LOCALIZING WITH RESPECT TO SELF-MAPS OF THE CIRCLE

We describe a general procedure to construct idempotent functors on th e pointed homotopy category of connected CW-complexes, some of which e xtend P-localization of nilpotent spaces, at a set of primes P. We foc us our attention on one such functor, whose local objects are CW-compl exes 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 algebra ic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield's homology localization, Bousfield-Ka n completion, and Quillen's plus-construction.