arXiv subjects
Publications and source records attributed to Mahmoud Benkhalifa.
For a rational elliptic space, this paper examines the relationship between its homotopy groups and its self-homotopy equivalence group. Moreover, we investigate how this group is embedded in general linear groups.
We prove that any group $G$ occurs as $\E(X)$, where $X$ is CW-complex of finite dimension and $\E(X)$ denotes its group of self-homotopy equivalence. Thus, we generalize a well know-theorem due to Costoya and Viruel \cite{CV} asserting that any finite group occurs as $\E(X)$, where $X$ is rational elliptic space.
In \cite{G}, G. Lupton conjectured that the group of self-homotopy equivalences of an $F_0$-space inducing the identity on the homotopy groups is finite. Thus, the aim of this paper is to establish this conjecture.
Let $X$ be a \text{\rm{2}}-connected and \text{\rm{6}}-dimensional CW-complex $X$ such that $H_{3}(X)\otimes\Z_2=0$. This paper aims to describe the group $\E(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup $\E_{*}(X)$ of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of $X$, denoted by WES$(X)$, we define the group $Γ\mathcal{S}(X)$ of $Γ$-automorphisms of WES$(X)$ and we prove that $\E(X)/\E_*(X)\cong Γ\mathcal{S}(X).$
In this paper we prove that any finite group of order $n$ can be viewed as the group of the solutions of a certain matrix equation $XB=BY$, where the unknowns $X,Y$ are two permutation matrices of order $n$ and $(1+k)n+2 $ respectively and where $k\in \Bbb N$ is given by Cayley's theorem. Moreover, we show that $G$ is isomorphic to a certain subgroup formed by permutation matrices of order $(1+k)n$ obtained by permuting all the rows of the identity matrix $I_{(1+k)n}$.
For a simply connected rationally elliptic CW-complex $X$, we show that the cohomology and the homotopy Euler-Poincaré characteristics are related to two new numerical invariants namely $η_{X}$ and $ρ_{X}$ which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of $X$.
Let $R$ be a (P.I.D) and let $T(V),\partial)$ be a free $R$-dga. The quasi-isomorphism type of $(T(V),\partial)$ is the set, denoted $\{(T(V),\partial)\}$, of all free dgas which are quasi-isomorphic to $(T(V),\partial)$. In this paper we investigate to characterize and to compute the set $\{(T(V),\partial)\}$ for a new class of free dgas called perfect (a special kind of a perfect dga is the Adams-Hilton model of simply connected CW-complex such that $H_{*}(X,R)$ is free). We show that if $(T(V),\partial)$ and $(T(W),δ)$ are two perfect dgas, then $(T(W),δ)\in \{(T(V),\partial)\}$ if and only if their Whitehead exact sequences are isomorphic. Moreover we show that every dga $(T(V),\partial)$ can be split to give a pair $\big((T(V),\widetilde{\partial}),(π_{n})_{n\geq 2}\big)$ consisting with a perfect dga $(T(V),\widetilde{\partial})$ and a family of extensions $(π_{n})_{n\geq 2}$ and we establish that if $(T(W),\widetildeδ)\in \{(T(V),\widetilde{\partial})\}$ and if the extensions $(π_{n})_{n\geq 2}$ and $(π'_{n})_{n\geq 2}$ are isomorphic (in a certain sense), then $(T(W),δ)\in \{(T(V),\partial)\}$.
Let $X$ be a simply connected rational elliptic space of formal dimension $n$ and let $\E(X)$ denote the group of homotopy classes of self-equivalences of $X$. If $X^{[k]}$ denotes the $k^{\text{th}}$ Postikov section of $X$ and $X^{k}$ denotes its $k^{\text{th}}$ skeleton, then making use of the models of Sullivan and Quillen we prove that $\E(X)\cong\E(X^{[n]})$ and if $n>m=max\big\{k \,| \,π_{k}(X)\neq 0\big\}$ and $\E(X)$ is finite, then $\E(X)\cong\E(X^{m+1})$. Moreover, in case when $X$ is 2-connected, we show that if $π_{n}(X)\neq0$, then the group $\E(X)$ is infinite.
Let $R$ be a principal ideal domain (PID). For a simply connected CW-complex $X$ of dimension $n$, let $Y$ be a space obtained by attaching cells of dimension $q$ to $X$, $q>n$, and let $A(Y)$ denote an Adams-Hilton model of $Y$. If $\mathcal E(A(Y))$ denotes the group of homotopy self-equivalences of $A(Y)$ and $\mathcal E_{*}(A(Y))$ its subgroup of the elements inducing the identity on $H_{*}( Y,R)$, then we construct two short exact sequences: $$\underset{i}{\oplus}\,H_{q}(ΩX,R)\rightarrowtail \mathcal{E}(A(Y))\overset{}{ \twoheadrightarrow}Γ^{q}_{n}\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\underset{i}{\oplus}\,H_{q}(ΩX,R) \rightarrowtail \E_{*}(A(Y))\overset{}{ \twoheadrightarrow}Π^{q}_{n} $$ where $i=\mathrm{rank} \,H_{q}(Y,X;R)$, $Γ^{q}_{n}$ is a subgroup of $\aut(\mathrm{Hom}_{}(H_{q}( Y,X;R))\times \E(A(X))$ and $Π^{q}_{n}$ is a subgroup of $\mathcal E_{*}(A(X))$.
This paper defines an invariant associated to Whitehead's certain exact sequence of a simply connected CW-complex which is much more elementary - and less powerful - than the boundary invariant of Baues. Nevertheless, in good cases, it classifies the homotopy types of CW-complexes.
Let R be a subring of the rationals with least non-invertible prime p. Let X = X^{n} \cup_α (\bigcup_{j \in J} e^{q}) be a cell attachment with J finite and q small with respect to p. Let E(X_R) denote the group of homotopy self-equivalences of the R-localization X_R. We use DG Lie models to construct a short exact sequence 0 \to \bigoplus_{j \in J}π_q(X^n)_R \to E(X_R) \to C^q \to 0 where C^q is a subgroup of GL_{|J|}(R) \times E(X^n_R). We obtain a related result for the R-localization of the nilpotent group E_*(X) of classes inducing the identity on homology. We deduce some explicit calculations of both groups for spaces with few cells.
For a 1-connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$. The aim of this paper is to prove that, for every $n\in\Bbb N$, there exists a 1-connected rational CW-complex $X_{n}$ such that $\mathcal{E}(X_{n})\cong \underset{2^{n+1}\mathrm{. times}}{\underbrace{\Bbb Z_{2}\oplus... \Bbb \oplus \Bbb Z_{2}}}$.
For a simply connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalence of $X$ and let $\mathcal{E}_{\sharp}(X)$ be its subgroup of homotopy classes which induce the identity on homotopy groups. As we know, the quotient group $\frac{\mathcal{E}(X)}{\mathcal{E}_{\sharp}(X)}$ can be identified with a subgroup of $Aut(π_{*}(X))$. The aim of this work is to determine this subgroup for rational spaces. We construct the Whitehead exact sequence associated with the minimal Sullivan model of $X$ which allows us to define the subgroup $\mathrm{Coh.Aut}(\mathrm{Hom}\big(π_{*}(X),\Bbb Q)\big)$ of self-coherent automorphisms of the graded vector space $\mathrm{Hom}(π_*(X),\Bbb Q)$. As a consequence we establish that $\mathcal{E}(X) / \mathcal{E}_{\sharp}(X) \cong \mathrm{Coh.Aut} (\mathrm{Hom}(π_*(X),\Bbb Q))$. In addition, by computing the group $\mathrm{Coh.Aut}\big(\mathrm{Hom}(π_{*}(X),\Bbb Q)\big)$, we give examples of rational spaces that have few self-homotopy equivalences.