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 $l,r\ge 2$ there exists an $l$-connected manifold $M\in \mathcal{F}$ with a nontrivial strictly defined $r$-fold Massey product in $H^{\ast }(M)$. 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 ${\mathcal{Z}}_P^J=(D^{2j_i},S^{2j_i-1}{)}^{\partial P^{\ast }}$, $J=(j_1,\dots ,j_m)$ over a graph-associahedron $P=P_{\Gamma }$ and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.

28 pages, 1 figure, minor changes in presentation, some applications of main results added, list of references updated

References in corpus (3)

• Algebraic invariants for right-angled Artin groups
• Moment-angle manifolds, 2-truncated cubes and Massey operations
• On free loop spaces of toric spaces

Cited by in corpus (2)

• Embeddings of moment-angle manifolds and sequences of Massey products
• Direct families of polytopes with nontrivial Massey products