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

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Abstract For the Grassmann manifold $$\widetilde G_{n,4}$$ G ~ n , 4 of oriented 4-planes in $$\mathbb{R}^{n}$$ R n no full description of its cohomology ring with coefficients in the two element field $$\mathbb {Z}_{2}$$ 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 $$\mathbb {Z}_{2}$$ Z 2 -cup-length of $$\widetilde G_{n,4}$$ G ~ n , 4 for $$n=2^t,2^t-1,2^t-2$$ n = 2 t , 2 t - 1 , 2 t - 2 for all $$t \geq 4$$ t ≥ 4 .