In this chapter we establish the following simple and useful sufficient conditions on a tower of fibrations {Ys}, in order that it can be used to obtain the homotopy type of the R-completion of a given space X: (i)
If f: X → {Ys is a map which induces, for every R-module M, an isomorphism $$\mathop {\lim }\limits_ \to H*\left( {{Y_S};M} \right) \approx H*\left( {X;M} \right)$$ then f induces a homotopy equivalence $${R_\infty }X \simeq \mathop {\lim }\limits_ \to {R_\infty }{Y_S}$$ .
(ii)
If, in addition, each Ys is R-complete (Ch.I, 5.1) , then the space $$\mathop {\lim }\limits_ \to {Y_S}$$ already has the same homotopy type as R∞X.
(iii)
If, in addition, each Ys satisfies the even stronger condition of being R-nilpotent (4.2), then, in a certain precise sense, the tower {Ys} has the same homotopy type as the tower {Rs}