# \(v_{1}\)-Periodic Homotopy Groups of \(SO(n)\)

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*Martin Bendersky; Donald M. Davis*

We compute the 2-primary \(v_1\)-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 \(\operatorname{Spin}(n)\). The
\(E_2\)-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 \(\mathbf{Z}/2\).

As the spectral sequence converges to the
\(v_1\)-periodic homotopy groups of the \(K\)-completion of a
space, one important part of the proof is that the natural map from
\(\operatorname{Spin}(n)\) to its \(K\)-completion induces an
isomorphism in \(v_1\)-periodic homotopy groups.

#### Table of Contents

# Table of Contents

## $v_{1}$-Periodic Homotopy Groups of $SO(n)$

- Contents v6 free
- 1. Introduction 110 free
- 2. The BTSS of BSpin(n) and the CTP 514 free
- 3. Listing of results 817
- 4. The 1-line of Spin(2n) 1827
- 5. Eta towers 2332
- 6. d[sub(3)] on eta towers 3039
- 7. Fine tuning 3645
- 8. Combinatorics 4453
- 9. Comparison with J-homology approach 5564
- 10. Proof of fibration theorem 6978
- 11. A small resolution for computing Ext[sub(A)] 7483
- Bibliography 8998