On the rationalization of the circle

Abstract

We give an example showing that, for a nilpotent group G G and a set of primes P P , the P P -localization homomorphism l : G → G P l:G \to {G_P} need not induce an isomorphism in cohomology with arbitrary (twisted) Z P {{\mathbf {Z}}_P} -module coefficients. From this fact we infer that, in the pointed homotopy category of connected CW-complexes, the inclusion of the subcategory of spaces whose higher homotopy groups are Z P {{\mathbf {Z}}_P} -modules and whose fundamental group is uniquely P ′ {P’} -radicable does not admit a left adjoint.