## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 5, Number 1 (2003), 297-324.

### 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 5, Number 1 (2003), 297-324.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139839936

**Mathematical Reviews number (MathSciNet)**

MR2006403

**Zentralblatt MATH identifier**

1031.55008

**Subjects**

Primary: 55Q52: Homotopy groups of special spaces

Secondary: 55Q51: $v_n$-periodicity 55T15: Adams spectral sequences 57T20: Homotopy groups of topological groups and homogeneous spaces

#### Citation

Davis, Donald M. Representation types and 2-primary homotopy groups of certain compact Lie groups. Homology Homotopy Appl. 5 (2003), no. 1, 297--324. https://projecteuclid.org/euclid.hha/1139839936