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oriented Grassmann manifold; characteristic rank; Stiefel-Whitney class
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.
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