From Representation Theory to Homotopy Groups

This work applys Bousfield's theorem for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3).

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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.