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

Donald M. Davis

55Q52, 55T15, 57T20

Department of Mathematics
Lehigh University
Bethlehem, PA 18015
dmd1@lehigh.edu

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 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.