Topological complexity of oriented Grassmann manifolds

Abstract

We study the ${\mathbb{Z}}_2$-zero-divisor cup-length, denoted by ${\mathrm{zcl}}_{{\mathbb{Z}}_2}({\tilde{G}}_{n,3})$, of the Grassmann manifolds ${\tilde{G}}_{n,3}$ of oriented $3$-dimensional vector subspaces in ${\mathbb{R}}^n$. Some lower and upper bounds for this invariant are obtained for all integers $n\ge 6$. For infinitely many of them the exact value of ${\mathrm{zcl}}_{{\mathbb{Z}}_2}({\tilde{G}}_{n,3})$ 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 ${\tilde{G}}_{n,3}$.