Motion planning in real flag manifolds

Jesús González ; Bárbara Gutiérrez ; Darwin Gutiérrez ; Adriana Lara

Homology, homotopy, and applications, Tome 18 (2016) no. 2, pp. 359-375.

Résumé

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).

Détail

DOI : 10.4310/HHA.2016.v18.n2.a20

Classification : 55M30, 57T15, 68T40, 70B15
Keywords: flag manifold, surface, topological complexity, zero-divisors cup-length, motion planning