The order of the Hopf bundle on projective Stiefel manifolds

Abstract

The projective Stiefel manifold X_n,k has a canonical line bundle ξ_n,k, called the Hopf bundle. The order of cξn, the complexification of ξn,k,as an element of (the abelian group) K(X_n,k), has been determined in [3], [5], [6]. The main result in the present work is that this order equals the order of ξn,k itself, as an element of KO(X_n,k), for n≡0,±1 (mod 8), or for is in the "upper range for n" (approximately k ≥ n/2). Certain applications are indicated.