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 Z K , which contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family F of polyhedral products being smooth closed manifolds such that for any l, r ≥ 2 there exists an l-connected manifold M ∈ 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 Γ to provide a (rationally) formal generalized moment-angle manifold Z J P = (D ¯2ji , S ¯2ji−1 ) ∂P * , J = (j 1 , . . . , j m ) over a graphassociahedron P = P Γ and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.