The rational (non-)formality of the non-3-equal manifolds

Let M^(k)d(n) be the manifold of n -tuples (x₁,…,xₙ)∈(Rᵈ)ⁿ having non- k -equal coordinates. We show that, for d≥2 , M^(3)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(Rᵈ,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, which are assessed geometrically through Poincaré duality.