On the cup-length of the oriented Grassmann manifolds \(\widetilde G_{n,4}\)

Abstract

For the Grassmann manifold \(\widetilde G_{n,4}\) of oriented 4-planes in \(\mathbb{R}^{n}\) no full description of its cohomology ring with coefficients in the two element field \(\mathbb {Z}_{2}\) is available. It is known however that it contains a subring that can be identified with a quotient of a polynomial ring by a certain ideal. Examining this quotient ring by means of Gröbner bases we are able to determine the \(\mathbb {Z}_{2}\)-cup-length of \(\widetilde G_{n,4}\) for \(n=2^t,2^t-1,2^t-2\)for all \(t \geq 4\).