤ kℤ )is an isomorphism for m > ; it is already proved in Proposition 2.2 that π m X n ; ℤ kℤ = 0 for m ≤ n.
This completes the proof of the Theorem 4.2.