On the characteristic rank of vector bundles over oriented Grassmannians
Volume 244 / 2019
Abstract
We study the cohomology algebra of the Grassmann manifold 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.
