A remark on James numbers of Stiefel manifolds
The James number \(W\{\) n,k\(\}\) is the index of the image of \[ q_*: \pi_{2n-1}(U(n)/U(n-k))\to \pi_{2n-1}(S^{2n-1}) \] where q: U(n)/U(n-k)\(\to U(n)/U(n-1)=S^{2n-1}\) is the canonical quotient map from the Stiefel manifold to the sphere. By using the stable homotopy \(\pi^ s_{2n-1}\) instead of \(\pi_{2n-1}\) we define the stable James number \(W^ s\{n,k\}.\) By the Freudenthal suspension theorem, \(W\{n,k\}=W^ s\{n,k\}\) if \(n\geq 2k-1\). The main purpose of the paper is to investigate the case \(n=2k-2\). The quaternion case is also considered.
scientific article; zbMATH DE number 3935974