Heights of Stiefel--Whitney classes and zero-divisor cup-length of some Grassmann manifolds

We calculate the heights of Stiefel–Whitney classes of the canonical vector bundle over the oriented Grassmannians ${\tilde{G}}_{n,4}\cong SO(n)\mathrm{/}(SO(4)\times SO(n-4))$ in the cases $n\in \{2^t-2,2^t-1,2^t,2^t+1\}$, $t\ge 4$. Using some additional computations in modulo $2$ cohomology of ${\tilde{G}}_{n,4}$ and the well-known connection between topological complexity and zero-divisor cup-length, we obtain lower bounds for topological complexity of these Grassmannians. We also extend recent results of Rusin, who computed the modulo $2$ cup-length of ${\tilde{G}}_{n,4}$ for $n\in \{2^t-2,2^t-1,2^t\}$, to the case $n=2^t+1$, $t\ge 3$.