Topological complexity of oriented Grassmann manifolds

Abstract

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 $\widetilde G_{n,3}$.