We prove that the moment-angle complex \(\mathcal{Z}_\mathcal{K}\) corresponding to a \(3\)-dimensional simplicial sphere \(\mathcal{K}\) has the
cohomology ring isomorphic to the cohomology 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.
The moment-angle complex is a topological space (a CW complex) with a torus action that features in toric topology and homotopy theory of polyhedral products [1].
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 [2], [1]. 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 cohomology 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 [1].
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 homotopy-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 [2] that \(\mathcal{Z}_P\) is diffeomorphic to a connected sum of products of spheres if and only if \(P\) is obtained from a 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 Proposition 8):
Proposition 1. 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:
\(P\) is obtained from a simplex \(\Delta^3\) by iterating the vertex cut operation, i. e. \(P^*\) is a stacked polytope;
\(\mathcal{Z}_P\) is diffeomorphic to a connected sum of products of spheres;
\(H^*(\mathcal{Z}_P)\) is isomorphic to the cohomology ring of a connected sum of products of spheres;
the one-dimensional skeleton of the nerve complex \(\mathcal{K}_P\) is a chordal graph;
\(\mathcal{K}_P\) 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 cohomology ring isomorphic to the cohomology ring of a connected sum of products of spheres (see Theorem 11):
Theorem 1. 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:
\(\mathcal{K}=S^0*S^0*S^0*S^0\) (the boundary of a \(4\)-dimensional cross-polytope);
\(\mathcal{K}^1\) is a chordal graph;
\(\mathcal{K}^1\) 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 [3].
When \(\dim P \ge 5\), 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 Example 7. A stronger sufficient condition valid for simplicial spheres of arbitrary dimension is given
in Theorem 10.
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 [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 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 the 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 homomorphism \(H^*(\mathcal{Z}_{\mathcal{K}_J})\to H^*(\mathcal{Z}_\mathcal{K})\) in cohomology. ◻
Theorem 1 ([1]). 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 \[\label{multH} \mathcal{H}^{k,I}\otimes\mathcal{H}^{l,J}\longrightarrow \mathcal{H}^{k+l+1,I\sqcup J},\qquad k,l\ge0,\; I\cap J=\varnothing.\tag{1}\]
Proposition 2. If \(\mathcal{K}\) is a simplicial complex of dimension \(n-1\), then the cohomological product length of \(\mathcal{Z}_\mathcal{K}\) is at most \(n\).
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 Theorem 1, 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 \[n-1=\dim\mathcal{K}\geq l = \Bigl(\sum^r_{i = 1} l_i - |J_i| - 1\Bigr) + r -
1 \geq r - 1,\] hence \(n \geq r\), 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 \(n\le3\), 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 \(n\ge4\); 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 [1]).
Theorem 3 ([1]). 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 of 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 [4], see [2]. See also [5] for a different approach.
Theorem 4 (see [1]). 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 a perfect elimination order if for any vertex \(\{i\}\) all its neighbours with indices less than \(i\) are pairwise adjacent.
Theorem 5 ([6]). 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 Theorem 5.
Proposition 6. 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 a perfect elimination
order.
Lemma 2. 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 Lemma 1. By Theorem 4\(\mathcal{Z}_C\) is also a connected sum of products of spheres, so there
are nontrivial products \(a'_j \cdot b'_j = c'\) in the ring \(H^*(\mathcal{Z}_C)\), where \(c'\) is the fundamental class of \(\mathcal{Z}_C\) and \(\deg c' = |C| + 2 \leq m + 2 < m + n = \deg c\), 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 Lemma 2 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 7. Let \(P\) be a three-dimensional polytope obtained by cutting two vertices of a tetrahedron \(\Delta^3\). By Theorem 4, \[\mathcal{Z}_P \cong (S^3 \times S^{6})^{\#3} \# (S^4 \times S^{5})^{\#2} \,.\] Now let \(P' = P \times\Delta^d\), where
\(d>1\), so that \(\mathcal{K}_{P'}\) is a simplicial sphere of dimension \(d+2>3\). We have \(\mathcal{Z}_{P'}=\mathcal{Z}_P\times \mathcal{Z}_{\Delta^d} \cong \mathcal{Z}_P \times S^{2d - 1}\), which is not a connected sum of products of spheres. However, \(\mathcal{K}_{P'}^1\)
is a chordal graph. Indeed, \(\mathcal{K}_{P'} = \mathcal{K}_P * \partial\Delta^d\). Hence, each missing edge of \(\mathcal{K}_{P'}\) is a missing edge of \(\mathcal{K}_P\). There are only three missing edges in \(\mathcal{K}_{P'}\), 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.
Lemma 3. 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 [3] 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 [3] 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 Theorem 1. 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 Theorem 1, 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 cohomology 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 \(a_i\cdot a_j\ne 0\) for any \(i\ne j\), we have \(\mathop{\mathrm{rank}} A^6=\mathop{\mathrm{rank}}
R^6=\frac{r(r-1)}{2}\). This implies that \(s_i\cdot s_j\ne 0\) for \(i\ne j\). 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 cohomology classes corresponding to sphere factors in different summands of the connected sum \(M_1\# M_2\#\cdots\# M_k\) is zero. Therefore, \(s_1\cdot s_2\cdots s_r\ne0\) 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 Theorem 1 that \(\mathcal{K}_{I_1\sqcup I_2\sqcup\cdots\sqcup I_r}=\mathcal{K}_{I_1} * \mathcal{K}_{I_2} * \cdots * \mathcal{K}_{I_r}\). ◻
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 [1].) 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 [2], where the equvalence of conditions (a), (b) and (c) was proved:
Proposition 8. 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:
\(P\) is obtained from a simplex \(\Delta^3\) by iterating the vertex cut operation, i. e. \(P^*\) is a stacked polytope;
\(\mathcal{Z}_P\) is diffeomorphic to a connected sum of products of spheres;
\(H^*(\mathcal{Z}_P)\) is isomorphic to the cohomology ring of a connected sum of products of spheres;
the one-dimensional skeleton of the nerve complex \(\mathcal{K}_P\) is a chordal graph;
\(\mathcal{K}_P\) 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).
(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
cohomological product length of \(\mathcal{Z}_P\) is at most \(3\) (Proposition 2), 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 [3]. This contradicts the assumption.
Now, \(\mathcal{K}^1_P\) is a chordal graph by Lemma 2.
(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 [7]. 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 \(P'\) such that \(P\) is obtained from \(P'\) by cutting a vertex with formation of a new facet \(F_m\). Then \(\mathcal{K}_{P'}\) 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 \(\mathcal{K}_{P'}\) is also a chordal graph by Proposition 6. Now \(P'\) 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 in \(\mathcal{K}_P \setminus\{v\}\). Therefore, \(H^*(\mathcal{Z}_C)\) is a subring of \(H^*(\mathcal{Z}_{\mathcal{K}_P \setminus\{v\}})\) by Lemma 1. On the other hand, there are nontrivial products is \(H^*(\mathcal{Z}_C)\) by Theorem 4, 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 [8]: if an \(n\)-dimensional simple polytope \(P\) is obtained from \(P'\) by a vertex cut, and \(\mathcal{K}_{P'}\) is minimally non-Golod, then \(\mathcal{K}_P\) is
also minimally non-Golod. ◻
Recall that the product in \(H^*(\mathcal{Z}_\mathcal{K})=\mathcal{H}^{*,*}(\mathcal{K})\) is given by 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 homology class in \(\widetilde{H}_{|I|-2}(\mathcal{K})\), which we continue to denote by \(\partial\Delta_I\). We denote by \(\mathop{\mathrm{MF}}_n(\mathcal{K})\) the set of missing faces \(I\) of dimension \(n\), that
is, with \(|I|=n+1\).
Lemma 4. Let \(I\in \mathop{\mathrm{MF}}_l(\mathcal{K})\) be a missing face of \(\mathcal{K}\). Then any cohomology class \(c \in \mathcal{H}^{l-1,
*}(\mathcal{K})\) such that \(\langle c,\partial\Delta_I\rangle \neq 0\) is indecomposable.
Proof.. Let \(\mathcal{K}'\) be the simplicial complex obtained from \(\mathcal{K}\) by filling in all missing faces of dimension \(l\) with
simplices, so that \(\mathop{\mathrm{MF}}_l(\mathcal{K}') = \varnothing\) and \(\mathcal{K}^{l-1}=(\mathcal{K}')^{l-1}\). Then the inclusion \(i:
\mathcal{K}\hookrightarrow \mathcal{K}'\) induces a ring homomorphism \(i^*: \mathcal{H}^{*,*}(\mathcal{K}') \rightarrow \mathcal{H}^{*,*}(\mathcal{K})\) and \(\mathcal{H}^{r,*}(\mathcal{K}') \cong \mathcal{H}^{r,*}(\mathcal{K})\) for \(r \leq l-2\). Also, \(i_*(\partial\Delta_I) = 0\) for \(I\in\mathop{\mathrm{MF}}_l(\mathcal{K})\).
Suppose \(c\) is decomposable, that is, \(c = \sum_{i=1}^p a_i \cdot b_i\). Choose \(a_i', b_i'\) such that \(i^*(a_i') = a_i\) and \(i^*(b_i') = b_i\) and define \(c' := \sum_{i=1}^p a_i' \cdot b_i'\). Then \(i^*(c') =
c\) and \[\langle c,\partial\Delta_I\rangle = \langle i^*(c'),\partial\Delta_I\rangle = \langle c',i_*(\partial\Delta_I)\rangle = 0.\] This is a contradiction. ◻
Theorem 9. Let \(\mathcal{K}\) be a three-dimensional simplicial sphere such that \(\mathcal{K}\ne\partial\Delta^4\) 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 1 . We have \(\mathcal{H}^{k,*}=0\) for \(k\geq4\) 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 [1]), 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 [1]) 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 cohomology 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 \(\lambda_i\ne0\). Since \(\mathcal{K}^1\) is chordal, \(\gamma_i \in \mathop{\mathrm{MF}}_2(\mathcal{K})\). 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 Lemma 4. 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 \(a' \in \widetilde{H}^0(\mathcal{K}_{[m] \setminus(I \cup J)})\) such that \(0 \neq a' \cdot
c^2 = a' \cdot a^0 \cdot b^1 \in \widetilde{H}^3(\mathcal{K})\). Then \(a^0 \cdot a' \neq 0\), 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 cohomology ring of a connected sum of products of 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 cohomology ring of a connected sum of products of spheres, as shown by Example 7. 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 \(I \in \mathop{\mathrm{MF}}_{l+1}(\mathcal{K})\) such that \(\langle c, \partial\Delta_I \rangle \neq
0\).
Theorem 10. Let \(\mathcal{K}\) be a simplicial sphere of dimension \(d\) such that \(\mathcal{K}\ne\partial\Delta^{d+1}\) 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 cohomology ring of a connected sum of products of spheres with two spheres in each product.
Proof.. We can assume that \(d\ge2\), as otherwise \(\mathcal{K}\) is the boundary of polygon and the result follows from Theorem 4. As in the proof of Theorem 9, we analyse possible nontrivial products in 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 \(i+j \ge d\) 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 \(I \in \mathop{\mathrm{MF}}_{i+1}(\mathcal{K})\) 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
homology groups \(\widetilde{H}_j(\mathcal{K}_J)\) are torsion-free for \(j\ge d-1-q\). Since \(d-1-q\le q\), we obtain that \(\widetilde{H}_j(\mathcal{K}_J)\) is torsion-free for \(j\ge q\), whereas \(\widetilde{H}^j(\mathcal{K}_J)\) is torsion-free for \(j\le
q\). 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 cohomology class with \(l \leq q\) is indecomposable by Lemma 4.
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 \(a' \in \widetilde{H}^{d-i-j-2}(\mathcal{K}_{[m]
\setminus(I \cup J)})\) such that \(0 \neq a' \cdot c = a' \cdot a \cdot b \in \widetilde{H}^d(\mathcal{K})\) by Poincaré duality. Therefore, \(a \cdot a' \neq 0\) 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, \[i + k = d-j-2 \le d-2-\frac{q}{2} < q.\] Thus, \(a' \cdot
a\) 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 cohomology ring of a connected sum of products of spheres with two spheres in each product. ◻
The next theorem extends the result of Theorem 9 to a complete characterisation of three-dimensional spheres \(\mathcal{K}\) such that \(H^*(\mathcal{Z}_\mathcal{K})\) is isomorphic to the cohomology ring of a connected sum of products of spheres.
Theorem 11. 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:
\(\mathcal{K}=S^0*S^0*S^0*S^0\) (the boundary of a \(4\)-dimensional cross-polytope);
\(\mathcal{K}^1\) is a chordal graph;
\(\mathcal{K}^1\) 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 Lemma 3 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 \(r\le 4\), 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 Theorem 9. 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 Theorem 9 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 cohomology ring of a connected sum of products of spheres. ◻
Remark 1. Note that under condition (c) of Theorem 11 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 [3]. Later it was shown in [9] that the corresponding
moment-angle manifold \(\mathcal{Z}_\mathcal{K}\) is diffeomorphic to \(M\).
Remark 2. 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 \(\mathcal{K}\ne\partial\Delta^4\) is minimally non-Golod if and only if \(\mathcal{K}^1\) is a chordal graph. We elaborate on this in a subsequent paper.