On the rationalization of the circle

Casacuberta, Carles

doi:10.1090/S0002-9939-1993-1176479-3

On the rationalization of the circle
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Proc. Amer. Math. Soc. 118 (1993), 995-1000 Request permission

Abstract:

We give an example showing that, for a nilpotent group $G$ and a set of primes $P$, the $P$-localization homomorphism $l:G \to {G_P}$ need not induce an isomorphism in cohomology with arbitrary (twisted) ${{\mathbf {Z}}_P}$-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 ${{\mathbf {Z}}_P}$-modules and whose fundamental group is uniquely ${P’}$-radicable does not admit a left adjoint.

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