Topology of polyhedral products over simplicial multiwedges

Abstract: We prove that certain conditions on multigraded Betti numbers of a simplicial complex $K$ imply existence of a higher Massey product in cohomology of a moment-angle-complex $\mathcal Z_K$, which contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family $\mathcal{F}$ of polyhedral products being smooth closed manifolds such that for any $l,r\geq 2$ there exists an $l$-connected manifold $M\in\mathcal F$ 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 Koszul homology of their Stanley--Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph $\Gamma$ to provide a (rationally) formal generalized moment-angle manifold $\mathcal Z_{P}^{J}=(D^{2j_{i}},S^{2j_{i}-1})^{\partial P^*}$, $J=(j_{1},\ldots,j_m)$ over a graph-associahedron $P=P_{\Gamma}$ and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.