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

Abstract

Bousfield has shown how the 2-primary $v_1$-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 $E_6$ or Spin$(4k+2)$ with $k$ not a 2-power. We then use his description to give an explicit determination of the 2-primary $v_1$-periodic homotopy groups of $E_7$ and $E_8$. This completes a program, suggested to the author by Mimura in 1989, of computing the $v_1$-periodic homotopy groups of all compact simple Lie groups at all primes.