ON THE RATIONALIZATION OF THE CIRCLE

We give an example showing that, for a nilpotent group G and a set of primes P, the P-localization homomorphism l:G --> G(P) need not induce an isomorphism in cohomology with arbitrary (twisted) Z(P)-module coe fficients. From this fact we infer that, in the pointed homotopy categ ory of connected CW-complexes, the inclusion of the subcategory of spa ces whose higher homotopy groups are Z(P)-modules and whose fundamenta l group is uniquely P'-radicable does not admit a left adjoint.