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

Let CW26/similar or equal to 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) circle times Z(2) = 0 and Tor(H-3(X); Z(2)) = 0. In this paper, we define an "algebraic" category D whose objects are certain exact sequences, a functor F : CW26/similar or equal to -> D such that F(X) is the Whitehead exact sequence of X, and we prove that F is a "detecting functor", a notion introduced by Baues [1], which implies that the homotopy types of objects in the category CW26 are in bijection with the isomorphic classes of objects of D. Consequently, we show that two objects ofCW(2)(6) are homotopic if and only if theirWhitehead exact sequences are isomorphic in D.