THE 1-LINE OF THE K-THEORY BOUSFIELD-KAN SPECTRAL SEQUENCE FOR Spin(2n+ 1)

The p-primary v1-periodic homotopy groups, v −1 1 π∗(X; p), of a topological space X, as defined in [13], are a localization of the portion of the actual homotopy groups of X detected by K-theory. In [10, 4, 6], v−1 1 π∗(X; p) was calculated for classical groups X and primes p in all cases except (SO(n), 2). In this paper we make a first step toward the calculation of v−1 1 π∗(SO(n); 2), which is of course isomorphic to v−1 1 π∗(Spin(n); 2). In [7], a Bousfield-Kan-type spectral sequence E r (X) based on periodic K-theory was introduced. It converges to the homotopy groups of the K-completion X∧ K of the space X. In this paper, we will deal exclusively with the localization of this