Localizing with respect to self-maps of the circle

Abstract

We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected CW {\text {CW}} -complexes, some of which extend P P -localization of nilpotent spaces, at a set of primes P P . We focus our attention on one such functor, whose local objects are CW {\text {CW}} -complexes X X for which the p p th power map on the loop space Ω X \Omega X is a self-homotopy equivalence if p ∉ P p \notin 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.