LAUSR

Abstract We develop strong lower bounds for the span of the projective Stiefel manifolds X n , r = O ( n ) / ( O ( n − r ) × 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.