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.
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