On the Homotopy Types of 2-Connected and 6-Dimensional CW-Complexes

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 $\mathcal{D}$.