LOCALIZING WITH RESPECT TO SELF-MAPS OF THE CIRCLE

We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected CW-complexes, some of which extend P-localization of nilpotent spaces, at a set of primes P. We focus our attention on one such functor, whose local objects are CW-complexes X for which the pth power map on the loop space OMEGAX is a self-homotopy equivalence if p is-not-an-element-of P. We study its algebraic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield's homology localization, Bousfield-Kan completion, and Quillen's plus-construction.