On the cohomology of oriented Grassmann manifolds

摘要

This paper presents a new approach to studying the kernel of the additive homomorphism from $H^q(G_{n,k})$ to $H^{q+1}(G_{n,k})$ given by the cup-product with the first Stiefel–Whitney class of the canonical $k$-plane bundle over the Grassmann manifold $G_{n,k}$ of all $k$-dimensional vector subspaces in Euclidean $n$-space. This method enables us to improve the understanding of the $mathbb{Z}_2$-cohomology of the “oriented” Grassmann manifold $widetilde{G}_{n,k}$ of oriented $k$-dimensional vector subspaces in Euclidean $n$-space. In particular, we derive new information on the characteristic rank of the canonical oriented $k$-plane bundle over $widetilde{G}_{n,k}$ and the $mathbb{Z}_2$-cup-length of $widetilde{G}_{n,k}$. Our results on the cup-length for three infinite families of the manifolds $widetilde{G}_{n,3}$ confirm the corresponding claims of Fukaya’s conjecture from 2008.