Статьи, опубликованные в английской версии журнала
On the Homotopy Types of 2-Connected and 6-Dimensional CW-Complexes
M. Benkhalifa Department of Mathematics, College of Sciences, University of Sharjah
Аннотация:
Let
$\mathbf{CW^6_2}/_{ \simeq}$ be the homotopy category of
{2}-connected \rm{6}-dimensional CW-complexes $X$ such that
$H_{3}(X)$ is uniquely 2-divisible; i.e.,
$H_{3}(X)\otimes \mathbb{Z}_2=0$ and
$\operatorname{Tor} (H_{3}(X);\mathbb{Z}_2)=0$. In this paper, we define an "algebraic" category
$\mathscr{D}$ whose objects are certain exact sequences, a functor $\mathcal{F}\colon \mathbf{CW^6_2}/_{ \simeq} \to\mathscr{D}$ such that
$\mathcal{F}(X)$ is the Whitehead exact sequence of
$X$, and we prove that
$\mathcal{F}$ is a “detecting functor”, a notion introduced by Baues
[1:x129], which implies that the homotopy types of objects in the category
$\mathbf{CW^6_2}$ are in bijection with the isomorphic classes of objects of
$\mathscr{D}$. Consequently, we show that two objects of
$\mathbf{CW^6_2}$ are homotopic if and only if their Whitehead exact sequences are isomorphic in
$\mathcal{D}$.
Ключевые слова:
2-connected 6-dimensional CW-complex, homotopy types, Whitehead's certain exact sequence, detecting functor.
MSC: 55P15 Поступило: 20.04.2023
Исправленный вариант: 11.07.2023
Язык публикации: английский