Topological complexity of oriented Grassmann manifolds

Abstract

We study the topological complexity of the Grassmann manifolds $\widetilde G_{n,3}$ of oriented $3$-dimensional vector subspaces in $\mathbb R^n$. By a result of Farber, for any field $K$, the topological complexity of a space $X$ is greater than $\zcl_{K}(X)$, where $\zcl_{K}(X)$ is the $K$-zero-divisor cup-length of $X$. In this paper we examine $\zcl_{\mathbb{Z}_2}(\widetilde G_{n,3})$. Some lower and upper bounds for this invariant are obtained for all integers $n\ge6$. For infinitely many of them the exact value of $\zcl_{\mathbb Z_2}(\widetilde G_{n,3})$ 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 $\widetilde G_{n,3}$.