The number of summands in [formula omitted]-periodic homotopy groups of [formula omitted

Abstract

The v 1 -periodic homotopy groups can be roughly described as the portions of the actual homotopy groups localized at a prime p that are detected by K-theory. In 1991 Davis showed that if p is an odd prime, v 1 −1 π 2 k ( SU ( n ) ; p ) is cyclic, gave the formula for its order, and proved that v 1 −1 π 2 k − 1 ( SU ( n ) ; p ) has the same order, but is not always cyclic. In this work we determine the number of summands of v 1 −1 π 2 k − 1 ( SU ( n ) ; p ) for all values of p, k, and n, where p is an odd prime. The method being used involves finding the rank of a family of matrices generated by the Adams operations. Determining the group structure of v 1 −1 π 2 k − 1 ( SU ( n ) ; p ) groups still remains an open question.