On the topological complexity and zero-divisor cup-length of real Grassmannians

Abstract

Topological complexity naturally appears in the motion planning in robotics. In this paper we consider the problem of finding topological complexity of real Grassmann manifolds $G_k(\mathbb {R}^{n})$. We use cohomology methods to give estimates on the zero-divisor cup-length of $G_k(\mathbb {R}^{n})$ for various $2\leqslant k< n$, which in turn give us lower bounds on topological complexity. Our results correct and improve several results from Pavešić (Proc. Roy. Soc. Edinb. A151 (2021), 2013–2029).