From representation theory to homotopy groups
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Bibliographic Information
From representation theory to homotopy groups
（Memoirs of the American Mathematical Society, no. 759）
American Mathematical Society, 2002
Available at 16 libraries
Note
"November 2002, volume 160, number 759 (first of 5 numbers)"
Includes bibliographical references : p. 48-50
Description and Table of Contents
Description
A formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations has been obtained by Bousfield. This work applies this theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989. The method is different to that used by the author in previous works. There is no homotopy theoretic input, and no spectral sequence calculation. The input is the second exterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8). Eigenvectors of psi^2 must also be eigenvectors of psi^k for any k. The matrix of these eigenvectors is the key to the analysis. Its determinant is closely related to the homotopy decomposition of E8 localized at each prime. By taking careful combinations of eigenvectors, a set of generators of K(E8) can be obtained on which there is a nice formula for all Adams operations. Bousfield's theorem (and considerable Maple computation) allows the v1-periodic homotopy groups to be obtained from this.
Table of Contents
- Introduction Representation theory and $\psi^2$ in $K$-theory Nice form for $\psi^2$ in $PK^1(E_8)_{(5)}$ and $PK^1(X)$ Determination of $v_1^{-1}\pi_{2m}(E_8
- 5)$ Determination of $v_1^{-1}\pi_{2m-1}(E_8
- 5)$ Calculation of $v_1^{-1}\pi_\ast(E_8
- 3)$ LiE program for computing $\lambda^2$ in $R(E_8)$ Analysis of $F_4$ and $E_7$ at the prime $3$ References.
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