Localization and periodicity in unstable homotopy theory

Abstract

In this paper, we develop a hierarchy of natural localizations of spaces, called the vn-periodizations for n > 0, which may be used to expose and study periodic phenomena in unstable homotopy theory. These vn-periodizations act less radically than the corresponding homological localizations [2] and respect fibrations to a very considerable extent. A major part of this paper is devoted to developing the general theory of periodizations of spaces, thereby providing a foundation for the study of the vn-periodization and many others. Some of this general theory has been developed independently by Dror Farjoun [13], [14], and we refer the reader to his work for an alternative approach with other interesting general results. During the past decade, remarkable progress was made by Ravenel, Hopkins, Devinatz, and Smith [12], [16], [17], [33] toward a global understanding of stable periodic phenomena, and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena. An excellent exposition of localization and periodicity in stable homotopy theory is now available in Ravenel's book [35]. Major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena. These phenomena belong to a hierarchy starting with those detected rationally, followed by those detected in classical K-theory and in the successive Morava K-theories. Most fundamentally, each finite CW-spectrum has an intrinsic periodicity given by a vn self-map which becomes a selfequivalence after suitable localization. For simplicity, we describe our results in the pointed homotopy category Hoo of connected CW-complexes. For spaces W, Y E Hoo , we call Y W-periodic or W-local when the pointed mapping space map. (W, Y) is contractible, or equivalently when [Xt W, Y] * for t > 0. As shown more generally in [3, Corollary 7.2], [10], or [13], there is a natural initial example X -+ PwX of a map from X to a W-periodic space PwX in Hoo, and we call this the W-periodization or W-localization of X. Our notation is derived from the classical example where W = Sn+l and PwX is the n-th Postnikov section