v1-periodic homotopy groups of SO(n)
Martin Bendersky and Donald M. Davis
55Q52, 55T15, 57T20
Hunter College, CUNY, NY, NY 10021
Lehigh University, Bethlehem, PA 18015
Abstract
We compute the 2-primary v1-periodic homotopy groups of the special
orthogonal groups SO(n). The method is to calculate
the Bendersky-Thompson spectral sequence, a K*-based unstable homotopy
spectral sequence, of Spin(n). The E2-term is an
Ext group in a category of Adams modules. Most of the differentials
in the spectral sequence are determined by naturality from
those in the spheres.
The resulting groups consist of two main parts. One is summands whose
order depends on the minimal exponent of 2 in several
sums of binomial coefficients times powers. The other is a sum of
roughly [log_2(2n/3)] copies of Z/2.
As the spectral sequence converges to the v1-periodic homotopy groups
of the K-completion of a space, one important part of
the proof is that the natural map from Spin(n) to its K-completion
induces an isomorphism in v1-periodic homotopy groups.