Mathematical Sciences: Lie Groups up to Homotopy

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9505006 Wilkerson Ever since work of Serre in the 1950's, algebraic topology has often been studied one prime at a time. This idea, borrowed from algebra and number theory, has resulted in new questions being raised. For example, what does it mean for a topological space to be a Lie group at a particular prime p? One can answer with a reformulation that combines the group structure on G and the topological structure on G into a single classifying space, BG. Further abstraction of the properties true for all BG leads to the definition of a p-compact group which is a suitable homotopical analogue of a Lie group at the prime p. Remarkably, these p-compact groups possess analogues of standard Lie theoretical constructions such as maximal tori, Weyl groups, and root systems. The goal is to use these structures to produce a classification that is similar in spirit to the classification of compact connected Lie groups, but different in detail. A by-product would be the complete solution to the Steenrod question (1961) of which spaces have mod-p cohomology algebras that are finitely generated polynomial rings. Algebraic topology studies geometric objects by abstracting more tractable algebraic information. One rich source of such data is the group of symmetries of a geometric configuration. These so- called Lie groups, after the 19th century analyst and geometer, Sophus Lie, have had profound applications in algebra, physics, and number theory. One advantage of algebraic topology and homotopy theory is that its natural tools, such as homology and homotopy groups, are impervious to small deformations or changes in the geometric structure under study. The goal of this work is to create a classification of the homotopical analogues of Lie groups by the study of the algebraic topology of their classifying spaces. ***