Homology, Homotopy and Applications

Volume 18 (2016)

Number 2

Motion planning in real flag manifolds

Pages: 359 – 375

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n2.a20

Authors

Jesús González (Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico City, Mexico)

Bárbara Gutiérrez (Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico City, Mexico)

Darwin Gutiérrez (Departamento de Formación Básica, Escuela Superior de Cómputo del Instituto Politécnico Nacional, México City, Mexico)

Adriana Lara (Departamento de Matemáticas, Escuela Superior de Física y Matemáticas del Instituto Politécnico Nacional, Mexico City, Mexico)

Abstract

Starting from Borel’s description of the mod-2 cohomology of real flag manifolds, we give a minimal presentation of the cohomology ring of semi-complete flag manifolds $F_{k,m}:=F(1,\ldots,1,m)$, where $1$ is repeated $k$ times. This is used to estimate Farber’s topological complexity of $F_{k,m}$ 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. We also get corresponding results for the $s$-th higher topological complexity of these spaces, proving the surprising fact that, as $s$ increases, our cohomological estimates become stronger. Indeed, we get a full description of the higher motion planning problem of some of these manifolds. As a byproduct, we get a complete computation of the higher topological complexity of all closed surfaces (orientable or not).

Keywords

flag manifold, surface, topological complexity, zero-divisors cup-length, motion planning

2010 Mathematics Subject Classification

55M30, 57T15, 68T40, 70B15

Full Text (PDF format)

Published 29 November 2016