Topological localization and nilpotent groups

Peter Hilton, Guido Mislin, Joseph Roitberg
1972 Bulletin of the American Mathematical Society  
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
more » ... . 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.
doi:10.1090/s0002-9904-1972-13110-0 fatcat:742hfyk6hbhk3kj2gesao245c4