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 $\tilde{G}_{n, k}$ 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 $k\geq 5$. In the second part, these results enable us to improve on the general upper bounds for the $\mathbb{Z}_{2}$-cup-length of $\tilde{G}_{n, k}$. In particular, for $\tilde{G}_{2^{t}, 3}$ ($t\geq 3$) we prove that the cup-length is equal to $2^{t}-3$, which verifies the corresponding claim of Tomohiro Fukaya's conjecture from 2008.