  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.
