Homology, Homotopy and Applications

Volume 5 (2003)

Number 1

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

Pages: 297 – 324

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n1.a13

Author

Donald M. Davis (Lehigh University, Bethlehem, Pennsylvania, U.S.A.)

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.

Keywords

homotopy groups, exceptional Lie groups, representation theory

2010 Mathematics Subject Classification

55Q52, 55T15, 57T20

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