On towers approximating homological localizations

Our object of study is the natural tower which, for any given map f: A → B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E*, yielding, for example, an analogue of Quillen's plus-construction for E*,. We discuss in detail the case of ordinary homology with coefficients in ℤ/p or ℤ[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f: A → pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(ℤ[1/p], 1).