On Higher Massey Products and Rational Formality for Moment—Angle Manifolds over Multiwedges

Abstract

We prove that certain conditions on multigraded Betti numbers of a simplicial complex K imply the existence of a higher Massey product in the cohomology of a moment-angle complex \({{\cal Z}_K}\), and this product contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family \({\cal F}\) of polyhedral products being smooth closed manifolds such that for any l, r ≥ 2 there exists an l-connected manifold \(M \in {\cal F}\) with a nontrivial strictly defined r-fold Massey product in H*(M). As an application to homological algebra, we determine a wide class of triangulated spheres K such that a nontrivial higher Massey product of any order may exist in the Koszul homology of their Stanley–Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph Γ to provide a (rationally) formal generalized moment-angle manifold \(\mathcal{Z}_{P}^{J}=\left(\underline{D}^{2 j_{i}}, \underline{S}^{2 j_{i}-1}\right)^{\partial P^{*}}\)J = (j₁,…,j_m), over a graph-associahedron P = P_Γ, and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.