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.