We compute the Lusternik-Schnirelmann category (LS-cat) and higher topological complexity (TC s , s 2) of the "nok-equal" configuration space Conf (k) (R, n). With k = 3, this yields the LS-cat and the higher topological complexity of Khovanov's group PP n of pure planar braids on n strands, which is an R-analogue of Artin's classical pure braid group on n strands. Our methods can be used to describe optimal motion planners for PP n provided n is small.The second and third authors were supported by a Conacyt scholarship and a Conacyt Postdoctoral Fellowship, respectively.

The aim of this paper is to give a (discrete) Morse theoretic proof of the fact that the k-th skeleton of the flag complex F , associated to the lattice of subspaces of a finite dimensional vector space, is homotopy equivalent to a wedge of spheres of dimension min{k, dim(F )}. The tight control provided by Morse theoretic methods allows us to give an explicit formula for the number of spheres appearing in each of these wedge summands.

d (n) is rationally formal if and only if n ≤ 6. This stands in sharp contrast with the fact that all classical configuration spaces M (2) d (n) = Conf (R d , n) are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non formality of M (3) d (n) for n > 6 is established via detection of non-trivial triple Massey products assessed through Poincaré duality.

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