THE GROUP OF SELF-HOMOTOPY EQUIVALENCES OF A SIMPLY CONNECTED AND 4-DIMENSIONAL CW-COMPLEX

Let X be a CW complex, E(X) the group of homotopy classes of self-homotopy equivalences of X and E∗(X) its subgroup of the elements that induce the identity on homology. This paper deals with the problem 19 in [Contemp. Math., 519 (2010), 217-230]. Given a group G, find a space X such that E(X) E∗(X) = G. For a simply connected and 4-dimensional CW-complex X we define a group B4 ⊂ aut(H∗(X,Z)) in term of the Whitehead exact sequence of X and we show that this problem has a solution if G ∼= B4 for some space X. Mathematics Subject Classification (2010): 55P10


Introduction
Let X be a CW complex, and let E(X) denote the group of homotopy classes of self-homotopy equivalences of X.The determination of the group E(X) presents a challenging problem of computation with a long history of progress on special cases (cf.[1,3,5,6,8,9,10,13,15,16]).Several problems related to the group E(X) are given in the literature especially the realizability of E(X) as a given group G [13] and the (in)finiteness of the nilpotent group E * (X) [2,6,9].

A variant of the realizability problem is the following:
Problem [12,Problem 19]: Given a group G, find X such that E(X) = G.
Here E(X) is a distinguished subgroup or quotient of E(X).It may be the subgroup E (X) of self-equivalences that induce the identity on the homotopy groups, the subgroup E * (X), or the derived subgroup, or E(X) may be the quotient E(X) E * (X) .
The aim of this paper is to investigate the problem quoted above for a simply connected and 4-dimensional CW-complex X.For this purpose we define the group B 4 in terms of the Whitehead exact sequence of X [17, page 72]: and a "certain" notion of automorphisms, called the Γ-automorphisms, of this sequence given in Definition 2.6.
Our main result is the following: Theorem 1.If X is a simply connected and 4-dimensional CW-complex, then The idea of using rational homotopy methods to translate the problem of computing or at least getting some informations regarding the (in)finiteness of the groups E(X) and E * (X) within the framework of minimal commutative differential graded algebras and algebraic homotopy of DGA maps traces back to the results of Arkowitz-Lupton [2] in which they exhibited conditions under which E * (X) is finite or infinite where X is a rational space having a 2 -stage Postnikov-like decomposition (for example, rationalizations of homogeneous spaces).Using rational homotopy theory we show the following result: Theorem 2. Let X be a simply connected 4-dimensional CW-complex having 4cells.Then E * (X) is finite in the following two cases: In Section 2, we recall the basic definitions of Whitehead's certain exact sequence and his theorem about 4-dimensional simply-connected CW-complexes and in Section 3, we define the group B 4 and give some of its important properties, moreover we formulate and prove the main theorem.In Section 4 we end this work by giving some applications.

The certain exact sequence of Whitehead
2.1.The definition of Whitehead's certain exact sequence.All the materials of this section which is essential and fundamental in this work can be found in details in [4,17].
Let X be a simply connected CW-complex defined by the collection of its skeleta (X n ) n≥0 , where we can suppose X 0 = X 1 = .
The long exact sequence of the pair (X n , X n−1 ) in homotopy and in homology are connected by the Hurewicz morphism h * in order to give the following commutative diagram: where β n = β n,n and j n = j n,n , defines the cellular chain complex of X. Moreover represents by adjunction the attaching map for the n-cells Now Whitehead [17, page 72] inserted the Hurewicz homomorphism in a long exact sequence connecting homology and homotopy.First he defined the following abelian group We notice that β n+1 • d n+1 = 0 and so With this map, Whitehead [17] defined the following sequence: and proved the following.
Theorem 2.2.The sequence (2), called the Whitehead exact sequence of X, is a natural exact sequence.
2.2.4-dimensional CW-complexes.Let Ab be the category of abelian groups.
Using the notion of quadratic maps, Whitehead constructed a functor Γ : Ab → Ab called Whitehead's quadratic functor [17].
Proposition 2.3.The Whitehead's quadratic functor has the following properties (see for example [4, page 448]) for more details): (5) Let A an abelian group and let n. : A → A denote the multiplication by n, that means a −→ na, If n. is an isomorphism of abelian groups, then Γ(n.) : Γ(A) → Γ(A) is the multiplication by n 2 i.e., Γ(n.) = n 2 and it is also an isomorphism of abelian groups.Definition 2.4.Given four abelian groups H 4 , H 3 , H 2 , π where H 4 is free.A Γsequence is an exact sequence of abelian groups: where Γ is the Whitehead's quadratic functor.
Example 2.5.According to Proposition 2.3, if X is a simply connected 4-dimensional CW-complex, then its Whitehead exact sequence can be written as follows: thus its a Γ-sequence.
Notice that in this case H 4 (X, Z) is a free abelian groups which admits the set of all the 4-cells as a basis.
Definition 2.6.Let X be a simply connected 4-dimensional CW-complex and let there exists an automorphism Ω : π 3 (X) → π 3 (X) making the following diagram commutes: In order to state Whitehead's theorems on 4-dimensional CW-complexes.We need the following two definitions.
able if there exists a simply connected 4-dimensional CW-complex X such that its Whitehead exact sequence coincides with the given Γ-sequence.
Definition 2.9.Let X be a simply connected 4-dimensional CW-complex and let (f 4 , f 3 , f 2 ) be a Γ-automorphism of the Whitehead exact sequence of X.We say CW-complex extending the notion of the Γ-automorphisms.

The main results
3.1.The group B 4 .In this paragraph we introduce the group B 4 given in the introduction and which plays a crucial role in this paper.
Definition 3.1.Let X be a simply connected 4-dimensional CW-complex.We define B 4 to be the set of all the Γ-automorphisms of the Whitehead exact sequence of X.
, then the composition: from Definition 2.6 we deduce that there exist two automorphisms Ω, Ω : π 3 (X) → π 3 (X) making the diagrams (1) and (2) commute: (1) . Therefore the commutativity of the diagrams ( 1) and ( 2) implies that the following diagram commutes: Finally if (f 4 , f 3 , f 2 ) ∈ B 4 , then by definition f 4 , f 3 , f 2 are automorphisms so we get the triple ( 4 there is an automorphism Ω : π 3 (X) → π 3 (X) making the diagram (1) commutes which implies that the following diagram is also commutative: Let X be a simply connected 4-dimensional CW-complex.Example 2.7 allows us to define a map Ψ : E(X) → B 4 by setting: Proposition 3.4.The map Ψ is a surjective homomorphism of groups whose kernel is E * (X).
Proof.First let [α], [α ] ∈ E(X).Using the formula (4), an easy computation shows that: it follows that Ψ is a homomorphism of groups.Clearly kerΨ = E * (X) and finally the surjection of the homomorphism Ψ is given by Theorem 2.10.
Accordingly we are now ready the announce our main result.
Theorem 3.5.If X is a simply connected and 4-dimensional CW-complex, then Corollary 3.6.Let G be a group.If G ∼ = B 4 , then the problem quoted in the introduction has a solution.

Applications
Let X be a simply connected 4-dimensional CW-complex.Our first application deals with the question of the (in)finiteness of the groups E(X) and E * (X).More precisely from Theorem 3.5 we derive the following corollary which is is straightforward.
(1) E(X) is finite if and only if E * (X) and B 4 are finite; (2) if B 4 is an infinite group, then so is E(X).
Next the following theorem concerns the finiteness of the group E * (X).
Theorem 4.2.Let X be a simply connected 4-dimensional CW-complex having 4-cells.Then E * (X) is finite in the following two cases: Proof.First let us consider the space X Q which is the rationalized of the space X.
That is a simply connected CW-complex which satisfies: By rational homotopy theory X Q admits a Quillen model.That means there exists a free differential graded Lie algebra (L(V ), ∂), where V is a graded vector space such that each V i−1 admits the set of the i-cells of X as a basis.In addition we have: As the Quillen model determines completely the rational homotopy type of a simply connected CW-complex X, we can derive that: where E (L(V ), ∂) denotes the group of DG Lie homotopy self-equivalences of (L(V ), ∂) and where E * (L(V ), ∂) denotes the subgroup of E (L(V ), ∂) consists of maps inducing the identity automorphism of the indecomposables.
Next according to Dror-Zabrodsky [11], we know that E * (X) is a nilpotent group and in [14] Maruyama proved that 6) we get: Then let {v 1 , . . ., v n } be a basis of the vector space V 3 (here we assume that the CW-complex has n 4-cells).For every r ∈ Q, we define α r : (L(V ), ∂) → (L(V ), ∂) as follows: As the differential ∂ is quadratic, the following diagram is obviously commutative: so α r is a DG Lie morphism which induces the identity on the indecomposables.It ] is nil.So the elements x i and y i , given in the formula (8), are also nil and obviously E * (L(V ), ∂) is trivial.It follows by ( 7) that E * (X) So the elements x i can be chosen non-zero so that α r and α r are not homotopic provides that r = r .
) is infinite and by (7) E * (X) is also infinite.
Theorem 4.3.Let X be a simply connected 4-dimensional CW-complex.If the groups H * (X, Z) are finite, then so is E(X).
Proof.First since the groups H * (X, Z) are finite, the group B 4 is also finite.Next the finiteness of H * (X, Z) implies that the Quillen model of X is trivial so the group E * (X) Q is also trivial.Therefore by Maruyama Theorem we deduce that E * (X) is finite.As a result E(X) is also finite.
Theorem 4.2 implies the following corollary.
Corollary 4.4.Let X be a simply connected 4-dimensional CW-complex having if and only if the group B 4 is finite.
The next result relates the finiteness of the E(X) to the Hurewicz homomorphism.
Proof.First according to the exact sequence of Whitehead of X, the surjectivity of the Hurewicz homomorphism h 4 implies that the homomorphism b 4 is nil.It follows that every automorphism f 4 ∈ aut(H 4 (X, Z)) makes the following diagram commutes: therefore for every f 4 ∈ aut(H 4 (X, Z)) the triple (f 4 , id H3(X,Z) , id H2(X,Z) ) belongs to the group B 4 .Thus aut(H 4 (X, Z)) × {id H3(X,Z) } × {id H2(X,Z) } is a subgroup of B 4 .
Finally we conclude Theorem 4.5 by observing that the two groups aut(H 4 (X, Z)) and aut(H 4 (X, Z)) × {id H3(X,Z) } × {id H2(X,Z) } are isomorphic.Now as X is a simply connected 4-dimensional CW-complex, then the group H 4 (X, Z) is free of rank n, where n is the number of the 4-cells of X.
Corollary 4.6.Let X be a simply connected 4-dimensional CW-complex.Assume that h 4 is surjective.
(1) If n ≥ 2, then the index [E(X) : E * (X)] is infinite and the quotient group E * (X) contains an element of order 2.

Examples.
In the following examples we give explicit computations of the group B 4 showing that it may be finite or infinite.
Example 4.7.Let X be a simply connected 4-dimensional CW-complex such that: First using the properties of Whitehead's quadratic functor given in Proposition 2.3 we obtain that: Therefore the Whitehead exact sequence of X which is an example of a Γ-sequence can be written as follows: It is important to notice that by virtue of Theorem 2.10 this Γ-sequence is realizable, so there exists a simply connected 4-dimensional CW-complex X having (10) as the Whitehead exact sequence.
Next let us compute the group B 4 in this case.Since the group aut(Z) then: It follows, using the properties of Whitehead's quadratic functor given in Proposition 2.3, that Γ(f 2 ) = 1.Therefore we seek the automorphisms f 4 = ±1 and f 3 = ±1 for which there exists an automorphism Ω : π 3 (X) → π 3 (X) making the following diagram commutes; ? ??so we have to treat two cases.
This implies that any automorphism Ω : commutes splits also i.e., Ω = 1 ⊕ f 3 .As a result we deduce that: Consequently we get only 4 triples which are: As every triple is obviously of order 2 we conclude, in this case, that: Case 2: The homomorphism Z b −→ Z is nil.In this case any automorphism f 4 makes the diagram (5) commutes.Moreover we get the extension: which also splits and we conclude as in the case 1. Consequently we get only 8 triples which are: As every triple is obviously of order 2 we conclude, in this case, that: Notice that in this example, according to Theorem 4.2, the groups E * (X) and E(X) are both infinite.
Remark 2.11.Theorem 2.10 is not valid for CW-complexes of higher dimensions.Nevertheless the author[7] generalize Whitehead's theorems for simply connected n-dimensional CW-complexes where n ≥ 5 by introducing the notion of strong automorphisms of the Whitehead exact sequences of simply connected n-dimensional