Motion planning in real flag manifolds

TL;DR

This paper provides a minimal cohomology presentation for certain real flag manifolds, estimates their topological complexity, and explores the behavior of higher topological complexities, revealing new insights into motion planning problems.

Contribution

It introduces a minimal cohomology ring presentation for semi complete flag manifolds and analyzes their topological complexities, including higher complexities, with new computational results.

Findings

Almost sharp estimates for topological complexity of $F_{2,2^e-1}$

Higher topological complexities increase the estimates' strength

Complete computations of higher topological complexity for all closed surfaces

Abstract

Starting from Borel's description of the mod-2 cohomology of real flag manifolds, we give a minimal presentation of the cohomology ring for semi complete flag manifolds $F_{k,m}:=F(1,\ldots,1,m)$ where $1$ is repeated $k$ times. The information is used in order to estimate Farber's topological complexity of these spaces when $m$ approaches (from below) a 2-power. In particular, we get almost sharp estimates for $F_{2,2^e-1}$ which resemble the known situation for the real projective spaces $F_{1,2^e}$. Our results indicate that the agreement between the topological complexity and the immersion dimension of real projective spaces no longer holds for other flag manifolds. More interestingly, we also get corresponding results for the $s$-th (higher) topological complexity of these spaces. Actually, we prove the surprising fact that, as $s$ increases, the estimates become stronger. Indeed, we get several full computations of the higher motion planning problem of these manifolds. This property is also shown to hold for surfaces: we get a complete computation of the higher topological complexity of all closed surfaces (orientable or not). A homotopy-obstruction explanation is included for the phenomenon of having a cohomologically accessible higher topological complexity even when the regular topological complexity is not so accessible.