Unstable localization and periodicity

In the 1980's, remarkable advances were made by Ravenel, Hopkins, Devinatz, and Smith toward a global understanding of stable homotopy theory, showing that some major features arise "chromatically" from an interplay of periodic phenomena arranged in a hierarchy (see [20], [21], [28]). We would like very much to achieve a similar understanding in unstable homotopy theory and shall describe some progress in that direction. In particular, we shall explain and extend some results of our papers [4], [11], and some closely related results of Dror Farjoun and Smith [17], [18], [19]. Periodic phenomena in stable homotopy theory are quite effectively exposed by localizations with respect to various periodic homology theories such as the Morava K-theories [6]' [27]. This approach remains promising in unstable homotopy theory, but a different sort of localization, called the W-nulliJication or W-periodization for a chosen space W, now seems more fundamental and effective. It simply trivializes the [W, -]*-homotopy of spaces in a universal way. In Section 1 of this article, we recall the general theory of nullifications, including some crucial properties which have only recently been discovered. In Section 2, we introduce a corresponding theory of nullifications for spectra which we apply to determine nullifications of Eilenberg-MacLane spaces and other infinite loop spaces. In Section 3, we begin to classify spaces according to the nullification functors which they produce, and prove a classification theorem for finite suspension complexes similar to the Hopkins-Smith classification theorem for finite spectra. In Section 4, we study the arithmetic nullifications, which act very much like classical localizations and completions of spaces. We apply them to determine arbitrary nullifications of Postnikov spaces and to extend the classification results of Section 3 beyond finite suspension complexes. In Section 5, we present an unstable chromatic tower providing successive approximations to a space, incorporating higher and higher types of periodicity. In Section 6, we introduce a sequence of monochromatic homotopy categories containing the successive fibres of chromatic towers. Using work of Kuhn [22] and others, we show that the nth stable monochromatic homotopy category embeds as a categorical retract of its unstable counterpart. Finally, in Section 7, we apply some of the preceding work to obtain general results on E*-acyclicity and E*-equivalences of spaces for various spectra E.