Motion planning in real flag manifolds

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,., 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_2,2e-1 which resemble the known situation for the real projective spaces F_1,2e. 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).