On the 2-primary -periodic homotopy groups of spaces

We develop foundations of a general approach for calculating p-primary v 1 -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 v 1 -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 v 1 -stabilization functor Φ 1 from spaces to spectra. Roughly speaking, we obtain the p-primary v 1 -periodic homotopy of a space X from the p-adic KO-cohomology of Φ 1 X , which we obtain from the p-adic KO-cohomology and K-cohomology of X by a v 1 -stabilization process under suitable conditions.