On the characteristic rank of vector bundles over oriented Grassmannians

Abstract

We study the cohomology algebra of the Grassmann manifold Gek,n of oriented k-dimensional subspaces in Rn+k via the characteristic rank of the canonical vector bundle γek,n over Gek,n (denoted by charrank(γek,n)). Using Gröbner bases for the ideals determining the cohomology algebras of the “unoriented” Grassmannians Gk,n we prove that charrank(γek,n) increases with k. In addition, we calculate the exact value of charrank(γe4,n), and for k ≥ 5 we improve a general lower bound for charrank(γek,n) obtained by Korbaš. Some corollaries concerning the cup-length of Ge4,n are also given.