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 respec t to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homolog y theory E, yielding, for example, an analogue of Quillen's plus-cons truction for E. We discuss in detail the case of ordinary homology wi th coefficients in Z/p or Z[1/p]. Our main tool is a comparison theore m for nullification functors (that is, localizations with respect to m aps of the form f: A --> pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-conn ected spaces coincides with nullification with respect to a Moore spac e M(Z[1/p], 1).