### v1-periodic homotopy groups of SO(n)

This 120-page paper, coauthored with Martin Bendersky, will appear in the
Memoirs of the American Math Society.
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.

dvi file and ps file