Localizations and evaluation subgroups

Abstract

If G n(X) is the nth evaluation subgroup of a simple connected finite CW-complex, then G (Xp) Gn(X)p for p = 0 or a prime. Let X be a connected simple finite CW-complex, Xp its localization at p for p prime or 0 [51, and Gn(X) the evaluation subgroup of 7rn(X) [1]. If epX -X is the canonical map, we show that ca e G (X) if and only if e *(a) E G (X ) for all p. As corollaries we obtain that G (X ) G (Xp) n and X is a G-space [1] if and only if X is a G-space for all p, where Gn(X)P is the localization of the group Gn(X). This is analogous to results obtained for H-spaces [31, [5]. All H-spaces are G-spaces, and many properties of H-spaces are shared by G-spaces. 1. Preliminaries. Spaces X and W are assumed to be pointed, simple (abelian fundamental groups acting trivially on the homotopy and homology groups), connected, finite CW-complexes. We will not distinguish between a map and its homotopy class. For p a prime let Qp= tk/qlk, q integers, p t q} and QO the rationals. Qp is the localization of the integers at the prime p. The general reference for localization theory is [5]. We review some of these results here. Definition 1.1. A space X is p-local if 7r*(X) admits a Q p-module structure extending the usual Z-module structure. For each X there is a p-local space X p and canonical map ep: X Xp such that if g:X -4 Y, where Y is p-local, there is a unique (up to homotopy) g * Xp -+Y such that g !g ep. This is equivalent to the map pb: 7Tn(X) 0 Qp 7r (X ) being an isomorphism, where ?p(a 8 r) = re *(ax), where the multiplication is the Q p-module structure on 7 n(Xp) (ep p-localizes in homotopy). Finally we point out that localization is functorial. Key results about the evaluation subgroups can be found in [1]. We will establish the notations needed in this paper. L(W, X; /) will be the Presented to the Society, January 26, 1975 under the title Localizations and Gn(X); received by the editors July 2, 1973 and, in revised form, February 27, 1974. AMS (MOS) subject classifications (1970). Primary 55E05; Secondary 55F05.