Mathematics – Algebraic Topology
Scientific paper
1998-12-09
Mathematics
Algebraic Topology
50 pages; available at http://www.lehigh.edu/~dmd1/e8.html
Scientific paper
Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. We apply Bousfield's 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 compact simple Lie groups, a project suggested by Mimura in 1989. The method involves no homotopy theoretic input and no spectral sequences. The input is the second eexterior 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, we obtain a set of generators of K(E8) on which we have a nice formula for all Adams operations. Bousfield's theorem (and much Maple computation) allows us to obtain the v1-periodic homotopy groups.
No associations
LandOfFree
From Representation Theory to Homotopy Groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with From Representation Theory to Homotopy Groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and From Representation Theory to Homotopy Groups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-310675