Mathematical Sciences: Function Complexes And The Steenrod Algebra In Homotopy Theory

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The three investigators, Wilkerson, Smith, and McClure, are involved in three connected general investigations in homotopy theory. A major goal of Wilkerson's research is to complete the classification of finite loop spaces as a generalization of compact Lie group theory. The tools are function complexes and calculations with the Steenrod algebra going back to work of Adams- Wilkerson, Miller, Carlsson, and Lannes. Part of the work generalizes known facts about group actions and fixed points to homotopy fixed points. Smith seeks an explicit construction of the Morava K-theories. These homology theories play a vital role in the Devinatz-Hopkins-Smith nilpotence theorem. Previous constructions rely on geometric techniques which are hard to interpret homotopically. The resulting classifying spaces can really be constructed only "up to homotopy," leaving many questions about their nature open. Finally, in joint work with S. Jackowski and R. Oliver, McClure intends to try to find a simplified proof of the immersion conjecture. His other three problems all involve some aspect of the theory of topological Hochschild homology. The details of these three parts vary, but all are concerned either with reducing geometric information to a subject for calculation or to perfecting one of the principal algebraic tools used for this purpose. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation.