v 1 - periodic homotopy groups of Sp ( n )

y In this paper we calculate the 2-primary v 1-periodic homotopy groups of the sym-plectic groups Sp(n). The proof utilizes new methods of calculating the unstable Novikov spectral sequence. One corollary is that some homotopy group of Sp(n) contains an element of order 2 2n?1. 1 Main theorem In this paper we calculate the 2-primary v 1-periodic homotopy groups of the sym-plectic groups Sp(n). The odd-primary v 1-periodic homotopy groups of Sp(n) were determined in 13]; they are an immediate consequence of the groups of SU(n). In Section 5, we apply our results to James numbers and exponents of actual homotopy groups. The v 1-periodic homotopy groups of a space, whose deenition we recall in the next paragraph, are a localization of the actual homotopy groups. Very roughly, they are the portion of the homotopy groups detectable by real and complex K-theory and their operations. A rst attempt at computing the v 1-periodic homotopy groups AMS Subject Classiication 55T15.