Localization and Periodicity in Unstable Homotopy Theory

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9803601 Bousfield The investigator will continue his research in algebraic topology, using localization methods to expose and analyze periodic phenomena in the unstable homotopy theory of spaces. During the past two decades, remarkable progress has been made 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. This project is part of an effort to develop a similar understanding in unstable homotopy theory. The investigator has introduced a hierarchy of localizations called periodizations, which serve to resolve a given space into a chromatic tower with monochromatic fibers. He will work to understand these periodizations as well as the closely related localizations with respect to Morava K-theories. He recently discovered surprising new functors between the stable and unstable monochromatic layers of homotopy theory. Using these functors and other tools, he will explore the structure of the monochromatic layers. At the classical K-theoretic level, this work is leading to solutions of old problems concerning periodic homotopy groups and the K-theory of spaces. Topological or geometric spaces arise, for instance, as sets of solutions to systems of equations, and they play a central role in mathematics. Beyond their usual measurements, such spaces have deeper properties that persist even after drastic deformations. These properties begin very simply with the number of separate components and the number of "holes" in a low dimensional space but become tremendously rich and informative in higher dimensions. The field of homotopy theory is devoted to their study, using powerful algebraic and geometric techniques. In recent years, an exciting chromatic approach to the subject has emerged, allowing decompositions of complicated homotopical phenomena into much simpler periodic parts. However, this chromatic approach was originally rest ricted to certain stable phenomena. In the present project, the investigator is working to extend it to a full range of unstable homotopy theoretic phenomena. ***