Topological complexity of real Grassmannians

TL;DR

This paper investigates the topological complexity of real Grassmannians by analyzing their cohomology rings, providing estimates for motion planning complexity and exploring how these invariants change with dimension.

Contribution

It introduces new computations of zero-divisor cup-length and topological complexity for real Grassmannians, enhancing understanding of their motion planning properties.

Findings

Computed zero-divisor cup-length for real Grassmannians

Estimated topological complexity for motion planning

Analyzed monotonicity of topological invariants with dimension

Abstract

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds $G_k(\mathbb{R}^n)$ to compute zero-divisor cup-length and estimate topological complexity of motion planning for $k$-linear subspaces in $\mathbb{R}^n$. In addition, we obtain results about monotonicity of Lusternik-Schnirelmann category and topological complexity of $G_k(\mathbb{R}^n)$ as a function of $n$.