Homotopical localizations of spaces

For a map f of spaces, Dror Farjoun and the author have constructed an f-localization functor, where a space Y is called f-local when map(f, Y) is an equivalence. This very general construction gives all known idempotent homotopy functors of spaces. The main theorem of this paper shows that f-localization functors always preserve fiber sequences of connected H-spaces up to small error terms. For instance, the localization with respect to the nth Morava K-theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups. This implies, for example, that a K(1)-homology equivalence of H-spaces must induce an isomorphism of v₁-periodic homotopy groups. Results are also obtained on the A-nullification or A-periodization functors, which are just the f-localization functors for the maps f from spaces A to points. Two spaces are said to have the same nullity when they give the same nullification functors, and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds. The A-nullifications of nilpotent Postnikov spaces are completely determined.