A note on the cohomology ring of the oriented Grassmann manifolds $widetilde{G}_{n,4}$

摘要

We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold $widetilde{G}_{n,4}$ to compute the generators of the $mathbb{Z}_2$–cohomology groups $H^j(widetilde{G}_{n,4})$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of $widetilde{G}_{n,3}$ we conjecture some predictions.