The 1-line of the \(K\)-theory Bousfield-Kan spectral sequence for \(\text{Spin}(2n+1)\) (Q2765020)

The computation of the \(v_1\)-periodic homotopy groups of a space \(X\), \(v_1^{-1}\pi_*(X;p)\) at a prime \(p\) has been carried out successfully by the authors with \textit{M. Mahowald} [Pac. J. Math. 170, No. 2, 319--378 (1995; Zbl 0851.55021)] for the case \((G,p)\), where \(G\) is a classical group and \(p\) any prime, except in the case of the oriented orthogonal group \(\text{SO}(n)\) and the prime 2. The method of computation is a Bousfield-Kan type of spectral sequence, suitably localized at \(v_1\). The input for this spectral sequence involves an unstable cobar construction and the \(K_*K\)-coaction on \(K_*(X)\). Since \(v_1^{-1}\pi_*(\text{SO}(n);2)\) is isomorphic to \(v_1^{-1}\pi_*(\text{Spin}(n);2)\), the authors concentrate on \(\text{Spin}(n)\). In this paper, they make the first step in determining \(v_1^{-1}\pi_*(\text{Spin}(2n+1);2)\) by computing completely the 1-line \(E_2^{1,t}\) of the spectral sequence converging to these groups. Some results are given generally for the case of a simply-connected finite H-space whose K-homology \(K_*(X)\) is an exterior algebra on odd dimensional classes of bounded degree. The determination of the 1-line involves primitives in \(K^1(X)\) as modules over the Adams operations. A toehold is provided by the authors' previous computation of the spectral sequence for the symplectic group \(\text{Sp}(n)\) [Lond. Math. Soc. Lect. Note Ser. 176, 73--86 (1992; Zbl 0778.55003)]. The ranks of the groups in the answer are determined by a combinatorial tour-de-force.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00054].