Abstract
Let \(\mathbf{CW^6_2}/_{ \simeq}\) be the homotopy category of 2-connected 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], 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 \(\mathcal{D}\).
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The author is deeply grateful to the referee for carefully reading the article and for valuable suggestions, which have greatly improved the manuscript.
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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
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Benkhalifa, M. On the Homotopy Types of 2-Connected and 6-Dimensional CW-Complexes. Math Notes 114, 687–703 (2023). https://doi.org/10.1134/S0001434623110068
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DOI: https://doi.org/10.1134/S0001434623110068