[Paper Review] Topology of polyhedral products over simplicial multiwedges

Ivan Limonchenko|arXiv (Cornell University)|Nov 1, 2017

Homotopy and Cohomology in Algebraic Topology34 references3 citations

TL;DR

This paper establishes a combinatorial criterion for rational formality of generalized moment-angle manifolds over graph-associahedra by linking higher Massey products in cohomology to the structure of simplicial multiwedges. It proves that a moment-angle manifold $π_P^J$ is formal if and only if its associated graph $̳$ is a disjoint union of segments, pentagons, and hexagons, and classifies all diffeomorphism types of such formal manifolds as connected sums of sphere products or odd-dimensional spheres.

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 $Γ$ 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 $P=P_Γ$ and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.

Motivation & Objective

• To determine a combinatorial criterion for rational formality of generalized moment-angle manifolds $π_P^J$ over graph-associahedra $P=P_\Gamma$.
• To classify all diffeomorphism types of formal moment-angle manifolds over graph-associahedra.
• To establish conditions under which higher Massey products exist in the cohomology of polyhedral products and Stanley–Reisner rings.
• To connect multigraded Betti numbers of a simplicial complex $K$ to the existence of strictly defined higher Massey products in $H^*(\mathcal{Z}_K)$.
• To extend the theory of simplicial multiwedges to construct smooth closed manifolds with arbitrary connectivity and nontrivial $r$-fold Massey products.

Proposed method

• Using the simplicial multiwedge construction to generate a family of polyhedral products that are smooth closed manifolds.
• Applying the simplicial multiwedge operation to preserve joins of simplicial complexes, enabling control over the topology of resulting moment-angle manifolds.
• Establishing a link between the multigraded Betti numbers of $K$ and the existence of nontrivial higher Massey products in $H^*(\mathcal{Z}_K)$.
• Using the fact that $\mathcal{Z}_{P(J)}$ is diffeomorphic to a product of $\mathcal{Z}_{I^1(J)}$, $\mathcal{Z}_{P_5(J)}$, and $\mathcal{Z}_{P_6(J)}$ for polytopal spheres to analyze formality.
• Leveraging known results on the topology of moment-angle manifolds over polygons and products of polytopes to classify diffeomorphism types.
• Analyzing the effect of vertex truncations ($\operatorname{vc}^k(P)$) on formality and Massey product structure via graph-theoretic and simplicial complex invariants.

Experimental results

Research questions

• RQ1 When is a generalized moment-angle manifold $\mathcal{Z}_P^J$ over a graph-associahedron $P=P_\Gamma$ rationally formal?
• RQ2 What are the diffeomorphism types of formal moment-angle manifolds over graph-associahedra?
• RQ3 Under what conditions on the simplicial complex $K$ does the moment-angle complex $\mathcal{Z}_K$ admit a nontrivial strictly defined higher Massey product?
• RQ4 How does the simplicial multiwedge construction generate smooth closed manifolds with prescribed connectivity and nontrivial Massey products?
• RQ5 What is the relationship between the 1-skeleton of $K_P$ and the existence of nontrivial triple Massey products in $H^*(\mathcal{Z}_P)$?

Key findings

• A moment-angle manifold $\mathcal{Z}_P^J$ over a graph-associahedron $P=P_\Gamma$ is rationally formal if and only if $\Gamma$ is a disjoint union of segments, pentagons, and hexagons.
• The diffeomorphism type of a formal moment-angle manifold $\mathcal{Z}_P^J$ is a connected sum of products of spheres or odd-dimensional spheres, specifically $\mathcal{Z}_{I^1(J)} = S^{2|J|-1}$, $\mathcal{Z}_{P_5(J)}$ is a connected sum of $S^3 \times S^4$'s, and $\mathcal{Z}_{P_6(J)}$ is a connected sum of $S^3 \times S^5$'s and $S^4 \times S^4$'s.
• For any $l, r \geq 2$, there exists an $l$-connected smooth closed manifold $M$ in the family $\mathcal{F}$ of polyhedral products constructed via simplicial multiwedges, with a nontrivial strictly defined $r$-fold Massey product in $H^*(M)$.
• The existence of a nontrivial triple Massey product in $H^*(\mathcal{Z}_P)$ depends only on the 1-skeleton (graph) of $K_P$, and is preserved under vertex truncations $\operatorname{vc}^k(P)$.
• The simplicial multiwedge construction preserves the join structure of simplicial complexes, allowing the construction of $\mathcal{Z}_{K(J)}$ as a product of moment-angle manifolds over simpler components.
• The equivalence of formality, the absence of nontrivial Massey products, and the combinatorial type of $P$ (being a product of segments, pentagons, and hexagons) is established for generalized moment-angle manifolds over graph-associahedra.