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

Abstract

We derive an inequality for the $\mathbb Z_2$-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 $\tilde G_{n,k}\cong SO(n)/SO(k)\times SO(n-k)$ $(6\leq 2k\leq n)$ 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 $\tilde G_{n,k}$. For $\tilde G_{2^t-1,3}$ $(t\geq 3)$ our upper and lower bounds coincide, giving that the $\mathbb Z_2$-cup-length is $2^t-3$ and the characteristic rank equals $2^t-5$. Some applications to the Lyusternik-Shnirel'man category are also presented.