On the characteristic rank of vector bundles over oriented Grassmannians
Volume 244 / 2019
Abstract
We study the cohomology algebra of the Grassmann manifold $\widetilde G_{k,n}$ of oriented $k$-dimensional subspaces in $\mathbb R^{n+k}$ via the characteristic rank of the canonical vector bundle $\widetilde\gamma_{k,n}$ over $\widetilde G_{k,n}$ (denoted by $\mathrm{charrank}(\widetilde{\gamma}_{k,n})$). Using Gröbner bases for the ideals determining the cohomology algebras of the “unoriented” Grassmannians $G_{k,n}$ we prove that $\mathrm{charrank}(\widetilde{\gamma}_{k,n})$ increases with $k$. In addition, we calculate the exact value of $\mathrm{charrank}(\widetilde{\gamma}_{4,n})$, and for $k\geq5$ we improve a general lower bound for $\mathrm{charrank}(\widetilde{\gamma}_{k,n})$ obtained by Korbaš. Some corollaries concerning the cup-length of $\widetilde G_{4,n}$ are also given.
