A moment-angle manifold whose cohomology is not torsion free

TL;DR

This paper presents a method to construct moment-angle manifolds with cohomology containing torsion, and describes the associated simplicial sphere via its non-faces, advancing understanding of their topological properties.

Contribution

It introduces a novel construction technique for moment-angle manifolds with torsion in cohomology and provides a way to describe the related simplicial sphere.

Findings

Constructed explicit examples of moment-angle manifolds with torsion cohomology

Developed a method to describe simplicial spheres by their non-faces

Enhanced understanding of the topological complexity of moment-angle manifolds

Abstract

In this paper we give a method to construct moment-angle manifolds whose cohomologies are not torsion free. We also give method to describe the corresponding simplicial sphere by its non-faces.

Equations

$$\mathbb{R}{\mathcal{Z}}_L=$$

$${\mathcal{Z}}_L=$$

$$H^{\ast }(\mathbb{R}{\mathcal{Z}}_L)$$

$$H^{\ast }({\mathcal{Z}}_L)$$

$$\begin{array}[]{cccccccccccc}L_{1}=ss_{\sigma_{1}}L_{0},&L_{2}=ss_{\sigma_{2}}L_{1},&\cdots,&L_{s}=ss_{\sigma_{s}}L_{s-1}.\end{array}$$

$$\mathbb{A}=2^I\setminus K$$

$$K=2^I\setminus \mathbb{A}$$

$$\mathbb{M}=\{\sigma\in 2^{I}|\mbox{$\sigma$ is a missing face of $K$}\}$$

$$K=\{\tau\in 2^{I}|\mbox{$\tau$ does not contain any $\sigma\in\mathbb{M}$ }\}.$$

$$\mathbb{P}=\{{\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_s\},$$

$$\mathbb{M}\subseteq \mathbb{P}\subseteq \mathbb{A}.$$

$$K_{\mathbb{P}}(I)=\{\tau\subset I|\mbox{$\tau$ does not contain any $\sigma_{i}\in\mathbb{P}$}\}$$

$$2^{I}\setminus K_{\mathbb{P}}(I)=\{\tau\subset I|\mbox{$\tau$ contains a $\sigma_{i}\in\mathbb{P}$}\}.$$

$${\mathbb{P}}^{\prime }=\{{\sigma }_1,{\sigma }_2,\cdots ,{\hat{\sigma }}_j,\cdots ,{\sigma }_s\}$$

$$\displaystyle\mathbb{P}=\left\{\begin{array}[]{lllll}\sigma_{1}=(1,2,4,5),\sigma_{2}=(1,2,3),\sigma_{3}=(3,4),\\ \sigma_{4}=(3,5),\sigma_{5}=(1,3,4),\sigma_{6}=(3,4)\end{array}\right\}$$

$$lin{k}_K\sigma =$$

$$Intsta{r}_K\sigma =\{\tau \in K|\sigma \subset \tau \}$$

$$\partial sta{r}_K\sigma =sta{r}_K\sigma \setminus Intsta{r}_K\sigma =\{\tau \in K|\sigma \cup \tau \in K,\sigma \not\subset \tau \}.$$

$$K_1\ast K_2=\{\sigma \cup \tau \in 2^{I\cup J}|\sigma \in K_1,\tau \in K_2\}.$$

$$\mathbb{P}-\sigma =\{{\sigma }_1\setminus \sigma ,{\sigma }_2\setminus \sigma ,\cdots ,{\sigma }_s\setminus \sigma \}$$

$$link_{K}\sigma=K_{\mathbb{P}-\sigma}(I\setminus\sigma)=\{\tau\subset(I\setminus\sigma)|\mbox{$\sigma$ does not contain any $\sigma_{i}\setminus\sigma\in\mathbb{P}-\sigma$}\}.$$

$$star_{K}\sigma=K_{\mathbb{P}-\sigma}(I)=\{\tau\subset I|\mbox{$\sigma$ does not contain any $\sigma_{i}\setminus\sigma\in\mathbb{P}-\sigma$}\}.$$

$$2^{I\setminus \sigma }\setminus lin{k}_K\sigma =$$

$$2^{I}\setminus star_{K}\sigma=2^{I}\setminus K_{\mathbb{P}-\sigma}(I)=\{\tau\subset I|\mbox{$\tau$ contains a $\sigma_{i}\setminus\sigma\in\mathbb{P}-\sigma$}\}.$$

$$2^{I\setminus \sigma }\setminus lin{k}_K\sigma \subseteq 2^{I\setminus \sigma }\setminus K_{\mathbb{P}-\sigma }(I\setminus \sigma ).$$

$$2^{I\setminus \sigma }\setminus K_{\mathbb{P}-\sigma }(I\setminus \sigma )\subseteq 2^{I\setminus \sigma }\setminus lin{k}_K\sigma .$$

$$2^{I}\setminus star_{K}\sigma=2^{I}\setminus K_{\mathbb{P}-\sigma}(I)=\{\tau\subset I|\mbox{$\tau$ contains a $\sigma_{i}\setminus\sigma\in\mathbb{P}-\sigma$}\}.$$

$$\mathbb{M}-(1,2)=$$

$$=$$

$$\simeq$$

$${\mathbb{P}}_1\ast {\mathbb{P}}_2=\{{\sigma }_i\cup {\tau }_j|{\sigma }_i\in {\mathbb{P}}_1,{\tau }_j\in {\mathbb{P}}_2\},$$

$$K_{1}\cup K_{2}=K_{\mathbb{P}_{1}\ast\mathbb{P}_{2}}(I)=\{\tau\subset I|\mbox{$\tau$ does not contain any $\sigma_{i}\cup\tau_{j}\in\mathbb{P}_{1}\ast\mathbb{P}_{2}$}\}.$$

Full text

This project is supported by NSFC No.11471167 and No.11761072

1. Introduction

Corresponding to every abstract simplicial complex $L$ on the vertex set $[m]=\{1,2,\cdots,m\}$, there are the real and complex moment-angle complexes $\mathbb{R}\mathcal{Z}_{L}$ and $\mathcal{Z}_{L}$ (cf. [3, 4]). They are defined as

[TABLE]

The cohomology groups of $\mathbb{R}\mathcal{Z}_{L}$ and $\mathcal{Z}_{L}$ are given by Hochster’s theorem:

Theorem [1, 3, 4] Let $L$ be a simpicial complex on the vertex set [m], then

[TABLE]

*where $L|_{I}$ is the full subcomplex of $L$ on subset $I$ and $I$ runs over all the subsets of $[m]$. *

From [5, 6, 7] we know that both $\mathbb{R}\mathcal{Z}_{L}$ and $\mathcal{Z}_{L}$ are topological manifolds if $L$ is a simplicial sphere, referred to as moment-angle manifolds. Furthermore if $L$ is a polytopal sphere (the boundary complex of a simplicial polytope), then $\mathcal{Z}_{L}$ is a transverse intersection of real quadratic hypersurfaces (cf. [2]), while both $\mathbb{R}\mathcal{Z}_{L}$ and $\mathcal{Z}_{L}$ are framed differentiable manifolds.

F.Bosio and L.Meersseman in [2] announced that the cohomology groups of differentiable complex moment-angle manifolds may have any torsion $\mathbb{Z}/m$. Furthermore if $L$ is $\mathbb{Z}/2$ colourable, L.Cai, S.Choi and H. Park in [8, 9] proved that the small cover under $\mathbb{R}\mathcal{Z}_{L}$ may have any torsion $\mathbb{Z}/m$.

From Hochster’s theorem, it is easy to construct a moment-angle complex whose cohomology has torsion. But it is harder to construct such moment-angle manifolds (at least, the cohomology of all the moment-angle manifolds corresponding to dimensional 1, 2 and 3 simplicial spheres are torsion free (cf. [2] Corollary 11.1).

Based on Hochster’s theorem, our goal is to find a simplicial complex $K$ whose cohomology has torsion and $K$ is embedded in a polytopal sphere $L$ as a full subcomplex. Then both the real and complex moment-angle complexes corresponding to $L$ are differentiable manifolds and the cohomology of $\mathbb{R}\mathcal{Z}_{L}$ and $\mathcal{Z}_{L}$ have $\widetilde{H}^{*}(K)$ as a summand and then have torsion.

Theorem 3.2 (Construction) Let $K$ be a subcomplex (not a full subcomplex) of a simplicial sphere $L_{0}$ on the vertex set $[m]$, $\mathbb{M}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ be the set of missing faces of $K$, which are also simplices of $L_{0}$. As following, make stellar subdivisions at $\sigma_{1},\sigma_{2},\cdots,\sigma_{s}$ on $L_{0}$ one by one

[TABLE]

Then $K$ becomes a full subcomplex of $L_{s}$, $K=L_{s}|_{[m]}$.

In fact after making stellar subdivision on a polytopal (simplicial) sphere, it is still polytopal (simplicial) (see [10]). If $L_{0}$ is also a polytopal sphere, we thus obtain a polytopal sphere $L_{s}$ by Theorem $3.2$ such that $K$ is a full subcomplex of $L_{s}$. The real and complex moment-angle complexes corresponding to $L_{s}$ are differentable manifolds. By Hochster’s theorem both $H^{*}(\mathbb{R}\mathcal{Z}_{L_{s}})$ and $H^{*}(\mathcal{Z}_{L_{s}})$ have torsion if $\widetilde{H}^{*}(K)$ has torsion.

At last in section 4, we give a differentiable moment-angle manifold whose cohomology has $\mathbb{Z}/3$ as a summand. This is done as follows:

Triangulate the $mod$ $3$ Moore space $K$ which has $8$ vertices, $17$ $2$-dimensional facets and $22$ missing faces (see figure 3). It can be embedded in $\partial\Delta^{7}=L_{0}$. After making $22$ stellar subdivisions on it, $K$ becomes a full subcomplex of the polytopal sphere $L_{22}$. Then $L_{22}$ is a $6$ dimensional polytopal sphere with $30$ vertices. $\mathcal{Z}_{L_{22}}$ is a $37$ dimensional differentiable manifold and $H^{11}(\mathcal{Z}_{L_{22}})$ has $\widetilde{H}^{2}(K)=\mathbb{Z}/3$ as a summand.

It is notable that F.Bosio and L.Meersseman’s construction in [2] (Theorem $11.12$) applied to the same example does not give a moment-angle manifold whose cohomology has $\mathbb{Z}/3$ torsion.

The authors are grateful to professor Zhi. L$\ddot{u}$ for his helpful suggestion during this research. This work was done under the supervision of professor Xiangjun. Wang.

2. Simplicial Complement

An abstract simplicial complex $K$ on the vertex set $I$ is a collection of simplices that satisfies: for any simplex (face) $\sigma\in K$, all of its proper subsets (proper faces) are simplices of $K$.

An abstract simplicial complex $K$ could also be given by all of its non-faces

[TABLE]

and

[TABLE]

that satisfies: if $\sigma\in\mathbb{A}$ is not a simplex of $K$ and $\sigma^{\prime}\supset\sigma$ then $\sigma^{\prime}\in\mathbb{A}$ is not a simplex of $K$.

A simplex $\sigma=(i_{1},i_{2},\cdots,i_{k})\in 2^{I}$ is called a missing face (or minimal non-face) of $K$ if it is not a face of $K$, but all of its proper subsets are faces of $K$, i.e. $\sigma\notin K$ but every $\sigma_{j}=(i_{1},\cdots,\widehat{i_{j}},\cdots,i_{k})\in K$ , $j=1,2,\cdots,k$. An abstract simplicial complex could also be given by its set of missing faces

[TABLE]

and

[TABLE]

A subset $\sigma^{\prime}$ of $I$ is not a simplex of $K$ if and only if it contains a missing face $\sigma\in\mathbb{M}$ as a subset.

Definition 2.1 Let $K$ be a simplicial complex on the vertex set $I$ and $\mathbb{M}$, $\mathbb{A}$ be the sets of missing faces and non-faces of $K$ respectively. We define a simplicial complement of $K$, denoted by

[TABLE]

to be a collection of non-faces that includes all the missing faces $\mathbb{M}$ i.e.

[TABLE]

Similar to the set of missing faces $\mathbb{M}$, given a simplicial complement $\mathbb{P}$ (collection of non-faces) on the vertex set $I$, one can obtain a simplicial complex $K_{\mathbb{P}}$ on $I$ by:

[TABLE]

or by all of its non-faces

[TABLE]

A subset $\sigma$ of $I$ is not a simplex of $K_{\mathbb{P}}(I)$ if and only if it contains a non-face $\sigma_{i}$ in the simplicial complement $\mathbb{P}$.

Definition 2.4 Let $\mathbb{P},\mathbb{P}^{\prime}$ be two simplicial complements on the vertex set $I$, if they can obtain the same simplicial complex i.e. $K_{\mathbb{P}}(I)=K_{\mathbb{P}^{\prime}}(I)$, we say that $\mathbb{P}$ and $\mathbb{P}^{\prime}$ are equivalent, denoted by $\mathbb{P}\simeq\mathbb{P}^{\prime}$.

It is easy to see that: Two simplicial complements $\mathbb{P}$, $\mathbb{P}^{{}^{\prime}}$ on $I$ are equivalent if and only if for every non-face $\sigma\in\mathbb{P}$ there exists a $\sigma^{\prime}\in\mathbb{P}^{\prime}$ such that $\sigma^{\prime}\subseteq\sigma$ and for every non-face $\sigma^{\prime}\in\mathbb{P}^{\prime}$ there exists a $\sigma\in\mathbb{P}$ such that $\sigma\subseteq\sigma^{\prime}$.

Proposition 2.5 Let $\mathbb{P}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ be a simplicial complement of $K$ on $I$. For a non-face $\sigma_{j}\in\mathbb{P}$ if there exists a $\sigma_{i}\in\mathbb{P},i\neq j$ such that $\sigma_{i}\subseteq\sigma_{j}$, then we can remove $\sigma_{j}$ from $\mathbb{P}$ and the resulting simplicial complement

[TABLE]

is equivalent to $\mathbb{P}$. In this case we call that $\mathbb{P}$ is reduced to $\mathbb{P}^{\prime}$.

Every simplicial complement of $K$ could be reduced to the set of missing faces by removing all the larger non-faces.

Example 1 The simplicial complex $K$ is determined by the maximal simplices $(1,3)$, $(2,3)$, $(1,2,4)$, $(1,2,5)$, $(1,4,5)$, $(2,4,5)$ and their proper subsets on the vertex set $[5]=\{1,2,3,4,5\}$ (see Figure 1)

1$$2$$3$$4$$5

Figure 1

[TABLE]

is a simplicial complement of $K$ on vertex set $[5]$ where $\sigma_{3}=(3,4)=\sigma_{6}$ appeared twice and $\sigma_{3}=(3,4)\subset\sigma_{5}=(1,3,4)$. So $\sigma_{5}=(1,3,4)$ and $\sigma_{6}=(3,4)$ could be removed from $\mathbb{P}$ to reduce to the set of missing faces $\mathbb{M}=\{(1,2,4,5),(1,2,3),(3,4),(3,5)\}$.

The readers should be aware that the empty simplex $\{\emptyset\}$ (only the empty set is a simplex) is different from the empty complex $\emptyset$ (the empty set is not a simplex of $\emptyset$). $\mathbb{M}=\{(1),(2),\cdots,(m)\}$ is the set of missing faces of the empty simplex $\{\emptyset\}$ while $\mathbb{M}_{1}=\{\emptyset\}$ is the set of missing faces of the empty the empty complex $\emptyset$.

Let $K$ be a simplicial complex on the vertex set $I$ and $\sigma$ be a simplex of $K$. The $link$ and $star$ of $\sigma$ are defined to be the simplicial complexes

[TABLE]

The interior (open) $star$ is defined to be a subset of $2^{I}$ (cf. [12] §62 p.371)

[TABLE]

and the boundary of $star$ is the simplicial complex

[TABLE]

Let $K_{1}$ and $K_{2}$ be two simplicial complexes on vertex set $I$ and $J$, where $I\cap J=\emptyset$. The join of $K_{1}$ and $K_{2}$ is defined to be the simplicial complex on vertex set $I\cup J$

[TABLE]

Let $\mathbb{P}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ be a simplicial complement of $K$ on the vertex set $I$ and $\sigma\in K$ be a simplex. We difine

[TABLE]

which is a sequence of subsets on $I\setminus\sigma$.

Lemma 2.6 Let $\mathbb{P}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ be a simplicial complement of $K$ on the vertex set $I$. Then

1. (1)

$\mathbb{P}-\sigma=\{\sigma_{1}\setminus\sigma,\sigma_{2}\setminus\sigma,\cdots,\sigma_{s}\setminus\sigma\}$* is a simplicial complement of $link_{K}\sigma$ on the vertex set $I\setminus\sigma$, i.e. by $(2.2)$*

[TABLE] 2. (2)

If we consider $\mathbb{P}-\sigma$ as a sequence of non-faces on the vertex set $I$, then it is a simplicial complement of $star_{K}\sigma$ on I, i.e. by $(2.2)$

[TABLE]

Proof: We prove this Lemma by showing that they have the same non-faces

[TABLE]

and

[TABLE]

1. (1)

From its definition, we know that a simplex $\tau$ on the vertex set $I\setminus\sigma$ is not a simplex of $link_{K}\sigma$ if and only if $\sigma\cup\tau$ is not a simplex of $K$. In other words, there exists a $\sigma_{i}\in\mathbb{P}$ such that $\sigma_{i}\subseteq\tau\cup\sigma$. This is equivalent to say that $\sigma_{i}\setminus\sigma\subseteq\tau\cup\sigma\setminus\sigma=\tau$, every non-face $\tau\in 2^{I\setminus\sigma}\setminus link_{K}\sigma$ is a non-face of $K_{\mathbb{P}-\sigma}(I\setminus\sigma)$, i.e. $\tau\in 2^{I\setminus\sigma}\setminus K_{\mathbb{P}-\sigma}(I\setminus\sigma)$, so

[TABLE] 2. (2)

If a simplex $\tau$ on the vertex set $I\setminus\sigma$ contains a $\sigma_{i}\setminus\sigma$, then $\tau\cup\sigma\supseteq(\sigma_{i}\setminus\sigma)\cup\sigma\supseteq\sigma_{i}$, so such $\tau$ is not a simplex of $link_{K}\sigma$. This is equivalent to say that every non-face $\tau\in 2^{I\setminus\sigma}\setminus K_{\mathbb{P}-\sigma}(I\setminus\sigma)$ is a non-face of $link_{K}\sigma$, i.e. $\tau\in 2^{I\setminus\sigma}\setminus link_{K}\sigma$, so

[TABLE]

Thus $\mathbb{P}-\sigma=\{\sigma_{1}\setminus\sigma,\sigma_{2}\setminus\sigma,\cdots,\sigma_{s}\setminus\sigma\}$ is a simplicial complement of $link_{K}\sigma$ on the vertex set $I\setminus\sigma$.

Similarly, if we consider $\mathbb{P}-\sigma$ as a simplicial complement on the vertex set $I$, then

[TABLE]

The Lemma follows. $\Box$

Example 2 In Example 1, the $link$ of the simplex $(1,2)$ is two vertices $link_{K}(1,2)=\{(4),(5)\}$ and $star_{K}(1,2)$ is composed by two $2$-simplices $(1,2,4)$, $(1,2,5)$ and its proper subsets.

[TABLE]

is a simplicial complement of $link_{K}(1,2)$ on the vertex set $\{3,4,5\}$. Consider $\mathbb{M}-\sigma$ as a sequence of non-faces on the vertex set $[5]=\{1,2,3,4,5\}$, it becomes the simplicial complement of $star_{K}(1,2)$.

Let $\mathbb{P}_{1}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ and $\mathbb{P}_{2}=\{\tau_{1},\tau_{2},\cdots,\tau_{t}\}$ be the simplicial complements of $K_{1}$ and $K_{2}$ on the vertex set $I$. We define their join $\mathbb{P}_{1}\ast\mathbb{P}_{2}$ to be

[TABLE]

which is a sequence of subsets on $I$.

Lemma 2.7 Let $K_{1}$ and $K_{2}$ be two simplicial complexes on the vertex set $I$, $\mathbb{P}_{1}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ and $\mathbb{P}_{2}=\{\tau_{1},\tau_{2},\cdots,\tau_{t}\}$ be simplicial complements of $K_{1}$ and $K_{2}$ respectively. Then $\mathbb{P}_{1}\ast\mathbb{P}_{2}=\{\sigma_{i}\cup\tau_{j}|\sigma_{i}\in\mathbb{P}_{1},\tau_{j}\in\mathbb{P}_{2}\}$ is a simplicial complement of $K_{1}\cup K_{2}$ on the vertex set $I$,

[TABLE]

Proof: We prove this Lemma in the same way as the proof of Lemma $2.6$.

1. (1)

It is easy to see that a simplex $\tau$ on the vertex set $I$ is not a simplex of $K_{1}\cup K_{2}$ if and only if it is not a simplex of either $K_{1}$ or $K_{2}$. This implies that there exists a $\sigma_{i}\in\mathbb{P}_{1}$ such that $\sigma_{i}\subseteq\tau$ and also exists a $\tau_{j}\in\mathbb{P}_{2}$ such that $\tau_{j}\subseteq\tau$. This is equivalent to say that $\sigma_{i}\cup\tau_{j}\subseteq\tau$, every non-face $\tau$ of $K_{1}\cup K_{2}$ contains a $\sigma_{i}\cup\tau_{j}\in\mathbb{P}_{1}\ast\mathbb{P}_{2}$, so

[TABLE] 2. (2)

If a simplex $\tau$ on $I$ contains a non-face $\sigma_{i}\cup\tau_{j}\in\mathbb{P}_{1}\ast\mathbb{P}_{2}$, then $\sigma_{i}\subseteq\tau$ and $\tau_{j}\subseteq\tau$. This $\tau$ is neither a simplex of $K_{1}$ nor a simplex of $K_{2}$, so

[TABLE]

The Lemma follows. $\Box$

Corollary 2.8 If the simplicial complement $\mathbb{P}$ is equivalent to $\mathbb{P}^{\prime}$, then for any simplex $\sigma$ and simplicial complement $\mathbb{P}_{2}$

[TABLE]

$\Box$

Let $\sigma$ be a simplex of a simplicial complex $K$ on $[m]$. The stellar subdivision at $\sigma$ on $K$ is defined to be the union of the simplicial complexes $K\setminus Intstar_{K}\sigma$ and the cone $cone\partial star_{K}\sigma$ along their boundary $\partial star_{K}\sigma$, denoted by

[TABLE]

where

[TABLE]

After stellar subdivision, one more vertex is added which is the vertex of the cone.(cf. [2])

In [4] (Definition $2.7.1$), the stellar subdivision is defined to be

[TABLE]

where $K\setminus star_{K}\sigma$ is not a simplicial complex and $\sigma$ can be any simplex of $K$. Note that

[TABLE]

and

[TABLE]

so our definition coincides with that in [4].

Recall that $K\setminus Intstar_{K}\sigma=\{\tau\in K|\sigma\not\subset\tau\}$ is a simplicial complex because of for every $\tau^{\prime}\subset\tau$ and $\sigma\not\subset\tau$, one has $\sigma\not\subset\tau^{\prime}$, then as $\tau\in K$ is a simplex of $K\setminus Intstar_{K}\sigma$ and $K$ is a simplicial complex, $\tau^{\prime}\subset\tau$ is a simplex of $K\setminus Intstar_{K}\sigma$.

Theorem 2.9 Let $\mathbb{P}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ be a simplicial complement of $K$. Then $\{\mathbb{P},\sigma,(\mathbb{P}-\sigma)\ast(m+1)\}$ is a simplicial complement of $ss_{\sigma}K$ on the vertex set $[m+1]$, where

[TABLE]

Proof: First we prove that $\{\sigma,\mathbb{P}-\sigma\}$ is a simplicial complement of $cone\partial star_{K}\sigma$ on the vertex set $[m+1]=\{1,2,\cdots,m,m+1\}$.

From Lemma $2.6$ we know that $\mathbb{P}-\sigma$ is a simplicial complement of $star_{K}\sigma$ on the vertex set $[m]$.

A simplex $\tau$ on vertex set $[m]$ is not a simplex of $\partial star_{K}\sigma=star_{K}\sigma\setminus Intstar_{K}\sigma$ if and only if $\tau\in Intstar_{K}\sigma$ or $\tau\notin star_{K}\sigma$, i.e. $\sigma\subset\tau$ or there exists a $\sigma_{i}\setminus\sigma$ such that $\sigma_{i}\setminus\sigma\subset\tau$, so

[TABLE]

$\{\sigma,\mathbb{P}-\sigma\}$ is a simplicial complement of $\partial star_{K}\sigma$ on the vertex set $[m]$.

Take the cone of $\partial star_{K}\sigma$ on the vertex set $[m+1]$, a simplex $\tau\subset[m]$ or $(\tau,m+1)\subset[m+1]$ is not a simplex of $cone\partial star_{K}\sigma=(m+1)\ast\partial star_{K}\sigma$ if and only if $\tau$ is not a simplex of $\partial star_{K}\sigma$, i.e.

[TABLE]

$\{\sigma,\mathbb{P}-\sigma\}$ is a simplicial complement of $cone\partial star_{K}\sigma$ on the vertex set $[m+1]$.

Second, we prove that $\{\mathbb{P},\sigma,(m+1)\}$ is a simplicial complement of $K\setminus Intstar_{K}\sigma$ on the vertex set $[m+1]$.

A simplex $\tau$ on the vertex set $[m]$ is not a simplex of $K\setminus Intstar_{K}\sigma$ if and only if $\tau\notin K$ or $\tau\in Intstar_{K}\sigma$, i.e. there exists a $\sigma_{i}\in\mathbb{P}$ such that $\sigma_{i}\subset\tau$ or $\sigma\subset\tau$. $\{\mathbb{P},\sigma\}$ is a simplicial complement of $K\setminus Intstar_{K}\sigma$ on the vertex set $[m]$.

Consider $K\setminus Intstar_{K}\sigma$ as a simplicial complex on the vertex set $[m+1]$, $(m+1)$ does not appear in $K\setminus Intstar_{K}\sigma$. It is a ghost vertex and $(m+1)$ is a missing face. So

[TABLE]

is a simplicial complement of $K\setminus Intstar_{K}\sigma$ on the vertex set $[m+1]$.

From Lemma $2.7$, we know that $\{\mathbb{P},\sigma,(m+1)\}\ast\{\sigma,\mathbb{P}-\sigma\}$ is a simplicial complement of $ss_{\sigma}K=(K\setminus Intstar_{K}\sigma)\cup(cone\partial star_{K}\sigma)$, where

[TABLE]

At last we complete the proof by showing that the simplicial complement $\{\mathbb{P},\sigma,(m+1)\}\ast\{\sigma,\mathbb{P}-\sigma\}$ is equivalent to $\{\mathbb{P},\sigma,\{\mathbb{P}-\sigma\}\ast(m+1)\}$, i.e.

[TABLE]

First,

[TABLE]

Every subset $\sigma_{i}\cup\sigma\in\mathbb{P}\ast\sigma$, $(\sigma,m+1)\in(m+1)\ast\sigma$ and $\sigma\cup(\sigma_{i}\setminus\sigma)\in\sigma\ast\{\mathbb{P}-\sigma\}$ contain $\sigma$. They could be removed from $\{\mathbb{P},\sigma,(m+1)\}\ast\{\sigma,\mathbb{P}-\sigma\}$, so

[TABLE]

Then for any $\sigma_{i}\in\mathbb{P}$, one has $\sigma_{i}\setminus\sigma\in\mathbb{P}-\sigma$. So

[TABLE]

Any other $\sigma_{i}\cup(\sigma_{j}\setminus\sigma)\in\mathbb{P}\ast\{\mathbb{P}-\sigma\}$ contains $\sigma_{i}$, they could be removed from $\mathbb{P}\ast\{\mathbb{P}-\sigma\}$. Thus $\mathbb{P}\ast\{\mathbb{P}-\sigma\}$ is equivalent to $\mathbb{P}$ and $\{\mathbb{P},\sigma,(m+1)\}\ast\{\sigma,\mathbb{P}-\sigma\}$ could be reduced to

[TABLE]

The Theorem follows. $\Box$

Remark: If $\sigma$ is not a simplex of $K$, we still have $\{\mathbb{P},\sigma,\{\mathbb{P}-\sigma\}\ast(m+1)\}$ as a simplicial complement of a simplicial complex $ss_{\sigma}K$. In that case, there exists a $\sigma_{i}\in\mathbb{P}$ such that $\sigma_{i}\subseteq\sigma$. So $\sigma$ could be removed from $\{\mathbb{P},\sigma,\{\mathbb{P}-\sigma\}\ast(m+1)\}$ and $\sigma_{i}\setminus\sigma=\emptyset\in\mathbb{P}-\sigma$. Thus $(\sigma_{i}\setminus\sigma,m+1)=(m+1)\in\{\mathbb{P}-\sigma\}\ast(m+1)$ and all the other $(\sigma_{j}\setminus\sigma,m+1)$ could be removed from $\{\mathbb{P},\sigma,\{\mathbb{P}-\sigma\}\ast(m+1)\}$. That is to say that $(m+1)$ is a missing face and

[TABLE]

is still a simplicial complement of $ss_{\sigma}K=K$ but on the vertex set $[m+1]$ and a ghost vertex $(m+1)$ is added.

We still call it the stellar subdivision at $\sigma$ on $K$. $\Box$

Example 3 In Example 1, we make stellar subdivision at $\sigma=(1,2)$ on $K$ (see Figure 2)

1$$2$$3$$4$$5$$\stackrel{{\scriptstyle ss_{\sigma}K}}{{\Longrightarrow}}\dashline3(50,-10)(86,0) 1$$2$$3$$4$$5$$6

Figure 2

$\mathbb{P}=\{(1,2,4,5),(1,2,3),(3,4),(3,5)\}$ is a simplicial complement of $K$, $\sigma=(1,2)$, so

[TABLE]

[TABLE]

is a simplicial complement of $ss_{\sigma}K$.

3. Construction

After given the simplicial complement of stellar subdivision, we construct our moment-angle manifolds whose cohomology has torsion.

Lemma 3.1 Let $K$ be a simplicial complex on the vertex set $[m]$ and

[TABLE]

be a simplicial complement of it. Let $I$ be a subset of the vertex set $[m]$. Then

[TABLE]

is a simplicial complement of the full subcomplex $K|_{I}$ on the vertex set I.

Proof: From its definition, we know that the full subcomplex

[TABLE]

is a simplicial complex on the vertex set $I$. A subset $\tau$ on the vertex set $I$ is not a simplex of $K|_{I}$ if and only if $\tau$ is not a simplex of $K$. i.e. there exists a non-face $\sigma_{i}\in\mathbb{P}$ such that $\tau_{i}\subset\tau$. Note that $\tau\subset I$, $\tau_{i}\subset\tau\subset I$. The Lemma follows. $\Box$

Theorem 3.2 (Construction) Let $K$ be a subcomplex (not a full subcomplex) of a simplicial sphere $L_{0}$ on the vertex set $[m]$, $\mathbb{M}=\{\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$ be the set of missing faces of $K$, which are also simplices of $L_{0}$. As following, make stellar subdivisions at $\sigma_{1},\sigma_{2},\cdots,\sigma_{s}$ on $L_{0}$ one by one

[TABLE]

Then $K$ becomes a full subcomplex of $L_{s}$, $K=L_{s}|_{[m]}$.

Proof: Let $\mathbb{P}_{0}=\{\tau_{1},\tau_{2},\cdots,\tau_{r}\}$ be a simplicial complement of $L_{0}$ on $[m]$. From Theorem $2.9$ we know that

[TABLE]

is a simplicial complement of $L_{1}=ss_{\sigma_{1}}L_{0}$ on $[m+1]$, where

[TABLE]

By induction, we get a simplicial complement of $L_{s}=ss_{\sigma_{s}}L_{s-1}$ on $[m+s]$ as

[TABLE]

where

[TABLE]

Note that every non-face in $\mathbb{P}_{i}^{\prime}$ contains $(m+i)$ as a vertex. From Lemma $3.1$ we know that

[TABLE]

is a simplicial complement of the full subcomplex $L_{s}|_{[m]}$.

Finally, we consider the simplicial complement $\mathbb{P}_{s}|_{[m]}$. Note that $K$ is a subcomplex of $L_{0}$, every non-face $\tau_{i}\in\mathbb{P}_{0}$ is not a simplex of $K$, so there exists a $\sigma_{j}\in\mathbb{M}$ such that $\sigma_{j}\subseteq\tau_{i}$. Then $\tau_{i}$ could be removed from $\{\mathbb{P}_{0},\sigma_{1},\sigma_{2},\cdots,\sigma_{s}\}$.

Thus

[TABLE]

which is the set of missing faces of $K$. The theorem follows. $\Box$

Remark: If $L_{0}$ is also a polytopal sphere, the stellar subdivision of $L_{0}$ is also polytopal. It has been proved in a geometric sense by G. Ewald and G. C. Shephard in [10].

Let $\widetilde{L}_{0}$ be the simplicial polytope and its boundary $\partial\widetilde{L}_{0}=L_{0}$ be the polytopal sphere. If $\sigma$ is a simplex of $L_{0}$ and $\sigma$ is the intersection of the facets (maximal simplices of $L_{0}$) $F_{i_{1}},F_{i_{2}},\cdots,F_{i_{r}}$, one can take any point ${p}$ beyond the facets $F_{i_{1}},F_{i_{2}},\cdots,F_{i_{r}}$ and beneath the other facets (See [11] p.78 for the definitions of beyond and beneath ). The stellar subdivision $ss_{\sigma}\partial\widetilde{L}_{0}$ is the boundary of the convex hall of $\widetilde{L}^{\prime}_{0}=conv(\widetilde{L}_{0}\cup{p})$.

It could also be proved from the duality of polytopes.

Let $\widetilde{L}_{0}$ be the simplicial polytope corresponding to $L_{0}$, and $P_{0}$ be the dual simple polytope, (the vertex of $L_{0}$ corresponding to the facet while the facet of $L_{0}$ corresponding to the vertex of $P_{0}$). Let $\sigma=(i_{1},i_{2},\cdots,i_{k})$ be a simplex of $L_{0}$, make a stellar subdivision at $\sigma$ on $L_{0}$ is equivalent, though the duality of polytopes, to cutting off the face $\sigma^{*}=F_{i_{1}}\cap F_{i_{2}}\cap\cdots\cap F_{i_{k}}$ in $P_{0}$ by a generic hyperplane. The cutting off operation on a simple polytope is still simple, so $ss_{\sigma}\partial\widetilde{L}_{0}$ is polytopal. $\Box$

4. Application

Proposition 4.1 The cohomology of differentiable moment-angle manifolds may have torsion of any order.

Proof: Let $L_{0}$ be a polytopal sphere and $K$ be a subcomplex of $L_{0}$, whose cohomology has torsion. Construct a new polytopal sphere $L_{s}$ by Theorem $3.2$, then $K$ becomes a full subcomplex of $L_{s}$, while both $\mathbb{R}\mathcal{Z}_{L_{s}}$ and $\mathcal{Z}_{L_{s}}$ are framed differentiable manifolds. From Hochster’s Theorem, the cohomology of $\mathbb{R}\mathcal{Z}_{L_{s}}$ and $\mathcal{Z}_{L_{s}}$ has $\widetilde{H}^{*}(K)$ as a summand and then have torsion.

At least, every simplicial complex $K$ with $m$ vertexes is a subcomplex of the polytopal sphere $\partial\Delta^{m-1}$. So the cohomology of differentiable moment-angle manifolds could have any torsion. $\Box$

Here is a example.

Example 4 Let $K$ be the triangulated $mod$ $3$ Moore space (see Figure 3) which can be embedded in $6$-dimensional polytopal sphere

[TABLE]

1$$1$$2$$3$$2$$3$$1$$2$$3$$4$$5$$6$$7$$8

Figure 3

The set of missing faces of $L_{0}$ is

[TABLE]

The set of missing faces of $K$ is

[TABLE]

and the set of maximal simplices of K is

[TABLE]

Making $22$ stellar subdivisions at missing faces of $K$ on $\partial\Delta^{7}$, we thus obtain a $6$-dimensional polytopal sphere $L_{22}$ with $30$ vertices which has $K$ as a full subcomplex. The real moment-angle manifold corresponding to $L_{22}$ is of $6$-dimensional while the complex one is of $37$-dimensional where $H^{3}(\mathbb{R}\mathcal{Z}_{L_{22}})$ and $H^{11}(\mathcal{Z}_{L_{22}})$ has $\widetilde{H}^{2}(K)=\mathbb{Z}/3$ as a summand.

Passing to the dual, $\Delta^{7}$ is the dual simple polytope of $\partial\Delta^{7}$ with facets numbered as vertexes of $\partial\Delta^{7}$. Making stellar subdivision on $\partial\Delta^{7}$ at $\sigma=(i_{1},i_{2},\cdots,i_{r})$ is dual to cutting off face $F_{i_{1}}\cap F_{i_{2}}\cap\cdots\cap F_{i_{r}}$ in $\Delta^{7}$,

[TABLE]

After cutting off the faces $F_{i_{1}}\cap F_{i_{2}}\cap\cdots\cap F_{i_{r}}$ numbered at $\mathbb{M}$ in (4.2), one get a simple polytope $P_{22}$. The cohomology of the moment-angle manifold corresponding to $P_{22}$ has $H^{2}(K)=\mathbb{Z}/3$ as a summand and then has torsion. If we only cut off $\{1,2,\cdots,8\}\setminus\sigma$ for every maximal simplex $\sigma$ of $K$ in $\Delta^{7}$ as F.Bosio and L.Meersseman did in [2] (Theorem $11.12$), we do not get torsion.

Compute the missing faces after making stellar subdivision at $(1,2,3)$ and $(3,7)$ on $\partial\Delta^{7}$ in different sequence, one has

1. (1)

We make stellar subdivision at $(1,2,3)$ on $L_{0}=\partial\Delta^{7}$ at first, then make stellar subdivision at $(3,7)$.

From Theorem $2.10$ we know that,

[TABLE]

is a simplicial complement of $L_{1}=ss_{(1,2,3)}L_{0}$. After removing the larger non-faces $(1,2,3,4,5,6,7,8)$, we get the set of missing faces of $L_{1}$

[TABLE]

Then we make stellar subdivision at $(3,7)$ on $L_{1}$ and get the set of missing faces of $L_{2}=ss_{(3,7)}L_{1}$

[TABLE] 2. (2)

Similarly we make stellar subdivision at $(3,7)$ on $L_{0}$ at first, then make stellar subdivision at $(1,2,3)$, the resulting set of missing faces of $L_{2}^{\prime}$ is

[TABLE]

It is easy to see that two simplicial complexes $K$ and $K^{\prime}$ on vertex set $I$ are combinatorial equivalent if and only if their sets of missing faces $\mathbb{M}$ and $\mathbb{M}^{\prime}$ are equivalent, i.e. there exists a one to one correspondence $\phi:I\rightarrow I$ that gives a one to one correspondence between $\mathbb{M}$ and $\mathbb{M}^{\prime}$.

Comparing with these two sequence, we can find that $L_{2}$ has one 2-vertex missing faces $(3,7)$ while $L_{2}^{\prime}$ has two $(3,7),\ (7,10)$. This implies that $L_{2}$ is not combinatorially isomorphic to $L_{2}^{\prime}$ and this difference might persist during the later stellar subdivisions.

Remark: Though $K$ will be a full subcomplex of $L_{s}$ in every sequence of making stellar subdivisions at $K$’s missing faces, the combinatorial structure of $L_{s}$ may not be combinatorially isomorphic in different sequences. $\Box$