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

Abstract

Bousfield has shown how the 2-primary v1-periodic homotopy groups of certain compact Lie groups 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 satisfied in order that he can prove that his description is valid. We prove that a simply-connected compact simple Lie group satisfies 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 homotopy 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.