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

Abstract Let $M^{({k})}_{d}(n)$ be the manifold of n -tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ having non- k -equal coordinates. We show that, for $d\geq2$ , $M^{({3})}_{d}(n)$ is rationally formal if and only if $n\leq6$ . This stands in sharp contrast with the fact that all classical configuration spaces $M^{({2})}_d(n)=\text{Conf}(\mathbb{R}^d,n)$ 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 $n \gt 6$ is established via detection of non-trivial triple Massey products, which are assessed geometrically through Poincaré duality.