On the cohomology of oriented Grassmann manifolds

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.

Keywords

Stiefel–Whitney class, characteristic rank, cup-length, Grassmann manifold

2010 Mathematics Subject Classification

55R25, 57R20

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Published 29 November 2016