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.

中文翻译：

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射影 Stiefel 流形上的向量场和 Browder-Dupont 不变量

摘要 我们为射影 Stiefel 流形 X n , r = O ( n ) / ( O ( n − r ) × Z / 2 ) 的跨度开发了强下界，这使得对跨度。该技术在很大程度上涉及向量丛的基本稳定性特性。然而，情况 X n , 2 与 n 奇数会带来额外的困难，使用 Browder-Dupont 不变量可以部分解决这些问题。在这个过程中，我们观察到由于 Sutherland 引起的对称升力并不一定存在于所有奇维封闭流形，因此 Browder-Dupont 不变量，正如他所表述的那样，没有被普遍定义。我们将刻画那些在 X n , 2 上有明确定义的 Browder-Dupont 不变量的 n。