Memoirs of the American Mathematical Society 2004; 90 pp; softcover Volume: 172 ISBN10: 0821835890 ISBN13: 9780821835890 List Price: US$63 Individual Members: US$37.80 Institutional Members: US$50.40 Order Code: MEMO/172/815
 We compute the 2primary \(v_1\)periodic homotopy groups of the special orthogonal groups \(SO(n)\). The method is to calculate the BenderskyThompson 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. Readership Graduate students and research mathematicians interested in algebraic topology, manifolds, and cell complexes. Table of Contents  Introduction
 The BTSS of \({\rm BSpin}(n)\) and the CTP
 Listing of results
 The 1line of \({\rm Spin}(2n)\)
 Eta towers
 \(d_3\) on eta towers
 Fine tuning
 Combinatorics
 Comparison with \(J\)homology approach
 Proof of fibration theorem
 A small resolution for computing \({\rm ext}_{\mathcal A}\)
 Bibliography
