Moment-angle manifolds corresponding to three-dimensional simplicial spheres, chordality and connected sums of products of spheres (2024)

Victoria OganisianDepartment of Mathematics and Mechanics, MoscowState University, Russia;
National Research University Higher School of Economics, Moscow, Russiapotchtovy_jashik@mail.ruandTaras PanovDepartment of Mathematics and Mechanics, MoscowState University, Russia;
National Research University Higher School of Economics, Moscow, Russiatpanov@mech.math.msu.su

Abstract.

We prove that the moment-angle complex $\mathcal{Z}_{\mathcal{K}}$ corresponding to a $3$ -dimensional simplicial sphere $\mathcal{K}$ has the cohom*ology ring isomorphic to the cohom*ology ring of a connected sum of products of spheres if and only if either (a) $\mathcal{K}$ is the boundary of a $4$ -dimensional cross-polytope, or (b) the one-skeleton of $\mathcal{K}$ is a chordal graph, or (c) there are only two missing edges in $\mathcal{K}$ and they form a chordless $4$ -cycle. For simplicial spheres $\mathcal{K}$ of arbitrary dimension, we obtain a sufficient condition for the ring isomorphism $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M)$ where $M$ is a connected sum of products of spheres.

2020 Mathematics Subject Classification:

57S12, 57N65

This work was carried out within the project “Mirror Laboratories” of HSE University, Russian Federation. Victoria Oganisian is supported by a stipend from the Theoretical Physics and Mathematics Advancement Foundation “BASIS”

1. Introduction

The moment-angle complex is a topological space (a CW complex) with a torus action that features in toric topology and hom*otopy theory of polyhedral products[BP]. The topology of a moment-angle complex $\mathcal{Z}_{\mathcal{K}}$ is determined by the combinatorics of the corresponding simplicial complex $\mathcal{K}$ .If $\mathcal{K}$ is the nerve complex of a simple polytope $P$ , then the corresponding moment-angle complex, which is denoted by $\mathcal{Z}_{P}$ , is a smooth manifold.

There are several different geometric constructions of moment-angle manifolds enriching their topology with remarkable and peculiar geometric structures. One of them arises in holomorphic dynamics, where the moment-angle manifold $\mathcal{Z}_{P}$ appears as the leaf space of a holomorphic foliation on an open subset of a complex space, and is diffeomorphic to a nondegenerate intersection of Hermitian quadrics[BM], [BP, Chapter6]. All early examples of moment-angle manifolds appearing in this context where diffeomorphic to connected sums of products of spheres. This is the case, for example, when $P$ is two-dimensional (a polygon). From the description of the cohom*ology ring of $\mathcal{Z}_{P}$ it became clear that the topology of moment-angle manifolds in general is much more complicated than that of a connected sum of sphere products; for instance, $H^{*}(\mathcal{Z}_{P})$ can have arbitrary additive torsion or nontrivial higher Massey products[BP, Chapter4].

Nevertheless, the question remained of identifying the class of simple polytopes $P$ (or more generally, simplicial spheres $\mathcal{K}$ ) for which the moment-angle manifold $\mathcal{Z}_{P}$ is homeomorphic to a connected sum of products of spheres. This question is also interesting from the combinatorial and hom*otopy-theoretic points of view, as it is related to the conditions for the minimal non-Golodness of $\mathcal{K}$ and the chordality of its one-skeleton. For three-dimensional polytopes $P$ (or two-dimensional spheres $\mathcal{K}$ ), it was proved in[BM, Proposition11.6] that $\mathcal{Z}_{P}$ is diffeomorphic to a connected sum of products of spheres if and only if $P$ is obtained from the 3-simplex by consecutively cutting off some $l$ vertices. This characterisation can be extended by adding two more equivalent conditions, the chordality and the minimal non-Golodness (see Proposition3.1):

Proposition.

Let $\mathcal{K}$ be a two-dimensional simplicial sphere and let $P$ be the a three-dimensional simple polytope such that $\mathcal{K}=\mathcal{K}_{P}$ . Suppose that $P$ is not a cube. The following conditions are equivalent:

(a)
$P$ is obtained from the simplex $\Delta^{3}$ by iterating the vertex cut operation, i. e. $P^{*}$ is a stacked polytope;
(b)
$\mathcal{Z}_{P}$ is diffeomorphic to a connected sum of products of spheres;
(c)
$H^{*}(\mathcal{Z}_{P})$ is isomorphic to the cohom*ology ring of a connected sum of products of spheres;
(d)
the one-dimensional skeleton of the nerve complex $\mathcal{K}_{P}$ is a chordal graph;
(e)
$\mathcal{K}_{P}$ is minimally non-Golod, unless $P=\Delta^{3}$ .

For three-dimensional simplicial spheres $\mathcal{K}$ (including the nerve complexes of four-dimensional simple polytopes) we characterise the moment-angle manifolds $\mathcal{Z}_{K}$ with the cohom*ology ring isomorphic to the cohom*ology ring of a connected sum of products of spheres (see Theorem4.4):

Theorem.

Let $\mathcal{K}$ be a three-dimensional simplicial sphere. There is a ring isomorphism $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M_{1}\#\cdots\#M_{k})$ where each $M_{i}$ is a product of spheres if and only if one of the following conditions is satisfied:

(a)
$\mathcal{K}=S^{0}*S^{0}*S^{0}*S^{0}$ (the boundary of a $4$ -dimensional cross-polytope);
(b)
$\mathcal{K}^{1}$ is a chordal graph;
(c)
$\mathcal{K}^{1}$ has exactly two missing edges which form a chordless $4$ -cycle.

We conjecture that under each of the conditions (b) and (c) above the moment-angle manifold $\mathcal{Z}_{\mathcal{K}}$ is homeomorphic to a connected sum of products of spheres. Under condition (c) we have $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M_{1}\#\cdots\#M_{k})$ where one of the summands $M_{i}$ is a product of three spheres. The first example of such $\mathcal{K}$ was constucted in[FCMW].

When $\dim P\geq 5$ , the chordality of $\mathcal{K}^{1}_{P}$ does not imply that $H^{*}(\mathcal{Z}_{P})\cong H^{*}(M)$ where $M$ is a connected sum of products of spheres, see Example2.9. A stronger sufficient condition valid for simplicial spheres of arbitrary dimension is given in Theorem4.3.

2. Preliminaries

Let $\mathcal{K}$ be a simplicial complex on the set $[m]=\{1,\ldots,m\}$ . We assume that $\mathcal{K}$ contains an empty set $\varnothing$ and all one element subsets $\{i\}\subset[m]$ . The dimension of a simplicial complex $\mathcal{K}$ is the maximal cardinality of its simplices minus one.

We denote the full subcomplex of $\mathcal{K}$ on a vertex set $J=\{j_{1},\ldots,j_{k}\}\subset[m]$ by $\mathcal{K}_{J}$ or by $\mathcal{K}_{\{j_{1},\ldots,j_{k}\}}$ .

The moment-angle complex $\mathcal{Z}_{\mathcal{K}}$ corresponding to $\mathcal{K}$ is defined as follows (see [BP, §4.1]):

\mathcal{Z}_{\mathcal{K}}=\underset{I\subset\mathcal{K}}{\bigcup}\Bigl{(}%\underset{i\in I}{\prod}D^{2}\times\underset{i\notin I}{\prod}S^{1}\Bigr{)}%\subset\prod_{i=1}^{m}D^{2}\,.

Lemma 2.1.

If $\mathcal{K}_{J}$ is a full subcomplex of $\mathcal{K}$ , then $\mathcal{Z}_{\mathcal{K}_{J}}$ is a retract of $\mathcal{Z}_{\mathcal{K}}$ , and $H^{*}(\mathcal{Z}_{\mathcal{K}_{J}})$ is a subring of $H^{*}(\mathcal{Z}_{\mathcal{K}})$ .

Proof..

Let $i\colon\mathcal{Z}_{\mathcal{K}}\hookrightarrow(D^{2})^{m}$ be canonical inclusion, and let $q\colon(D^{2})^{m}\rightarrow(D^{2})^{|J|}$ be the map that omits the coordinates corresponding to $[m]\setminus J$ . Then $r=q\circ i\colon\mathcal{Z}_{\mathcal{K}}\rightarrow\mathcal{Z}_{\mathcal{K}_{%J}}$ is the required retraction, and it induces an injective hom*omorphism $H^{*}(\mathcal{Z}_{\mathcal{K}_{J}})\to H^{*}(\mathcal{Z}_{\mathcal{K}})$ in cohom*ology.∎

Theorem 2.2 ([BP, Theorem 4.5.8]).

There are isomorphisms of groups

H^{l}(\mathcal{Z}_{\mathcal{K}})\cong\underset{J\subset[m]}{\bigoplus}%\widetilde{H}^{l-|J|-1}(\mathcal{K}_{J})

These isomorphisms combine to form a ring isomorphism $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong\underset{J\subset[m]}{\bigoplus}%\widetilde{H}^{*}(\mathcal{K}_{J})$ , where the ring structure on the right hand side is given by the canonical maps

H^{k-|I|-1}(\mathcal{K}_{I})\otimes H^{l-|J|-1}(\mathcal{K}_{J})%\longrightarrow H^{k+l-|I|-|J|-1}(\mathcal{K}_{I\cup J})\,,

which are induced by simplicial maps $\mathcal{K}_{I\cup J}\rightarrow\mathcal{K}_{I}*\mathcal{K}_{J}$ for $I\cap J=\varnothing$ and zero otherwise.

We denote

\mathcal{H}^{l,J}=\widetilde{H}^{l}(\mathcal{K}_{J}),\quad\mathcal{H}^{*,J}=%\widetilde{H}^{*}(\mathcal{K}_{J})\quad\text{and}\quad\mathcal{H}^{l,*}=%\underset{J\subset[m]}{\bigoplus}\widetilde{H}^{l}(\mathcal{K}_{J}).

The ring structure in $H^{*}(\mathcal{Z}_{\mathcal{K}})=\mathcal{H}^{*,*}(\mathcal{K})$ is given by the maps

(2.1)

\mathcal{H}^{k,I}\otimes\mathcal{H}^{l,J}\longrightarrow\mathcal{H}^{k+l+1,I%\sqcup J},\qquad k,l\geq 0,\;I\cap J=\varnothing.

Proposition 2.3.

If $\mathcal{K}$ is an $n$ -dimensional simplicial complex, then the cohom*ological product length of $\mathcal{Z}_{\mathcal{K}}$ is at most $n+1$ .

Proof..

Suppose there are elements $c_{i}\in H^{l_{i}}(\mathcal{Z}_{\mathcal{K}})$ , $i=1,\ldots,r$ , such that $c_{1}\cdots c_{r}=c\neq 0$ .This implies, by Theorem2.2, that there are elements $\widehat{c}\in\widetilde{H}^{l}(\mathcal{K}_{J})$ and $\widehat{c}_{i}\in\widetilde{H}^{l_{i}-|J_{i}|-1}(\mathcal{K}_{J_{i}})$ such that $\widehat{c}_{1}\cdots\widehat{c}_{r}=\widehat{c}\neq 0$ , where $l=(\sum^{r}_{i=1}l_{i}-|J_{i}|-1)+r-1$ , $l_{i}-|J_{i}|-1\geq 0$ and $J=J_{1}\sqcup\cdots\sqcup J_{r}$ . It follows that

n=\dim\mathcal{K}\geq l=\Bigl{(}\sum^{r}_{i=1}l_{i}-|J_{i}|-1\Bigr{)}+r-1\geq r%-1,

hence $n+1\geq r$ , as claimed.∎

A (convex) polytope $P$ is a bounded intersection of a finite number of halfspaces in a real affine space. A facet of $P$ is its face of codimension $1$ .

A polytope $P$ of dimension $n$ is called simple if each vertex of $P$ belongs to exactly $n$ facets. So if $P$ is simple, then the dual polytope $P^{*}$ is simplicial and its boundary $\partial P^{*}$ is a simplicial complex, which we denote by $\mathcal{K}_{P}$ . Then $\mathcal{K}_{P}$ is the nerve complex of the covering of $\partial P$ by its facets. The moment-angle complex $\mathcal{Z}_{\mathcal{K}_{P}}$ is denoted simply by $\mathcal{Z}_{P}$ .

A simplicial sphere (or triangulated sphere) is a simplicial complex $\mathcal{K}$ whose geometric realisation is homeomorphic to a sphere. If $P$ is a simple polytope of dimension $n$ , then the nerve complex $\mathcal{K}_{P}$ is a simplicial sphere of dimension $n-1$ .For $n\leq 3$ , any simplicial sphere of dimension $n-1$ is combinatorially equivalent to the nerve complex $\mathcal{K}_{P}$ of a simple $n$ -dimensional polytope $P$ . This is not true in dimensions $n\geq 4$ ; the Barnette sphere is a famous example of a $3$ -dimensional simplicial sphere with $8$ vertices that is not combinatorially equivalent to the boundary of a convex $4$ -dimensional polytope (see[BP, §2.5]).

Theorem 2.4 ([BP, Theorem 4.1.4, Corollary 6.2.5]).

Let $\mathcal{K}$ be a simplicial sphere of dimension $(n-1)$ with $m$ vertices. Then $\mathcal{Z}_{\mathcal{K}}$ is a closed topological manifold of dimension $m+n$ . If $P$ be a simple $n$ -dimensional polytope with $m$ facets, then $\mathcal{Z}_{P}$ is a smooth manifold of dimension $m+n$ .

A simple polytope $Q$ is called stacked if it can be obtained from a simplex by a sequence of stellar subdivisions of facets. Equivalently, the dual simple polytope $P=Q^{*}$ is obtained from a simplex by iterating the vertex cut operation.

A connected sum of products of spheres is a closed $n$ -dimensional manifold $M$ homeomorphic to a connected sum $M_{1}\#\cdots\#M_{k}$ where each $M_{k}$ is a product spheres $S^{n_{k1}}\times\cdots\times S^{n_{kl}}$ , where $n_{k1}+\cdots+n_{kl}=n$ .

The next theorem follows from the results of McGavran[M], see[BM, Theorem6.3]. See also[GL, §2.2] for a different approach.

Theorem 2.5 (see [BP, Theorem4.6.12]).

Let $P$ be a dual stacked $n$ -polytope with $m>n+1$ facets. Then the corresponding moment-angle manifold is homeomorphic to a connected sum of products of spheres with two spheres in each product, namely,

\mathcal{Z}_{P}\cong\underset{k=3}{\overset{m-n+1}{\#}}(S^{k}\times S^{m+n-k})%^{\#(k-2)\binom{m-n}{k-1}}

In particular, the moment-angle complex corresponding to a polygon (a two-dimensional polytope) is a connected sum of products of spheres.

A graph $\Gamma$ is a one-dimensional simplicial complex.A graph $\Gamma$ is called chordal if every cycle of $\Gamma$ with more than $3$ vertices has a chord, where a chord is an edge connecting two vertices that are not adjacent in the cycle.The vertices of a graph are in perfect elimination order if for any vertex $\{i\}$ all its neighbours with indices less than $i$ are pairwise adjacent.

Theorem 2.6 ([FG]).

A graph is chordal if and only if its vertices can be arranged in a perfect elimination order.

The following property of chordal graphs is immediate from Theorem2.6.

Proposition 2.7.

Let $\Gamma$ be a chordal graph on $m$ vertices, and suppose that the vertices of $\Gamma$ are arranged in a perfect elimination order. Then $\Gamma\setminus\{m\}$ is also a chordal graph, and the vertices of $\Gamma\setminus\{m\}$ are automatically arranged in the perfect elimination order.

Lemma 2.8.

Let $\mathcal{K}$ be a simplicial sphere of dimension greater than $1$ such that $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M_{1}\#M_{2}\#\cdots\#M_{k})$ where each $M_{i}$ is a product of two spheres. Then the one-skeleton $\mathcal{K}^{1}$ is a chordal graph.

Proof..

Let $\dim\mathcal{K}=n-1$ and $M_{i}=S^{l_{i}}\times S^{m+n-l_{i}}$ , $i=1,\ldots,k$ . We denote the corresponding generators of $H^{*}(\mathcal{Z}_{\mathcal{K}})$ by $a_{i}$ , $b_{i}$ , where $\deg a_{i}=l_{i}$ , $\deg b_{i}=m+n-l_{i}$ , $i=1,\ldots,k$ , and $c$ , $\deg c=m+n$ (the fundamental class). We have relations $a_{i}\cdot b_{i}=c$ for $i=1,\ldots,k$ , and all other products in $H^{*}(\mathcal{Z}_{\mathcal{K}})$ are trivial.

Suppose that there is a chordless cycle $C$ in $\mathcal{K}$ with $p>3$ vertices. Then $C$ is a full subcomplex in $\mathcal{K}$ , therefore $H^{*}(\mathcal{Z}_{C})$ is a subring of $H^{*}(\mathcal{Z}_{\mathcal{K}})$ by Lemma2.1. By Theorem2.5 $\mathcal{Z}_{C}$ is also a connected sum of products of spheres, so there are nontrivial products $a^{\prime}_{j}\cdot b^{\prime}_{j}=c^{\prime}$ in the ring $H^{*}(\mathcal{Z}_{C})$ , where $c^{\prime}$ is the fundamental class of $\mathcal{Z}_{C}$ and $\deg c^{\prime}=|C|+2\leq m+2<m+n=\deg c$ , which is impossible in $H^{*}(\mathcal{Z}_{\mathcal{K}})$ .Thus, there are no chordless cycles in $\mathcal{K}$ with more than three vertices, so $\mathcal{K}^{1}$ is a chordal graph.∎

The converse of Lemma2.8 holds for two- and three-dimensional spheres, as shown in the next two sections, but fails in higher dimensions, as shown by the example below. A missing edge of $\mathcal{K}$ is a pair of vertices that do not form a $1$ -simplex.

Example 2.9.

Let $P$ be the three-dimensional polytope obtained by cutting two vertices of the tetrahedron $\Delta^{3}$ .By Theorem2.5,

\mathcal{Z}_{P}\cong(S^{3}\times S^{6})^{\#3}\#(S^{4}\times S^{5})^{\#2}\,.

Now let $P^{\prime}=P\times\Delta^{d}$ , where $d>1$ , so that $\mathcal{K}_{P^{\prime}}$ is a simplicial sphere of dimension $d+2>3$ .We have $\mathcal{Z}_{P^{\prime}}=\mathcal{Z}_{P}\times\mathcal{Z}_{\Delta^{d}}\cong%\mathcal{Z}_{P}\times S^{2d-1}$ , which is not a connected sum of products of spheres. However, $\mathcal{K}_{P^{\prime}}^{1}$ is a chordal graph. Indeed, $\mathcal{K}_{P^{\prime}}=\mathcal{K}_{P}*\partial\Delta^{d}$ . Hence, each missing edge of $\mathcal{K}_{P^{\prime}}$ is a missing edge of $\mathcal{K}_{P}$ . There are only three missing edges in $\mathcal{K}_{P^{\prime}}$ , and no two of them form a chordless $4$ -cycle. Also, there can be no chordless cycles with more than $4$ vertices, as any such chordless cycle has at least $5$ missing edges.

The next lemma builds upon the results of[FCMW, §4].

Lemma 2.10.

Let $\mathcal{K}$ be a simplicial sphere of dimension $>1$ such that $H^{*}(\mathcal{Z}_{K})\cong H^{*}(M_{1}\#M_{2}\#\cdots\#M_{k})$ , where each $M_{i}$ is a product of spheres. Suppose that $\mathcal{K}^{1}$ is not a chordal graph. Then all missing edges $I_{1},\ldots,I_{r}$ of $\mathcal{K}$ are pairwise disjoint and

\mathcal{K}_{I_{1}\sqcup I_{2}\sqcup\cdots\sqcup I_{r}}=\mathcal{K}_{I_{1}}*%\mathcal{K}_{I_{2}}*\cdots*\mathcal{K}_{I_{r}}.

Proof..

By [FCMW, Lemma4.5] any chordless cycle in $\mathcal{K}^{1}$ has three or four vertices. Since $\mathcal{K}^{1}$ is not chordal, it contains a chordless $4$ -cycle. Then by[FCMW, Lemma4.6] missing edges of $\mathcal{K}$ are pairwise disjoint, i. e. each pair of missing edges forms a chordless $4$ -cycle.

We have $H^{3}(\mathcal{Z}_{\mathcal{K}})\cong\bigoplus_{|J|=2}\widetilde{H}^{0}(%\mathcal{K}_{J})=\bigoplus_{j=1}^{r}\widetilde{H}^{0}(\mathcal{K}_{I_{j}})$ by Theorem2.2. Choose a basis $a_{1},\ldots,a_{r}$ of $H^{3}(\mathcal{Z}_{\mathcal{K}})$ according to this decomposition, so that $a_{j}$ corresponds to a generator of $\widetilde{H}^{0}(\mathcal{K}_{I_{j}})=\widetilde{H}^{0}(S^{0})\cong\mathbb{Z}$ for $j=1,\ldots,r$ . Each product $a_{j}\cdot a_{k}$ is nonzero by Theorem2.2, because $\mathcal{K}_{I_{j}\sqcup I_{k}}$ is a $4$ -cycle.

Through the ring isomorphism $H^{*}(\mathcal{Z}_{K})\cong H^{*}(M_{1}\#M_{2}\#\cdots\#M_{k})$ , three-dimensional sphere factors $S^{3}_{ji}$ in the connected summands $M_{i}$ correspond to cohom*ology classes in $H^{3}(\mathcal{Z}_{K})$ , which we denote by $s_{1},\ldots,s_{r}$ . We have $H^{3}(\mathcal{Z}_{K})\cong\mathbb{Z}\langle a_{1},\ldots,a_{r}\rangle\cong%\mathbb{Z}\langle s_{1},\ldots,s_{r}\rangle$ . Furthermore, if we denote the subring of $H^{*}(\mathcal{Z}_{K})$ generated by $a_{1},\ldots,a_{r}$ by $A$ and denote the subring generated by $s_{1},\ldots,s_{r}$ by $R$ , then we have a ring isomorphism $A\cong R$ . Since $a_{i}\cdot a_{j}\neq 0$ for any $i\neq j$ , we have $\mathop{\mathrm{missing}}{rank}A^{6}=\mathop{\mathrm{missing}}{rank}R^{6}=%\frac{r(r-1)}{2}$ . This implies that $s_{i}\cdot s_{j}\neq 0$ for $i\neq j$ . It follows that all spheres $S^{3}_{ji}$ , $j=1,\ldots,r$ , belong to the same connected summand $M_{i}$ , because the product of the cohom*ology classes corresponding to sphere factors in different summands of the connected sum $M_{1}\#M_{2}\#\cdots\#M_{k}$ is zero. Therefore, $s_{1}\cdot s_{2}\cdots s_{r}\neq 0$ in $R$ . This implies, by the ring isomorphism $A\cong R$ , that $a_{1}\cdot a_{2}\cdots a_{r}$ is nonzero in $H^{*}(\mathcal{Z}_{\mathcal{K}})$ . Now it follows from the product description in Theorem2.2 that $\mathcal{K}_{I_{1}\sqcup I_{2}\sqcup\cdots\sqcup I_{r}}=\mathcal{K}_{I_{1}}*%\mathcal{K}_{I_{2}}*\cdots*\mathcal{K}_{I_{r}}$ .∎

3. Two-dimensional spheres

Here we consider moment-angle manifolds corresponding to two-dimensional simplicial spheres $\mathcal{K}$ or, equivalently, to three-dimensional simple polytopes $P$ .

The case $P=I^{3}$ (a three-dimensional cube) is special. In this case the nerve complex $\mathcal{K}_{P}$ is $S^{0}*S^{0}*S^{0}$ (the join of three $0$ -dimensional spheres, or the boundary of a three-dimensional cross-polytope) and $\mathcal{Z}_{P}\cong S^{3}\times S^{3}\times S^{3}$ .

A simplicial complex $\mathcal{K}$ is called Golod if the multiplication and all higher Massey products in $H^{*}(\mathcal{Z}_{\mathcal{K}})$ are trivial. (Equivalently, the Stanley–Reisner ring $\mathbf{k}[\mathcal{K}]$ is a Golod ring over any field $\mathbf{k}$ , see[BP, §4.9].) A simplicial complex $\mathcal{K}$ on $[m]$ is called minimally non-Golod if $\mathcal{K}$ is not Golod, but for any vertex $i\in[m]$ the complex $\mathcal{K}_{[m]\setminus\{i\}}$ is Golod.

The following result extends[BM, Proposition11.6], where the equvalence of conditions (a), (b) and (c) was proved:

Proposition 3.1.

(a)
$P$ is obtained from a simplex $\Delta^{3}$ by iterating the vertex cut operation, i. e. $P^{*}$ is a stacked polytope;
(b)
$\mathcal{Z}_{P}$ is diffeomorphic to a connected sum of products of spheres;
(c)
$H^{*}(\mathcal{Z}_{P})$ is isomorphic to the cohom*ology ring of a connected sum of products of spheres;
(d)
the one-dimensional skeleton of the nerve complex $\mathcal{K}_{P}$ is a chordal graph;
(e)
$\mathcal{K}_{P}$ is minimally non-Golod, unless $P=\Delta^{3}$ .

Proof..

We prove the implications (a) $\Rightarrow$ (b) $\Rightarrow$ (c) $\Rightarrow$ (d) $\Rightarrow$ (a), (e) $\Rightarrow$ (d) and (a) $\Rightarrow$ (e).

(a) $\Rightarrow$ (b) This is Theorem2.5.

(b) $\Rightarrow$ (c) is clear.

(c) $\Rightarrow$ (d) Let $H^{*}(\mathcal{Z}_{P})\cong H^{*}(M_{1}\#M_{2}\#\cdots\#M_{k})$ , where each $M_{i}$ is a product of spheres.Since the cohom*ological product length of $\mathcal{Z}_{P}$ is at most $3$ (Corollary2.3), there is at most $3$ sphere factors in each $M_{i}$ . If some $M_{i}$ has exactly $3$ factors, then $\mathcal{Z}_{P}=S^{3}\times S^{3}\times S^{3}$ and $P$ is a cube by[FCMW, Theorem4.3(a)]. This contradicts the assumption.Now, $\mathcal{K}^{1}_{P}$ is a chordal graph by Lemma2.8.

(d) $\Rightarrow$ (a) We use induction on the number $m$ of facets of $P$ . The base $m=4$ is clear, as $P$ is a simplex $\Delta^{3}$ in this case.

For the induction step, assume that the vertices of $\mathcal{K}_{P}$ are arranged in a perfect elimination order. Let $j_{1},\ldots,j_{s}$ be the vertices adjacent to the last vertex $m$ . First we prove that $s=3$ .

Let $F_{i}$ denote the $i$ th facet of $P$ . Since $\{j_{1},\ldots,j_{s}\}$ is a clique of $\mathcal{K}^{1}_{P}$ , the facets $F_{j_{1}},\ldots,F_{j_{s}}$ are pairwise adjacent.Suppose that $s\geq 4$ . Renumbering the facets if necessary, we may assume that $F_{j_{1}}$ , $F_{j_{2}}$ , $F_{j_{3}}$ , $F_{j_{4}}$ are consecutive facets in a cyclic order around $F_{m}$ , so that $F_{m}\cap F_{j_{1}}\cap F_{j_{3}}=\varnothing$ and $F_{m}\cap F_{j_{2}}\cap F_{j_{4}}=\varnothing$ .Since $F_{j_{1}}$ and $F_{j_{3}}$ are adjacent, the facets $F_{m}$ , $F_{j_{1}}$ and $F_{j_{3}}$ form a $3$ -belt (a prismatic $3$ -circuit). This $3$ -belt splits $\partial P$ into two connected components[BE, Lemma2.7.2]. The facets $F_{j_{2}}$ and $F_{j_{4}}$ lie in different components, so they cannot be adjacent. A contradiction. Hence, $s=3$ .

Since $F_{m}$ has $3$ adjacent facets, it is a triangle. If $F_{m}$ is adjacent to a triangular facet, then $P$ is a simplex. Otherwise, there exist a polytope $P^{\prime}$ such that $P$ is obtained from $P^{\prime}$ by cutting a vertex with formation of a new facet $F_{m}$ .Then $\mathcal{K}_{P^{\prime}}$ is obtained from $\mathcal{K}_{P}$ by removing the vertex $\{m\}$ and adding simplex $\{j_{1},j_{2},j_{3}\}$ . Hence, the $1$ -skeleton of $\mathcal{K}_{P^{\prime}}$ is also a chordal graph by Proposition2.7. Now $P^{\prime}$ has $m-1$ facets, so we complete the induction step.

(e) $\Rightarrow$ (d) Let $\mathcal{K}_{P}$ be minimally non-Golod, and suppose there is a chordless cycle $C$ in $\mathcal{K}_{P}^{1}$ with $p>3$ vertices. Then $C$ is a full subcomplex of $\mathcal{K}_{P}$ and $p<m$ (otherwise $\mathcal{K}_{P}=C$ , which is impossible for a $3$ -dimensional polytope). For any vertex $v\in[m]\setminus C$ , note that $C$ is also a full subcomplex also in $\mathcal{K}_{P}\setminus\{v\}$ . Therefore, $H^{*}(\mathcal{Z}_{C})$ is a subring of $H^{*}(\mathcal{Z}_{\mathcal{K}_{P}\setminus\{v\}})$ by Lemma2.1. On the other hand, there are nontrivial products is $H^{*}(\mathcal{Z}_{C})$ by Theorem2.5, whereas all products in $H^{*}(\mathcal{Z}_{\mathcal{K}_{P}\setminus\{v\}})$ must be trivial, since $\mathcal{K}_{P}\setminus\{v\}$ is Golod. A contradiction. Hence, there are no chordless cycles in $\mathcal{K}_{P}^{1}$ .

(a) $\Rightarrow$ (e) This follows from[L, Theorem3.9]: if an $n$ -dimensional simple polytope $P$ is obtained from $P^{\prime}$ by a vertex cut, and $\mathcal{K}_{P^{\prime}}$ is minimally non-Golod, then $\mathcal{K}_{P}$ is also minimally non-Golod.∎

4. Three-dimensional spheres

Recall that the product in $H^{*}(\mathcal{Z}_{\mathcal{K}})=\mathcal{H}^{*,*}(\mathcal{K})$ is given by(2.1).A nonzero element $c\in\mathcal{H}^{l,J}=\widetilde{H}^{l}(\mathcal{K}_{J})$ is decomposable if $c=\sum_{i=1}^{p}a_{i}\cdot b_{i}$ for some nonzero $a_{i}\in\widetilde{H}^{r_{i}}(\mathcal{K}_{I_{i}})$ , $b_{i}\in\widetilde{H}^{l-1-r_{i}}(\mathcal{K}_{J\setminus I_{i}})$ , where $0\leq r_{i}\leq l-1$ and $I_{i}\subset J$ are proper subsets for $i=1,\ldots,p$ .

A missing face (or a minimal non-face) of $\mathcal{K}$ is a subset $I\subset[m]$ such that $I$ is not a simplex of $\mathcal{K}$ , but every proper subset of $I$ is a simplex of $\mathcal{K}$ . Each missing face corresponds to a full subcomplex $\partial\Delta_{I}\subset\mathcal{K}$ , where $\partial\Delta_{I}$ denotes the boundary of simplex $\Delta_{I}$ on the vertex set $I$ . A missing face $I$ defines a simplicial hom*ology class in $\widetilde{H}_{|I|-2}(\mathcal{K})$ , which we continue to denote by $\partial\Delta_{I}$ .We denote by $\mathop{\mathrm{missing}}{MF}_{n}(\mathcal{K})$ the set of missing faces $I$ of dimension $n$ , that is, with $|I|=n+1$ .

Lemma 4.1.

Let $I\in\mathop{\mathrm{missing}}{MF}_{l}(\mathcal{K})$ be a missing face of $\mathcal{K}$ . Then any cohom*ology class $c\in\mathcal{H}^{l-1,*}(\mathcal{K})$ such that $\langle c,\partial\Delta_{I}\rangle\neq 0$ is indecomposable.

Proof..

Let $\mathcal{K}^{\prime}$ be the simplicial complex obtained from $\mathcal{K}$ by filling in all missing faces of dimension $l$ with simplices, so that $\mathop{\mathrm{missing}}{MF}_{l}(\mathcal{K}^{\prime})=\varnothing$ and $\mathcal{K}^{l-1}=(\mathcal{K}^{\prime})^{l-1}$ . Then the inclusion $i:\mathcal{K}\hookrightarrow\mathcal{K}^{\prime}$ induces a ring hom*omorphism $i^{*}:\mathcal{H}^{*,*}(\mathcal{K}^{\prime})\rightarrow\mathcal{H}^{*,*}(%\mathcal{K})$ and $\mathcal{H}^{r,*}(\mathcal{K}^{\prime})\cong\mathcal{H}^{r,*}(\mathcal{K})$ for $r\leq l-2$ . Also, $i_{*}(\partial\Delta_{I})=0$ for $I\in\mathop{\mathrm{missing}}{MF}_{l}(\mathcal{K})$ .

Suppose $c$ is decomposable, that is, $c=\sum_{i=1}^{p}a_{i}\cdot b_{i}$ . Choose $a_{i}^{\prime},b_{i}^{\prime}$ such that $i^{*}(a_{i}^{\prime})=a_{i}$ and $i^{*}(b_{i}^{\prime})=b_{i}$ and define $c^{\prime}:=\sum_{i=1}^{p}a_{i}^{\prime}\cdot b_{i}^{\prime}$ . Then $i^{*}(c^{\prime})=c$ and

\langle c,\partial\Delta_{I}\rangle=\langle i^{*}(c^{\prime}),\partial\Delta_{%I}\rangle=\langle c^{\prime},i_{*}(\partial\Delta_{I})\rangle=0.

This is a contradiction.∎

Theorem 4.2.

Let $\mathcal{K}$ be a three-dimensional simplicial sphere such that $\mathcal{K}\neq\partial\Delta^{4}$ and $\mathcal{K}^{1}$ is a chordal graph. Then $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M)$ , where $M$ is a connected sum of products of spheres with two spheres in each product.

Proof..

We use the notation $\mathcal{H}^{*,*}=H^{*}(\mathcal{Z}_{\mathcal{K}})$ and analyse possible nontrivial products in(2.1). We have $\mathcal{H}^{k,*}=0$ for $k\geq 4$ since $\mathcal{K}$ is a three-dimensional sphere.Products of the form $\mathcal{H}^{3,*}\otimes\mathcal{H}^{i,*}\to\mathcal{H}^{4+i,*}$ , $\mathcal{H}^{2,*}\otimes\mathcal{H}^{2,*}\to\mathcal{H}^{5,*}$ and $\mathcal{H}^{2,*}\otimes\mathcal{H}^{1,*}\to\mathcal{H}^{4,*}$ are therefore trivial for dimensional reasons.

Since $\mathcal{K}$ is a 3-dimensional sphere, $\mathcal{Z}_{\mathcal{K}}$ is an $(m+4)$ -dimensional manifold. Nontrivial products $\widetilde{H}^{i}(\mathcal{K}_{I})\otimes\widetilde{H}^{2-i}(\mathcal{K}_{J})%\to\widetilde{H}^{3}(\mathcal{K}_{I\cup J})$ come from Poincaré duality for $\mathcal{Z}_{\mathcal{K}}$ (see[BP, Proposition4.6.6]), because $\widetilde{H}^{3}(\mathcal{K}_{I\cup J})$ is nonzero only when $I\sqcup J=[m]$ . The Poincaré duality isomorphisms $\widetilde{H}^{i}(\mathcal{K}_{I})\cong\widetilde{H}_{2-i}(\mathcal{K}_{[m]%\setminus I})$ (or the Alexander duality isomorphisms for the $3$ -sphere $\mathcal{K}$ , see[BP, 3.4.11]) imply that the groups $\widetilde{H}^{i}(\mathcal{K}_{I})$ are torsion-free for any $i$ and $I\subset[m]$ .

Next we prove that all multiplications of the form $\mathcal{H}^{0,*}\otimes\mathcal{H}^{0,*}\longrightarrow\mathcal{H}^{1,*}$ are trivial.Assume that there are cohom*ology classes $a,b\in\mathcal{H}^{0,*}$ such that $0\neq a\cdot b=:c\in\widetilde{H}^{1}(\mathcal{K}_{I})$ . Since $c\neq 0$ there exists $\gamma\in H_{1}(\mathcal{K}_{I})$ such that $\langle c,\gamma\rangle\neq 0$ . We can write $\gamma=\lambda_{1}\gamma_{1}+\cdots+\lambda_{k}\gamma_{k}$ , where each $\gamma_{i}$ is a simple chordless cycle in $\mathcal{K}^{1}$ and $\lambda_{i}\neq 0$ . Since $\mathcal{K}^{1}$ is chordal, $\gamma_{i}\in\mathop{\mathrm{missing}}{MF}_{2}(\mathcal{K})$ . Now, $0\neq\langle c,\gamma\rangle=\sum_{j=1}^{k}\lambda_{i}\langle c,\gamma_{i}\rangle$ , so $\langle c,\gamma_{i}\rangle\neq 0$ for some $i$ . Hence, $c$ is indecomposable by Lemma4.1. A contradiction.

Finally, we prove that all multiplications of the form $\mathcal{H}^{0,*}\otimes\mathcal{H}^{1,*}\longrightarrow\mathcal{H}^{2,*}$ are trivial.Assume that there exists a nontrivial product $a^{0}\cdot b^{1}=c^{2}\neq 0$ for some $a^{0}\in\widetilde{H}^{0}(\mathcal{K}_{I})$ , $b^{1}\in\widetilde{H}^{1}(K_{J})$ , $c^{2}\in\widetilde{H}^{2}(\mathcal{K}_{I\cup J})$ . By Poincaré duality there exists an element $a^{\prime}\in\widetilde{H}^{0}(\mathcal{K}_{[m]\setminus(I\cup J)})$ such that $0\neq a^{\prime}\cdot c^{2}=a^{\prime}\cdot a^{0}\cdot b^{1}\in\widetilde{H}^{%3}(\mathcal{K})$ . Then $a^{0}\cdot a^{\prime}\neq 0$ , so we obtain a nontrivial multiplication of the form $\mathcal{H}^{0,*}\otimes\mathcal{H}^{0,*}\longrightarrow\mathcal{H}^{1,*}$ . A contradiction.

It follows that the only nontrivial multiplications in $\mathcal{H}^{*,*}(\mathcal{K})$ are

\mathcal{H}^{0,I}\otimes\mathcal{H}^{2,[m]\setminus I}\longrightarrow\mathcal{%H}^{3,[m]}\quad\text{and}\quad\mathcal{H}^{1,J}\otimes\mathcal{H}^{1,[m]%\setminus J}\longrightarrow\mathcal{H}^{3,[m]},

which arise from Poincaré duality. Therefore, the ring $H^{*}(\mathcal{Z}_{\mathcal{K}})$ is free as an abelian group with $\mathbb{Z}$ -basis

\{1,a^{0}_{1},\ldots,a^{0}_{k},a^{1}_{1},\ldots,a^{1}_{l},b^{1}_{1},\ldots,b^{%1}_{l},b^{2}_{1},\ldots,b^{2}_{k},c\},

where $a^{0}_{1},\ldots,a^{0}_{k}\in\mathcal{H}^{0,*}$ , $a^{1}_{1},\ldots,a^{1}_{l},b^{1}_{1},\ldots,b^{1}_{l}\in\mathcal{H}^{1,*}$ , $b^{2}_{1},\ldots,b^{2}_{k}\in\mathcal{H}^{2,*}$ , $c\in\mathcal{H}^{3,m}=H^{m+3}(\mathcal{Z}_{\mathcal{K}})$ is the fundamental class, and the product is given by $a^{0}_{i}\cdot b^{2}_{j}=\delta_{ij}c$ and $a^{1}_{p}\cdot b^{1}_{q}=\delta_{pq}c$ , where $\delta_{ij}$ is the Kronecker delta. At least one of the groups $\mathcal{H}^{0,*}$ and $\mathcal{H}^{1,*}$ is nonzero, as otherwise $\mathcal{K}=\partial\Delta^{4}$ and $\mathcal{Z}_{\mathcal{K}}\cong S^{9}$ . Then $H^{*}(\mathcal{Z}_{\mathcal{K}})=\mathcal{H}^{*,*}$ is isomorphic to the cohom*ology ring of a connected sum of products spheres with two spheres in each product.∎

For simplicial spheres $\mathcal{K}$ of dimension $>3$ , the condition that $\mathcal{K}^{1}$ is a chordal graph does not imply that $H^{*}(\mathcal{Z}_{\mathcal{K}})$ is isomorphic to the cohom*ology ring of a connected sum of spheres, as shown by Example2.9. The next result gives a sufficient condition in any dimension. We say that the group $\mathcal{H}^{l,*}(\mathcal{K})$ is generated by missing faces of $\mathcal{K}$ if for any nonzero $c\in\mathcal{H}^{l,*}(\mathcal{K})$ there exists $I\in\mathop{\mathrm{missing}}{MF}_{l+1}(\mathcal{K})$ such that $\langle c,\partial\Delta_{I}\rangle\neq 0$ .

Theorem 4.3.

Let $\mathcal{K}$ be a simplicial sphere of dimension $d$ such that $\mathcal{K}\neq\partial\Delta^{d+1}$ and the group $\mathcal{H}^{l,*}(\mathcal{K})$ is generated by missing faces of $\mathcal{K}$ for $l\leq\left\lfloor\frac{2d-1}{3}\right\rfloor$ . Then $H^{*}(\mathcal{Z}_{\mathcal{K}})$ is isomorphic to the cohom*ology ring of a connected sum of products of spheres with two spheres in each product.

Proof..

We can assume that $d\geq 2$ , as otherwise $\mathcal{K}$ is the boundary of polygon and the result follows from Theorem2.5.As in the proof of Theorem4.2, we analyse possible nontrivial products in(2.1).We denote $q:=\left\lfloor\frac{2d-1}{3}\right\rfloor$ .

We have $\mathcal{H}^{k,*}=0$ for $k>d$ since $\mathcal{K}$ is an $d$ -dimensional sphere.Therefore, products of the form $\mathcal{H}^{i,*}\otimes\mathcal{H}^{j,*}\to\mathcal{H}^{i+j+1,*}$ with $i+j\geq d$ are trivial.

Nontrivial products or the form $\mathcal{H}^{i,*}\otimes\mathcal{H}^{j,*}\to\mathcal{H}^{i+j+1,*}$ with $i+j=d-1$ are given by $\widetilde{H}^{i}(\mathcal{K}_{I})\otimes\widetilde{H}^{d-1-i}(\mathcal{K}_{J}%)\to\widetilde{H}^{d}(\mathcal{K}_{I\cup J})$ and come from Poincaré duality, because $\widetilde{H}^{d}(\mathcal{K}_{I\cup J})$ is nonzero only when $I\sqcup J=[m]$ .We prove by contradiction that the groups $\widetilde{H}^{i}(\mathcal{K}_{I})$ are torsion-free for $i\leq q$ . Assume that there is a cocycle $0\neq c\in\mathcal{H}^{i,*}(\mathcal{K})$ and a nonzero integer $k$ such that $k\cdot c=0$ . Let $\tilde{c}$ be a representing cochain for $c$ , then $k\cdot\tilde{c}$ is a coboundary and $k\cdot\tilde{c}=d\tilde{b}$ for some cochain $\tilde{b}$ . By assumption there exists $I\in\mathop{\mathrm{missing}}{MF}_{i+1}(\mathcal{K})$ such that $\langle c,\partial\Delta_{I}\rangle\neq 0$ , hence,

0\neq k\cdot\langle c,\partial\Delta_{I}\rangle=\langle k\cdot\tilde{c},%\partial\Delta_{I}\rangle=\langle d\tilde{b},\partial\Delta_{I}\rangle=\langle%\tilde{b},\partial(\partial\Delta_{I})\rangle=0

and we get a contradiction. Now the Alexander duality isomorphisms $\widetilde{H}^{i}(\mathcal{K}_{J})\cong\widetilde{H}_{d-1-i}(\mathcal{K}_{[m]%\setminus J})$ imply that the hom*ology groups $\widetilde{H}_{j}(\mathcal{K}_{J})$ are torsion-free for $j\geq d-1-q$ . Since $d-1-q\leq q$ , we obtain that $\widetilde{H}_{j}(\mathcal{K}_{J})$ is torsion-free for $j\geq q$ , whereas $\widetilde{H}^{j}(\mathcal{K}_{J})$ is torsion-free for $j\leq q$ . By the universal coefficient theorem we conclude that the groups $\widetilde{H}^{j}(\mathcal{K}_{J})$ are torsion-free for all $j$ and $J$ .

All products of the form $\mathcal{H}^{i,*}\otimes\mathcal{H}^{j,*}\longrightarrow\mathcal{H}^{i+j+1,*}$ are trivial for $i+j<q$ , since any $l$ -dimensional cohom*ology class with $l\leq q$ is indecomposable by Lemma4.1.

Finally, we prove that all products of the form $\mathcal{H}^{i,*}\otimes\mathcal{H}^{j,*}\longrightarrow\mathcal{H}^{i+j+1,*}$ are trivial for $q\leq i+j\leq d-2$ . Suppose there are classes $a\in\mathcal{H}^{i,I}$ and $b\in\mathcal{H}^{j,J}$ with $q\leq i+j\leq d-2$ such that $0\neq a\cdot b=:c\in\mathcal{H}^{i+j+1,I\cup J}$ . Without loss of generality we assume that $i\leq j$ . Then there exists an element $a^{\prime}\in\widetilde{H}^{d-i-j-2}(\mathcal{K}_{[m]\setminus(I\cup J)})$ such that $0\neq a^{\prime}\cdot c=a^{\prime}\cdot a\cdot b\in\widetilde{H}^{d}(\mathcal{%K})$ by Poincaré duality. Therefore, $a\cdot a^{\prime}\neq 0$ and so we obtain a nontrivial product of the form $\mathcal{H}^{i,*}\otimes\mathcal{H}^{k,*}\longrightarrow\mathcal{H}^{i+k+1,*}$ for $k=d-i-j-2$ . By assumption, $q\leq i+j\leq 2j$ and $q>\frac{2d-1}{3}-1$ , hence,

i+k=d-j-2\leq d-2-\frac{q}{2}<q.

Thus, $a^{\prime}\cdot a$ is a product of the form $\mathcal{H}^{i,*}\otimes\mathcal{H}^{k,*}\longrightarrow\mathcal{H}^{i+k+1,*}$ with $i+k<q$ , so it must be trivial. A contradiction.

We obtain that the only nontrivial products in $\mathcal{H}^{*,*}(\mathcal{K})$ arise from Poincaré duality. It follows that the ring $H^{*}(\mathcal{Z}_{\mathcal{K}})$ is isomorphic to the cohom*ology ring of a connected sum of products of spheres with two spheres in each product.∎

The next theorem extends the result of Theorem4.2 to a complete characterisation of three-dimensional spheres $\mathcal{K}$ such that $H^{*}(\mathcal{Z}_{\mathcal{K}})$ is isomorphic to the cohom*ology ring of a connected sum of products of spheres.

Theorem 4.4.

Let $\mathcal{K}$ be a three-dimensional simplicial sphere. Then $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M_{1}\#\cdots\#M_{k})$ where each $M_{i}$ is a product of spheres if and only if one of the following conditions is satisfied:

(a)
$\mathcal{K}=S^{0}*S^{0}*S^{0}*S^{0}$ (the boundary of a $4$ -dimensional cross-polytope);
(b)
$\mathcal{K}^{1}$ is a chordal graph;
(c)
$\mathcal{K}^{1}$ has exactly two missing edges which form a chordless $4$ -cycle.

Proof..

First we prove the “only if” statement. If $\mathcal{K}^{1}$ is a chordal graph, then (b) is satisfied. Otherwise, by Lemma2.10 the missing edges $I_{1},\ldots,I_{r}$ of $\mathcal{K}$ are pairwise disjoint and $\mathcal{K}_{I_{1}\sqcup\cdots\sqcup I_{r}}=\mathcal{K}_{I_{1}}*\cdots*%\mathcal{K}_{I_{r}}$ . We have $r\leq 4$ , since $\dim\mathcal{K}=3$ .

If $r=4$ , then $\mathcal{K}=\mathcal{K}_{I_{1}}*\cdots*\mathcal{K}_{I_{4}}$ , so that (a) holds.

If $r=3$ , then $\mathcal{K}_{I_{1}\sqcup I_{2}\sqcup I_{3}}=\mathcal{K}_{I_{1}}*\mathcal{K}_{I%_{2}}*\mathcal{K}_{I_{3}}$ is a two-dimensional simplicial sphere. We have $\widetilde{H}_{0}(\mathcal{K}\setminus\mathcal{K}_{I_{1}\sqcup I_{2}\sqcup I_{%3}})\cong\widetilde{H}^{2}(\mathcal{K}_{I_{1}\sqcup I_{2}\sqcup I_{3}})\cong%\mathbb{Z}$ by Alexander duality. Hence, $\mathcal{K}\setminus\mathcal{K}_{I_{1}\sqcup I_{2}\sqcup I_{3}}$ is not connected. It follows that there is at least one more missing edge in $\mathcal{K}$ besides $I_{1},I_{2},I_{3}$ . A contradiction.

If $r=2$ , then (c) holds.

If $r=1$ , then $\mathcal{K}^{1}$ is in fact a chordal graph, since any chordless cycle with more than three vertices has at least two missing edges. Hence, (b) holds.

Now we prove the “if” statement. If (a) holds, then $\mathcal{Z}_{\mathcal{K}}$ is a product of spheres. If (b) holds, then $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M_{1}\#\cdots\#M_{k})$ where each $M_{i}$ is a product of spheres by Theorem4.2. Suppose (c) holds. Then $\mathcal{H}^{0,*}(\mathcal{K})=\mathbb{Z}\langle a_{1},a_{2}\rangle$ , where $a_{1}$ and $a_{2}$ correspond to the two missing edges of $\mathcal{K}$ , and $a_{1}\cdot a_{2}\neq 0$ . We use the same argument as in the proof of Theorem4.2 with one exception: there is one nontrivial product of the form $\mathcal{H}^{0,*}(\mathcal{K})\otimes\mathcal{H}^{0,*}(\mathcal{K})\otimes%\mathcal{H}^{1,*}(\mathcal{K})\longrightarrow\mathcal{H}^{3,*}(\mathcal{K})$ . Namely, $a_{1}\cdot a_{2}\cdot b\mapsto c$ , where $b$ is Poincaré dual to $a_{1}\cdot a_{2}$ and $c$ is the fundamental class of $\mathcal{K}$ . All other nontrivial products in $H^{*}(\mathcal{Z}_{\mathcal{K}})$ arise from Poincaré duality. Thus the ring $H^{*}(\mathcal{Z}_{\mathcal{K}})$ is generated by elements $\{a_{1},a_{2},b,c,x_{i},y_{i}\colon i=1,2,\ldots,N\}$ , where $x_{i},y_{i}\in\mathcal{H}^{1,*}(\mathcal{K})$ , with the following multiplication rules: $a_{1}\cdot a_{2}\cdot b=c$ , $x_{i}\cdot y_{i}=c$ for $i=1,2,\ldots,N$ , and all other products of generators are zero. Clearly, $H^{*}(\mathcal{Z}_{\mathcal{K}})$ is isomorphic to the cohom*ology ring of a connected sum of products of spheres.∎

Remark.

Note that under condition (c) of Theorem4.4 we have $H^{*}(\mathcal{Z}_{\mathcal{K}})\cong H^{*}(M)$ , where $M$ is a connected sum of products of spheres in which one of the summands is a product of three spheres. The first example of such a simplicial sphere $\mathcal{K}$ was constructed in[FCMW]. Later it was shown in[I] that the corresponding moment-angle manifold $\mathcal{Z}_{\mathcal{K}}$ is diffeomorphic to $M$ .

Remark.

It can be shown that if $\mathcal{K}$ is a three-dimensional simplicial sphere such that $\mathcal{K}^{1}$ is a chordal graph, then all higher Massey products in $H^{*}(\mathcal{Z}_{\mathcal{K}})$ are trivial. This implies that a three-dimensional simplicial sphere $\mathcal{K}\neq\partial\Delta^{4}$ is minimally non-Golod if and only if $\mathcal{K}^{1}$ is a chordal graph. We elaborate on this in a subsequent paper.

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