arXiv subjects

Youssef Rami

Publications and source records attributed to Youssef Rami.

10 recordsLinked to original sources

arXiv2024

On Topological Complexity of $(r,\rho(R))$-mild spaces

In this paper, we first prove the existence of relative free models of morphisms (resp. relative commutative models) in the category of $DGA(R)$ (resp. $CDGA(R)$), where $R$ is a principal ideal domain containing $\frac{1}{2}$. Next, we restrict to the category of $(r,\rho(R))$-H-mild algebras and we introduce, following Carrasquel's characterization, $secat(-, R)$, the sectional category for surjective morphisms. We then apply this to the $n$-fold product of the commutative model of an $(r,\rho(R))$-mild CW-complex of finite type to introduce $TC_n(X,R)$, $mTC_n(X,R)$ and $HTC_n(X,R)$ which extend well known rational topological complexities. We do the same for $\operatorname{sc(-, \mathbb{Q})}$ to introduce analogous algebraic $\operatorname{sc(-,R)}$ in terms of their commutative models over $R$ and prove that it is an upper bound for $secat(-, R)$. This also yields, for any $(r,\rho(R))$-mild CW-complex, the algebraic $tc_n(X,R)$, $mtc_n(X,R)$ and $Htc_n(X,R)$ whose relation to the homology nilpotency is investigated. In the last section, in the same spirit, we introduce in $DGA(R)$, $secat(-, R)$, $\operatorname{sc(-,R)}$ and their topological correspondents. We then prove, in particular, that $ATC_n(X,R)\leq TC_n(X,R)$ and $Atc_n(X,R)\leq tc_n(X,R)$.

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arXiv2023

An upper bound for the rational topological complexity of a family of elliptic spaces

In this work, we show that, for any simply-connected elliptic space $S$ admitting a pure minimal Sullivan model with a differential of constant length, we have ${\rm TC}_0(S)\leq 2{\rm cat}_0(S)+\chi_{\pi}(S)$ where $\chi_{\pi}(S)$ is the homotopy characteristic. This is a consequence of a structure theorem for this type of models, which is actually our main result.

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arXiv2022

On the rational topological complexity of coformal elliptic spaces

We establish some upper and lower bounds of the rational topological complexity for certain classes of elliptic spaces. Our techniques permit us in particular to show that the rational topological complexity coincides with the dimension of the rational homotopy for some special families of coformal elliptic spaces.

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arXiv2022

On the rationality conjecture of some finite CW-complexes

In this paper, we establish the rationality conjecture raised in \cite{FKS} for any $(r-1)$-connected ($r\geq 2$) $kr$-dimensional CW-complex $X$ ($k\geq 2$) having a unique spherical cohomology class $u\in \tilde{H}^r(X, \mathbb{Z})$ such that $u^k\not =0$. %which is nilpotent with order of nilpotency equal to $k+1$. Next, we illustrate (topologically) our result by giving the minimal cell structure of such a CW-complex whose cohomology is a truncated polynomial algebra.

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arXiv2022

On Topological Complexity of Gorenstein spaces

In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X;\mathbb{Q})}(\mathbb{Q},C^{\ast}(X;\mathbb{Q}))$ with a graded commutative algebra structure. This leads us to introduce the $\mathcal{E}xt$-version of higher (resp. module, homology) topological complexity of $X_0$, the rationalization of $X$ (resp. of $X$ over $\mathbb{Q}$). We then make comparisons between these invariants and their respective ordinary ones for Gorenstein spaces. We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristics especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.

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arXiv2019

On the cohomology of elliptic coformal spaces

In this article, we pursue the study begun in \cite{Lup02} on the cohomology of rationally elliptic coformal spaces. Consequently, we complete, for such spaces, the proof of Lupton's conjecture and deduce Hilali's.

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arXiv2018

A variant of the topological comlexity of a map

In this paper, we associate to two given continuous maps $f,g: X\rightarrow Z$, on a path connected space $X$, the relative topological complexity $TC^{(f, g, Z)}(X):=TC_X(X\times _ZX)$ of their fiber space $X\times _ZX$. When $g=f$ we obtain a variant of the topological complexity $TC(f)$ of $f: X\longrightarrow Z$ generalizing Farber's topological complexity $TC(X)$ in the sens that $TC(X)=TC(cst_{x_0})$; being $cst_{x_0}$ the constant map on $X$. Moreover, we prove that $TC(f)$ is a fiberwise homotopy equivalence invariant. When $(X,x_0)$ is a pointed space, we prove that $TC^{(f, cst_{x_0}, Z)}(X)$ interpolates $cat(X)$ and $TC(X)$ for any continuous map $f$.

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arXiv2017

Gaps in the Milnor-Moore spectral sequence and the Hilali conjecture

In his study of Halperin's toral-rank conjecture, M. R. Hilali conjectured that for any simply connected rationally elliptic space $X$, one must have $dimπ_*(X)\otimes \mathbb{Q} \leq dimH^*(X,\mathbb{Q})$. Let $(ΛV, d)$ denote a Sullivan minimal model of $X$ and $d_k$ the first non-zero homogeneous part of the differential $d$. In this paper, we use spectral sequence arguments to prove that if $(ΛV, d_k)$ is elliptic, then, there is no gaps in the $E_{\infty}$ term of the Milnor-Moore spectral sequence of $X$. Consequently, we confirm the Hilali conjecture when $V = V^{odd}$ or else when $k\geq 3$ and $(ΛV, d_k)$ is elliptic.

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arXiv2015

LS-Category and the Depth of Rationally Elliptic Spaces

Let $X$ be a finite type simply connected rationally elliptic CW-complex with Sullivan minimal model $(ΛV, d)$ and let $k\geq 2$ the biggest integer such that $d=\sum_{i\geq k}d_i$ with $d_i(V)\subseteq Λ^iV$. We show that: $cat(X_{\mathbb{Q}}) = depht(ΛV, d_k)$ if and only if $(ΛV,d_{k})$ is elliptic. This result is obtained by introducing tow new spectral sequences that generalize the Milnor-Moore spectral sequence and its $\mathcal{E}xt$-version \cite{Mur94}. As a corollary, we recover a known result proved - with different methods - by L. Lechuga and A. Murillo in \cite{LM02} and G. Lupton in \cite{Lup02}: If $(ΛV,d_{k})$ is elliptic, then $cat(X_{\mathbb{Q}}) = dim(π_{odd}(X)\otimes\mathbb{Q}) + (k-2)dim(π_{even}(X)\otimes\mathbb{Q})$. In the case of a field ${IK}$ of $char({IK})=p$ (an odd prim) we obtain an algebraic approach for $e_{IK}(X)$ where $X$ is an $r$-connected ($r\geq 1$) finite CW-complex such that $p> dim(X)/r$.

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arXiv2015

A new invariant that's a lower bound of LS-category

Let $X$ be a simply connected CW-complex of finite type and $\mathbb{K}$ any field. A first known lower bound of LS-category $cat(X)$ is the Toomer invariant $e_{\mathbb{K}} (X)$ (\cite{Too}). In $1980$'s Félix et al. introduced the concept of {\it depth} in algebraic topology and proved the depth theorem: $depth (H_*(ΩX, \mathbb{K})) \leq cat(X)$. In this paper, we use the Eilenberg-Moore spectral sequence of $X$ to introduce a new numerical invariant, denoted by $\textsc{r}(X, \mathbb{K})$, and show that it has the same properties as those of $e_{\mathbb{K}} (X)$. When the evaluation map (\cite{FHT88}) is non-trivial and $char(\mathbb{K})\not = 2$, we prove that $\textsc{r}(X, \mathbb{K})$ interpolates $depth(H_*(ΩX, \mathbb{K}))$ and $e_{\mathbb{K}} (X)$. Hence, we obtain an improvement of L. Bisiaux theorem (\cite{Bis99}) and then of the depth theorem. Motivated by these results, we associate to any commutative differential graded algebra $(A,d)$, a purely algebraic invariant $\textsc{r}(A,d)$ and, via the theory of minimal models, we relate it with our previous topological results. In particular, if $(ΛV,d)$ is a Sullivan minimal algebra such that $d=\sum_{i\geq k}d_i$ and $d_i(V)\subseteq Λ^iV$, a greater lower bound is obtained, namely $e_0(ΛV, d)\geq \textsc{r}(ΛV, d) + (k-2)$.

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