Vector fields on projective Stiefel manifolds and the Browder-Dupont invariant

Abstract

We develop strong lower bounds for the span of the projective Stiefel manifolds X-n,X-r = O(n)/(O(n - r) x Z/2), which enable very accurate (in many cases exact) estimates of the span. The technique, for the most part, involves elementary stability properties of vector bundles. However, the case X-n,X-2 with n odd presents extra difficulties, which are partially resolved using the Browder-Dupont invariant. In the process, we observe that the symmetric lift due to Sutherland does not necessarily exist for all odd dimensional closed manifolds, and therefore the Browder-Dupont invariant, as he formulated it, is not defined in general. We will characterize those n's for which the Browder-Dupont invariant is well-defined on X-n,X-2. Then the invariant will be used in this case to obtain the lower bounds for the span as a corollary of a stronger result.