Mathematical Sciences: Localization and Periodicity in Homotopy Theory

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9504497 Bousfield Bousfield will continue his research in homotopy theory, using localization methods to expose and analyze unstable periodic phenomena. During the 1980's, remarkable progress was 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. In recent work, Bousfield introduced a hierarchy of localizations, called the vn-periodizations for positive integers n, which serve to resolve a given space into a chromatic tower with monochromatic fibres. He will work to understand these vn-periodizations of spaces as well as the closely related homological localizations. He has already established strong links between v1-periodizations and K-homology localizations of spaces, leading to solutions of old problems concerning the K-theory of iterated loop spaces. Geometric objects or spaces play a central role in mathematics, arising for instance from the solution sets of systems of equations. Beyond their usual measurements, such spaces have deeper properties which persist even after drastic deformations. These properties begin almost trivially with the number of separate components and the number of holes 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. Various periodic phenomena in homotopy theory have been discovered since the 1950's, and an exciting "chromatic" approach to the subject was developed in the 1980's, allowing resolutions of confusingly complicated phenomena into much simpler periodic parts. This approach was originally restricted to certain simpler phenomena known as stable phenomena, but it is now being extended to a full, fascinating, important range of unstable homotopy theoretic phenome na. ***