On the 2-adic $K$-localizations of $H$-spaces

Abstract

We determine the 2-adic $K$-localizations for a large class of $H$-spaces and related spaces. As in the odd primary case, these localizations are expressed as fibers of maps between specified infinite loop spaces, allowing us to approach the 2-primary $v_1$-periodic homotopy groups of our spaces. The present $v_1$-periodic results have been applied very successfully to simply-connected compact Lie groups by Davis, using knowledge of the complex, real, and quaternionic representations of the groups. We also functorially determine the united 2-adic $K$-cohomology algebras (including the 2-adic $KO$-cohomology algebras) for all simply-connected compact Lie groups in terms of their representation theories, and we show the existence of spaces realizing a wide class of united 2-adic $K$-cohomology algebras with specified operations.