Some homotopy groups of the rotation group \(R_n\) (Q1306516)

Let \(\pi_k(R_n)\) be the \(k\)-dimensional homotopy group of the \(n\)-th rotation group \(R_n\). For \(k\leq 15\), there are classical results. Mimura and Toda determined them for \(k\leq 22\) and \(n\leq 9\). The 2-primary components for \(k=15\), 16 are determined by the first author. In this paper, the 2-primary components for \(k=17\), 18 are determined. The computation is based on the results of the homotopy groups \(\pi_{n+k}(S^n)\) for \(k\leq 18\) given by Toda and Oda. The authors study relations among elements of homotopy groups by the composition methods developed by Toda, and then determine \(\pi_k(R_n)\) by the exact sequence \[ \pi_{k+1}(S^n)@>\Delta>>\pi_k(R_n)@>i_*>>\pi_k(R_{n+1})\to\pi_k(S^n)\to \pi_{k-1}(R_n) \] induced from the inclusion \(i:R_n\to R_{n+1}\).