The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds

作者：Korbas Julius^*

来源：Bulletin of the Belgian Mathematical Society Simon Stevin, 2010, 17(1): 69-81.

• Cup-length
• Lyusternik-Shnirel'man category
• oriented Grassmann manifold
• cobordism
• Stiefel-Whitney characteristic class

We derive an inequality for the Z(2)-cup-length of any smooth closed connected manifold unorientedly cobordant to zero In relation to this, we introduce a new numerical invariant of a smooth closed connected manifold, called the characteristic rank In particular, our inequality yields strong upper bounds for the cup-length of the oriented Grassmann manifolds (G) over tilde (n,k) congruent to SO(n)/SO(k) x SO(n - k) (6 <= 2k <= n) if n is odd; if n is even, we obtain new upper bounds in a different way We also derive lower bounds for the cup-length of (G) over tilde (n,k) For (G) over tilde (2t-1,3) (t >= 3) our upper and lower bounds coincide, giving that the Z(2)-cup-length is 21 3 and the characteristic rank equals 2(t) - 5 Some applications to the Lyusternik-Shnirel'man category are also presented

• 出版日期2010-3