Math Reviews: 57:1484
This paper is a attempt to understand the Bousfield integral homology localization of a space X (52:1676 Bousfield, A. K. The localization of spaces with respect to homology. Topology 14 (1975), 133--150). It is shown that if the n'th Postnikov stage of X is nilpotent, then through dimension 2n or so the homotopy groups of the localization of X can be expressed up to a spectral sequence in terms of certain additive functors applied to the homotopy groups of X. (In defining these additive functors, it is necessary to take into account the action of the fundamental group of X on the higher homotopy groups.) The model for this is the exact sequence in Bousfield and Kan for the homotopy groups of the p-completion of a nilpotent space X in terms of certain Hom and Ext functors applied to the homotopy groups of X. The formula of Bousfield and Kan works in all dimensions; because the diagonal action of a group on a tensor product of two modules can be complicated, the methods of this paper works only in a stable range.