On the cohomology of oriented Grassmann manifolds

Július Korbaš ; Tomáš Rusin

Homology, homotopy, and applications, Tome 18 (2016) no. 2, pp. 71-84.

Résumé

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.

Détail

DOI : 10.4310/HHA.2016.v18.n2.a4

Classification : 55R25, 57R20
Keywords: Stiefel–Whitney class, characteristic rank, cup-length, Grassmann manifold