Contents Online

# 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

#### 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