2-PRIMARY v1-PERIODIC HOMOTOPY GROUPS OF SU(n) REVISITED

In 1991, Bendersky and Davis used the BP -based unstable Novikov spectral sequence to study the 2-primary v1periodic homotopy groups of SU(n). Here we use a K-theoretic approach to add more detail to those results. In particular, whereas only the order of the groups v−1 1 π2k−1(SU(n)) was determined in the 1991 paper, here we determine the number of summands in these groups and much information about the orders of those summands. In addition, we give explicit conditions for certain differentials and extensions in a spectral sequence, which affect the homotopy groups. Finally, we give complete results for v−1 1 π∗(SU(n)) for n ≤ 13. 1. Statement of results The 2-primary v1-periodic homotopy groups v −1 1 π∗(X) of a space X are a localization of the portion of the actual homotopy groups of X detected by 2-local K-theory. They form a good first approximation to π∗(X); if X is a sphere or compact Lie group, every group v−1 1 πi(X) is a direct summand of some group πi+2k(X).([16]) In a 1991 paper ([2]), Bendersky and the first author used the BP -based unstable Novikov spectral sequence (UNSS) to study the 2-primary v1-periodic homotopy groups v−1 1 π∗(SU(n)) of the special unitary groups. During the subsequent 14 years, K-theoretic approaches to v1-periodic homotopy groups have been developed by Bendersky, Bousfield, Davis, and Thompson ([4, 6, 9, 8]. In this paper, we apply these methods to obtain some refinements of the results of [2]. The principal accomplishments of this paper are: Date: March 9, 2005. 1991 Mathematics Subject Classification. 55Q52,55T15,57T20.