Characteristic rank of vector bundles over Stiefel manifolds

Abstract

The characteristic rank of a vector bundle $\xi$ over a finite connected $CW$-complex $X$ is by definition the largest integer $k$, $0\le k\le \dim (X)$, such that every cohomology class $x\in H^j(X;{\mathbb{Z}}_2)$, $0\le j\le k$, is a polynomial in the Stiefel-Whitney classes $w_i(\xi )$. In this note we compute the characteristic rank of vector bundles over the Stiefel manifold $V_k({\mathbb{F}}^n)$, $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$.