A note on the ${\mathbb {Z}}_2$-cohomology algebra of oriented Grassmann manifolds

Abstract

The ${\mathbb {Z}}_2$-cohomology ring $H^*({\widetilde{G}}_{n,k})$ of the oriented Grassmann manifold ${\widetilde{G}}_{n,k}\cong \mathrm {SO}(n)/(\mathrm {SO}(k)\times \mathrm {SO}(n-k))$ is in general unknown. This paper demonstrates that the characteristic rank of vector bundles in the sense of Korbaš, Naolekar, and Thakur can be helpful in improving our knowledge. More precisely, using the full knowledge of the characteristic rank of the canonical orientable k-plane bundle ${\widetilde{\gamma }}_{n,k}$ over ${\widetilde{G}}_{n,k}$ for $k=2$, we completely calculate the cohomology ring $H^*({\widetilde{G}}_{n,2})$. In addition, we completely determine the generators of the cohomology ring $H^*({\widetilde{G}}_{n,3})$ for $n=6,\,7,\,8,\,9,\,10,\,11$.