The Bousfield-Kan Spectral Sequence for Periodic Homology Theories

We construct the Bousfield-Kan (unstable Adams) spectral sequence based on certain nonconnective periodic homology theories E such as complex periodic K-theory, and define an E-completion of a space X. For$X=S^{2n+1}$and E = K we calculate the${\rm E}_{2}$-term and show that the spectral sequence converges to the homotopy groups of the K-completion of the sphere. This also determines all of the homotopy groups of the (unstable) K-theory localization of$S^{2n+1}$including three divisible groups in negative stems.