The characteristic rank and cup-length in oriented Grassmann manifolds

Abstract

In the first part, this paper studies the characteristic rank of the canonical oriented $k$-plane bundle over the Grassmann manifold $SO(n)/(SO(k) \times SO(n-k))$ of oriented $k$-planes in Euclidean $n$-space. It presents infinitely many new exact values if $k = 3$ or $k = 4$, as well as new lower bounds for the number in question if $k > 4$. In the second part, these results enable us to improve on the general upper bounds for the $Z/2Z$-cup-length of $SO(n)/(SO(k) \times SO(n-k))$. In particular, for $SO(2^t)/(SO(3) \times SO(2^t-3))$ (with $t > 2$) we prove that the cup-length is equal to $2^t-3$, which verifies the corresponding claim of Tomohiro Fukaya's conjecture from 2008.