## From Representation Theory to Homotopy Groups

Donald M. Davis

Lehigh University

dmd1@lehigh.edu

http://www.lehigh.edu/~dmd1/dmd1.html

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. In this paper, 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 all compact simple Lie groups, a project suggested by Mimura in 1989.

The method is completely different than that used by the author in previous
papers. 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, we obtain
a set of generators of K(E8) on which we have a nice formula for all Adams
operations. Bousfield's theorem (and considerable Maple computation)
allows us to obtain from this the v1-periodic homotopy groups.

dvi file; ps file; pdf file