In his book, "Null Spaces, Cellular Spaces and Homotopy Localization", Emmanuel Dror Farjoun proves that simply connected K-theory acyclic p-local spaces have suspensions that are built out of V(1), and its suspensions, in a sense which is similar to the way CW-complexes are built out of S^0 and its suspensions.
I'll discuss a generalization of this result. It turns out that adequately connected S(n)-acyclic p-local spaces have suspensions that are built out of V(n) and its suspensions. At least, up to:
1) A little intentional sloppiness in terms of what I mean by V(n), which I'll clarify.
2) A slight fine tuning of the hypotheses for large n, which again I'll clarify.