Mathematical Sciences: Inverse Limit Problems in Algebraic K-Theory

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This project will study two important constructions in algebraic K-theory. The first is the assembly map for the K-theory and L-theory of group rings. The method will be homotopy- theoretic, using the bounded K-theory of Pedersen and Weibel. The investigator hopes to complete his study of the surjectivity question for cocompact, discrete, torsion-free subgroups of Lie groups, and to generalize the approach to include more general groups with finite classifying space. The second object of study will be the descent spectral sequence for the algebraic K-theory of a field F. He will study a homotopy colimit construction for the K-theory of the algebraic closure of F, in the case where the absolute Galois group is topologically cyclic. The component pieces in the colimit construction will be copies of the K-theory of the field F. The details of these parts vary, but each is concerned with reducing geometric information to a subject for calculation or to perfecting the algebraic machinery used for the calculations. 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, and two of the principal tools for this are homotopy theory and K-theory.