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

We develop strong lower bounds for the span of the projective Stiefel manifolds $X_{n,r}=O(n)/(O(n-r)\times \mathbb Z/2)$, 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 $X_{n,2}$ 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 $X_{n,2}$. Then the invariant will be used in this case to obtain the lower bounds for the span as a corollary of a stronger result.