Homology, Homotopy and Applications

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

Donald M. Davis

Full-text: Open access

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


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