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We give a description of the mod 2 cohomology algebra of the oriented Grassmann manifold $\widetilde G_{2^t,4}$ as the quotient of a polynomial algebra by a certain ideal. In the process we find a Gr\"obner basis for that ideal, which we then use to exhibit an additive basis for $H^*(\widetilde G_{2^t,4};\mathbb Z_2)$.

We study the $\mathbb Z_2$-zero-divisor cup-length, denoted by $\operatorname{zcl}_{\mathbb Z_2}(\widetilde G_{n,3})$, of the Grassmann manifolds $\widetilde G_{n,3}$ of oriented $3$-dimensional vector subspaces in $\mathbb R^n$. Some lower and upper bounds for this invariant are obtained for all integers $n\ge6$. For infinitely many of them the exact value of $\operatorname{zcl}_{\mathbb Z_2}(\widetilde G_{n,3})$ is computed, and in the rest of the cases these bounds differ by 1. We thus establish lower bounds for the topological complexity of Grassmannians $\widetilde G_{n,3}$.

For $n$ a power of two, we give a complete description of the cohomology algebra $H^*(\widetilde G_{n,3};\mathbb Z_2)$ of the Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-planes in $\mathbb R^n$. We do this by finding a reduced Gr\"obner basis for an ideal closely related to this cohomology algebra. Using this Gr\"obner basis we also present an additive basis for $H^*(\widetilde G_{n,3};\mathbb Z_2)$.

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the $\mathbb Z_2$-cup-length of the Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-planes in $\mathbb R^n$. Along the way, we calculate the heights of the Stiefel--Whitney classes of the canonical vector bundle over $\widetilde G_{n,3}$.