[Paper Review] Families of minimally non-Golod complexes and their polyhedral products

Ivan Limonchenko|arXiv (Cornell University)|Sep 14, 2015

Advanced Combinatorial Mathematics14 references3 citations

TL;DR

This paper constructs infinite families of Golod and minimally non-Golod simplicial complexes with moment-angle complexes that have free integral cohomology but are not homotopy equivalent to wedges of spheres or connected sums of products of spheres. It establishes a criterion for Golodness and minimal non-Golodness in simplicial multiwedges and compositions, and presents a 4-dimensional minimally non-Golod complex on 9 vertices with non-trivial triple Massey products and cohomology length 1.

ABSTRACT

We consider families of simple polytopes $P$ and simplicial complexes $K$ well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics of the underlying complexes and algebraic properties of their Stanley--Reisner rings. We introduce infinite series of Golod and minimally non-Golod simplicial complexes $K$ with moment-angle complexes $\mathcal Z_K$ having free integral cohomology but not homotopy equivalent to a wedge of spheres or a connected sum of products of spheres respectively. We then prove a criterion for a simplicial multiwedge and composition of complexes to be Golod and minimally non-Golod and present a class of minimally non-Golod polytopal spheres.

Motivation & Objective

• To construct infinite families of Golod and minimally non-Golod simplicial complexes with moment-angle complexes that are not homotopy equivalent to wedges of spheres or connected sums of products of spheres.
• To establish a criterion for a simplicial multiwedge or composition of complexes to be Golod or minimally non-Golod.
• To characterize minimally non-Golod complexes via the cohomology length of their moment-angle complexes and the presence of non-trivial Massey products.
• To provide a 4-dimensional minimally non-Golod simplicial complex on 9 vertices with non-trivial triple Massey products and cohomology length 1.

Proposed method

• Utilizes the Stanley–Reisner ring and Tor-algebra to analyze the cohomology of moment-angle complexes via the isomorphism $ H^{*,*}( ilde{Z}_K; kk) o igoplus_{I ot eq ext{simplex}} ilde{H}^{*-|I|-1}(K_I) $.
• Applies Theorem 1.1 to interpret the cohomology of $ ilde{Z}_K $ as a bigraded Tor-algebra over the polynomial ring $ kk[v_1, \dots, v_m] $.
• Employs the suspension theorem (Theorem 1.3) to relate $ ilde{Z}_K $ to wedges of suspensions over non-simplices.
• Uses the criterion from [1, Corollary 4.14] to inductively analyze compositions and multiwedges of simplicial complexes.
• Applies results from [13] on Massey products and Golodness to verify non-triviality of higher-order operations.
• Constructs a 9-vertex 4-dimensional simplicial complex $ kk $ with explicit minimal non-faces to realize a complex with cohomology length 1 and non-trivial triple Massey product.

Experimental results

Research questions

• RQ1 Can infinite families of Golod and minimally non-Golod complexes be constructed such that their moment-angle complexes have free integral cohomology but are not homotopy equivalent to wedges of spheres or connected sums of products of spheres?
• RQ2 What conditions on simplicial multiwedges and compositions of complexes ensure Golodness or minimal non-Golodness?
• RQ3 What is the relationship between the cohomology length $ cup( ilde{Z}_K) $ and the Golod property for minimally non-Golod complexes?
• RQ4 Can a minimally non-Golod complex be constructed with non-trivial triple Massey products and cohomology length 1?
• RQ5 How do the combinatorics of the underlying simplicial complex and the structure of its Stanley–Reisner ring determine the homotopy type of the moment-angle complex?

Key findings

• The paper constructs infinite families of Golod and minimally non-Golod simplicial complexes $ K $ such that $ ilde{Z}_K $ has free integral cohomology but is not homotopy equivalent to a wedge of spheres or a connected sum of products of spheres.
• It proves that for any minimally non-Golod complex $ K $, the cohomology length $ cup( ilde{Z}_K) $ is at most 2.
• A 4-dimensional simplicial complex $ kk $ on 9 vertices is constructed such that $ cup( ilde{Z}_{kk}) = 1 $, and $ H^{14}( ilde{Z}_{kk}) $ contains a non-trivial triple Massey product.
• The complex $ kk $ is shown to be minimally non-Golod because $ kk $ is not Golod, but $ kk_{[m] \setminus v} $ is Golod for every vertex $ v $, as the cup product in cohomology is trivial.
• The paper establishes a criterion for a simplicial multiwedge or composition of complexes to be Golod or minimally non-Golod using induction and the substitution formula from [1, Corollary 4.14].
• The construction confirms that both $ cup( ilde{Z}_K) = 1 $ and $ cup( ilde{Z}_K) = 2 $ can occur for minimally non-Golod complexes, and that non-trivial Massey products can exist in such cases.