Spaces Associated with Stiefel Manifolds

THIS article concludes the survey of properties of the family of Stiefel manifolds which we began with (3) and (4). It is primarily concerned with two families of auxiliary spaces, called stunted protective and stunted quasiprojective spaces, whose algebraic topology not only makes an interesting study in itself but also illuminates the classical problem of determining which Stiefel manifolds admit cross-sections. The theory is fairly selfcontained, although there are important connexions and resemblances of technique as regards previous papers in this series. The auxiliary spaces are denned in the introductory section and our main results are stated in the following one.