I. The theory of topological localization has been developed in various forms by several authors (cf. for example [1] , [2], [8] , [10] , [12] ). In this note, we place ourselves in the framework of [10] and announce some foundational results in the theory. The importance and usefulness of these results has already been well illustrated, for example, by the applications in [4], [5], [9] . Before stating our main result, we fix some notation. Let X be a connected, simple CW-complex of finite

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... . If P is a collection of primes, we denote by X P the localization of X at P, by e P : X -» X P the localization map, by X 0 the rationalization of X, and by r P : X P -• X 0 the rationalization map, characterized by the equation r P ° e P = e 0 [10]. We then have THEOREM 1. Let Wbea connected, finite CW-complex and X a connected, simple CW-complex of finite type. Then the natural map d>:[^X]->ri([^^]-^[^^o]) p from the (pointed) homotopy set [VKX] into the pullback, over all primes p, of the maps r' p , induced by r p , is a set-theoretic bijection. The surjectivity of O is an easy consequence of Theorem 2 below and known results about localization [10]. The injectivity of may be established by elementary arguments in abelian group theory provided W is a sphere, or more generally a double suspension. A more delicate argument, but still requiring only abelian group theory, was introduced by Mimura-Nishida-Toda [8] to prove the injectivity of O for W a suspension. We show that the general case, with no added restriction on W, leads naturally to a problem in the theory of nilpotent groups. The connection between our problem and the theory of nilpotent groups stems from a classical result of G. W. Whitehead [11] asserting that, for W a connected finite complex and G a topological group of finite type, the (pointed) homotopy set [P^G] admits naturally the structure of a finitely generated nilpotent group. The solution of our problem is strongly AM S 1970 subject classifications. Primary 55D99.