Determination of Grassmann manifolds which are boundaries

Let FG_nk denote the Grassmann manifold of all k-dimensional (left) F-vector subspace of Fⁿ for F = R, the reals, C, the complex numbers, or H the quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in [4]. Partial results were obtained in [4] for the case F = R, including a sufficient condition, due to A. Dold, on n and k for R G_nk to bound. Here, we show that Dold's condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well