Resolutions of Spaces by Cubes of Fibrations

J.-L. Loday has used w-cubes of fibrations, where n is a non-negative integer, in his study of spaces with finitely many non-trivial homotopy groups [4]. His main result is the construction of an algebraic category equivalent to the weak homotopy category of path-connected spaces Z with TI^Z = 0 for / > w+1 [4, 1.7]. One step in the proof [4, 3.5] requires the construction of certain «-cubes offibrations, and the object of the present paper is to give a simple and explicit method of doing this; R. Brown has noticed that the proof of this step in [4] is incorrect, although Loday has since found a correct proof. I first described the construction in a letter to Brown and Loday in 1984. I am grateful to R. Brown for his encouragement in publishing it, and for general advice.