Representation types and 2-primary homotopy groups of certain compact Lie groups

Abstract. Bouseld has shown how the 2-primary v1-periodic homotopy groups of certain compact Lie groups G can be obtained from their representation ring with its decomposition into types and its exterior power operations. He has formulated a Technical Condition which must be satised in order that he can prove that his description is valid. We prove that a simply-connected compact simple Lie group satis es his Technical Condition if and only if it is not E6 or Spin(4k+ 2) with k not a 2-power. We then use his description to give an explicit determination of the 2-primary v1-periodic homo-topy groups of E7 and E8. This completes a program, suggested to the author by Mimura in 1989, of computing the v1-periodic homotopy groups of all compact simple Lie groups at all primes. 1