On the 2-primary v1-periodic homotopy groups of spaces

We develop foundations of a general approach for calculating p-primary v1-periodic homotopy groups of spaces using their p-adic KO-cohomologies and K-cohomologies with particular attention to the case p=2. As a main application, we derive a method for calculating v1-periodic homotopy groups of simply connected compact Lie groups using their complex, real, and quaternionic representation theories. This method has been applied very effectively by Davis in recent work. We rely heavily on the v1-stabilization functor Φ1 from spaces to spectra. Roughly speaking, we obtain the p-primary v1-periodic homotopy of a space X from the p-adic KO-cohomology of Φ1X, which we obtain from the p-adic KO-cohomology and K-cohomology of X by a v1-stabilization process under suitable conditions.