Comparing Kac-Moody groups over the complex numbers and fields of positive characteristic via homotopy theory

Abstract

This dissertation concerns the homotopical group theory of Kac-Moody groups. Applications stem from homotopical expressions of infinite and noncompact group classifying spaces in terms of finite and compact group classifying spaces through local to global constructions. New homotopy decompositions for the "unipotent" factors of parabolic subgroups of a discrete Kac-Moody group are given in terms of unipotent algebraic groups. As in the Lie case [23], a map is constructed from the classifying space of the discrete Kac-Moody group over the algebraic closure of the field with p elements to the complex topological Kac-Moody group of the same type. Rank 2, non-Lie, Kac-Moody groups are studied to show that in contrast to the Lie case [22] the classifying space of the discrete Kac-Moody group over the field with the kth power of p elements and the homotopy fixed points of the complex topological Kac-Moody group of the same type with respect to a newly constructed kth power of p unstable Adams operation frequently have different homotopy types after localization with respect to homology with coefficients in a field of characteristic relatively prime to p.

*Please refer to dissertation for references/footnotes.