Decomposition of loop spaces and periodic problem onπ∗

Abstract

Recently, a new interesting problem in this area has been proposed. Namely, find spaces X whose stable homotopy groups are summands of the unstable homotopy groups. Beben and Wu [1] gave examples of such spaces and applied their results to the Moore Conjecture. They showed that for a fixed odd prime p and some p–localization of a CW–complex of finite type X , there exists a sequence flng that converges to infinity such that †nX is a homotopy retract of X . Hence kC1.†nX / is a retract of k.X /. Letting flng converge to infinity, the stable homotopy groups of X are seen to be summands of its unstable homotopy groups. Symbolically, the group s .X / is a summand of .X /. In this way, Beben and Wu reduced the aforementioned problem to finding spaces X together with a sequence flng that converges to infinity such that †nX is a retract of X . In this article, we consider the case when p D 2. Our results are given as follows.