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 $\widetilde G_{n,4}\cong SO(n)/(SO(4)\times SO(n-4))$ in the cases $n\in\{2^t-2,2^t-1,2^t,2^t+1\}$, $t\ge4$. Using some additional computations in modulo $2$ cohomology of $\widetilde G_{n,4}$ 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 $\widetilde G_{n,4}$ for $n\in\{2^t-2,2^t-1,2^t\}$, to the case $n=2^t+1$, $t\ge3$.
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