Whitehead Products in Stiefel Manifolds and Samelson Products in Classical Groups

Abstract

The first non zero homotopy group of the Stiefel manifold $O_{n+k,k}$ of orthonormal $k$-frames in $F^{n+k}$ is generated by the standard embedding $i_{n+k,k}=i^{F}_{n+k,k}:S^{d(n+1)-1}=O_{n+1,1}\to O_{n+k,k}$ where $F$ is the field of the real numbers $R$, complex numbers $C$, or quaternions $H$ and $d$ is the dimension of $F$ over $R$. We study the Whitehead products $[i_{n+k,k}, i_{n+k,k}]$ and $[i^{C}_{n+k,k} \circ \eta_{2n+1}, i^{C}_{n+k,k}]$ where $\eta_m \in \pi_{m+1}(S^m)$ is a generator. As consequences we determine the orders of a few Samelson products in the classical groups and obtain a relation between the stable and unstable James numbers of the complex Stiefel manifolds.