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

A.K. Bousfield
bous@uic.edu

AMS Classification Numbers: 55Q51(Primary),55N15,55P60,55S25,57T20

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-cohomolgies 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 
D.M. Davis in recent work.  We rely heavily on the p-primary 
v1-stabilization functor Phi 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 Phi X, which we obtain from the 
p-adic KO-cohomology and K-cohomology  of X by a v1-stabilization 
process under suitable conditions.