Motion planning in real flag manifolds

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, . . . , 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-theory explanation is included for the phenomenon of having a cohomologically accessible higher topological complexity even when the regular topological complexity is not so accessible.

Topological complexity of the Klein bottle

“…Our results also complete recent work of J. González, B. Gutiérrez, D. Gutiérrez and A. Lara [14] on the higher topological complexity of the non-orientable surfaces. Recall that the s-topological complexity of a space X , TC s (X ), introduced by Y. Rudyak in [15], satisfies TC 2 = TC.…”

“…Recall that the s-topological complexity of a space X , TC s (X ), introduced by Y. Rudyak in [15], satisfies TC 2 = TC. Using the (higher) zero-divisors cuplength with coefficients in Z 2 , the authors of [14] compute TC s (N g ) for any s ≥ 3 and g ≥ 1.…”

On the topological complexity and zero-divisor cup-length of real Grassmannians

“… By proposition 2.1 from [6] we have (we use this proposition for , and ). Since (by proposition 2.2(2)), we have Since are in the additive basis (from proposition 2.2(1)), the last sum is zero if and only if , i.e.…”

Amenable covers of right‐angled Artin groups

“…The topological complexity of RAAGs has been computed [5,17]. We recall the precise result for completeness, even though we will not need it in the sequel.…”