Mahmoud Benkhalifa

ON THE CONJUGACY PROBLEM OF POSITIVE BRAIDS

In the symmetric group Sn, we introduced the notion of crossing and linking numbers to each permutation. Then a unique factorization of a permutation is given due to its crossing number of its factors and how the factors are linked. Consequently we introduced a matrix associated to each permutation, which we used it as a tool to prove that positive braids with different matrices are not conjugate braids. Up to n≤5 it is proved that two positive permutation braids are conjugate if and only if they have the same matrix, and a complete calculation of these matrices with the associated link type is given.

WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA

Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

On the homotopy type of (n-1)-connected (3n+1)-dimensional free chain Lie algebra

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ⊒Q, then p=∞. Denote by DGLnnp, n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGLnnp. In this work we intend to answer the following two questions: Given an object (L(V), ϖ) in DGLn3n+2 and denote by S(L(V), ϖ) the class of objects homotopy equivalent to (L(V), ϖ). How we can characterize a free dgl to belong to S(L(V), ϖ)? Fix an object (L(V), ϖ) in DGLn3n+2. How many homotopy equivalence classes of objects (L(W), δ) in DGLn3n+2 such that H*(W, d′)≊H*(V, d) are there? Note that DGLn3n+2 is a subcategory of DGLnnp when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.

CARDINALITY OF RATIONAL HOMOTOPY TYPES OF SIMPLY CONNECTED CW COMPLEXES

The aim of this paper is to consider the following problem: given a simply connected CW complex X of finite type, find the cardinal of the set of rational homotopy types of CW complexes Y such that H*(Y, ℚ) is isomorphic, as a commutative graded algebra, to H*(X, ℚ). For this purpose we introduce a long exact sequence, called the Whitehead exact sequence, which we associated with the minimal Sullivan model of X and which allows us to define the notion of the adapted couple associated with X. Thus we show that the rational homotopy type of X is completely determined by the equivalence class of its adapted couple.

ON THE "CERTAIN" EXACT SEQUENCE OF WHITEHEAD

This paper is devoted to the classification problem of homotopy types of a new class of simply connected CW-complexes called diagonal spaces which contains all spaces X such that H*(X, ℤ) is free. We show that two simply connected diagonal CW-complexes X and Y are homotopic if and only if their Whitehead exact sequences are isomorphic.