On the Cohomology Ring of Real Moment-Angle Complexes

TL;DR

This paper investigates the cohomology ring structure of real moment-angle complexes derived from simplicial complexes, providing explicit descriptions for boundary of polygons and highlighting differences from complex cases.

Contribution

It offers a detailed combinatorial description of the cohomology ring for real moment-angle complexes, especially for polygon boundaries, revealing new structural insights.

Findings

Cohomology generators are combinatorially described.

Full multiplicative structure is characterized for polygon boundaries.

Generators do not form a symplectic basis, unlike complex cases.

Abstract

In this article, we study the cohomology ring of real moment-angle complexes over a simplicial complex $K$. Combinatorial generators for the cohomology can be given in terms of $K$. For $K$ the boundary of an $n$-gon, we give a full description of the multiplicative structure of the cohomology ring in terms of the combinatorial generators. As a consequence, it is evident that these generators do not form a symplectic basis, unlike the case for moment-angle complexes.

Equations

$$K_I:=\{\sigma \in K|\sigma \subset I\}.$$

$$D(\sigma )=\prod _{i\in [m]}{Y}_i\text{ where }{Y}_i=\{\begin{array}{ll}{X}_i& i\in \sigma \\ A_i& i\in [m]\backslash \sigma \end{array}$$

$$Z_K(\underline{X},\underline{A}):=\underset{\sigma \in K}{\mathrm{colim}}D(\sigma )=\bigcup _{\sigma \in K}D(\sigma )$$

$$\begin{array}[]{ccl}Z_{K}(D^{1},S^{0})&=&D(\{1,2\})\cup D(\{1,3\})\cup D(\{2,3\})\\ &=&D^{1}\times D^{1}\times S^{0}\cup D^{1}\times S^{0}\times D^{1}\cup S^{0}\times D^{1}\times D^{1}\\ &=&\partial(D^{1}\times D^{1}\times D^{1})\\ &\cong&S^{2}\end{array}$$

$$\hat{D}(\sigma )=\wedge Y_i\text{ where }{Y}_i=\{\begin{array}{ll}{X}_i& i\in \sigma \\ A_i& i\notin \sigma \end{array}$$

$${\hat{Z}}_K(\underline{X},\underline{A})=\bigcup \hat{D}(\sigma )$$

$$\Sigma Z_K(\underline{X},\underline{A})\simeq \Sigma \underset{I\subset [m]}{\bigvee }{\hat{Z}}_{K_I}(\underline{X_I},\underline{A_I}))$$

$${\hat{Z}}_K(\underline{X},\underline{A})\simeq \Sigma |K|\wedge A^{\wedge [m]}\simeq |K|\ast A^{\wedge [m]}$$

$$H^{\ast }(Z_K(D^1,S^0))=\underset{I\subset [m]}{⨁}{H}^{\ast }(\Sigma {\mathcal{K}}_{\mathcal{I}})$$

$$\mathbb{Z}[K]:=\mathbb{Z}[m]/⟨v_{i_1}{v}_{i_2}\dots v_{i_k}|\{i_1,i_2,\dots ,i_k\}\notin K⟩$$

$$H^{\ast }(Z_K(D^2,S^1))\cong {\text{Tor}}_{\mathbb{Z}[m]}(\mathbb{Z}[K],\mathbb{Z})$$

$$H^k(Z_K(D^2,S^1))\cong \underset{J\subset [m]}{⨁}{\tilde{H}}^{k-|J|-1}(K_J)$$

$$\alpha ⌣\beta =\{\begin{array}{ll}\gamma & \text{ if }J\cap L=\varnothing \\ 0& \text{ if }J\cap L\ne \varnothing \end{array}$$

$$\dots \stackrel{\delta }{\to }{\tilde{H}}^{\ast }(X_i/A_i)\stackrel{g}{\to }{H}^{\ast }(X_i)\stackrel{f}{\to }{H}^{\ast }(A_i)\stackrel{\delta }{\to }{\tilde{H}}^{\ast +1}(X_i/A_i)\stackrel{g}{\to }\dots$$

$$\begin{array}[]{ccccc}H^{*}(A_{i})&=&B_{i}\oplus E_{i}\\ H^{*}(X_{i})&=&B_{i}\oplus C_{i}\\ \widetilde{H}^{*}(X_{i}/A_{i})&=&C_{i}\oplus W_{i}\\ \end{array}$$

$$Y^{I,\sigma }:=\underset{i\in \sigma }{⨂}{C}_i\otimes \underset{i\in I-\sigma }{⨂}{B}_i$$

$$E_r^{s,t}\to H^{\ast }(Z_K(\underline{X},\underline{A}))$$

$${\hat{E}}_r^{s,t}\to H^{\ast }({\hat{Z}}_K(X,A))$$

$$Z_K(\underline{X},\underline{A})\to {\hat{Z}}_K(\underline{X},\underline{A})$$

$$\begin{array}[]{ccccc}H^{*}(A_{i})&=&B_{i}\oplus E_{i}\\ H^{*}(X_{i})&=&B_{i}\oplus C_{i}\\ \end{array}$$

$$H^{\ast }(Z_K(\underline{X},\underline{A}))=\underset{I\subset [m],\sigma \subset I}{⨁}{E}^{I^c}\otimes Y^{I,\sigma }\otimes {\tilde{H}}^{\ast }(\Sigma |lk(\sigma {)}_{I^c}|)$$

$$\begin{array}[]{ccccc}\widetilde{H}^{*}(A_{i})&=&\widetilde{B}_{i}\oplus E_{i}\\ \widetilde{H}^{*}(X_{i})&=&\widetilde{B}_{i}\oplus C_{i}\\ \end{array}$$

$$H^{\ast }({\hat{Z}}_K(\underline{X},\underline{A}))=\underset{I\subset [m],\sigma \subset I}{⨁}{E}^{I^c}\otimes Y^{I,\sigma }\otimes {\tilde{H}}^{\ast }(\Sigma |lk(\sigma {)}_{I^c}|)$$

$${\tilde{H}}^{\ast }(\Sigma |lk({\sigma }_1){|}_{I_1^c})\otimes {\tilde{H}}^{\ast }(\Sigma |lk({\sigma }_2){|}_{I_2^c})\to {\tilde{H}}^{\ast }(\Sigma |lk({\sigma }_3){|}_{I_3^c})$$

$$\alpha ,\beta \in$$

$$=\underset{I\subset [m],\sigma \subset I}{⨁}{E}^{[m]-I}\otimes C^{\sigma }\otimes B^{I-\sigma }\otimes {\tilde{H}}^{\ast }(\Sigma \text{lk}(\sigma {)}_{I^c}),$$

$$\begin{array}[]{ccc}\alpha&=&a_{1}\otimes a_{2}\otimes\ldots\otimes a_{m}\otimes n_{\alpha}\\ \beta&=&b_{1}\otimes b_{2}\otimes\ldots\otimes b_{m}\otimes n_{\beta}\end{array}$$

$$y_i=\{\begin{array}{ll}{s}_i& i\in \sigma \\ t_i& i\in I-\sigma \\ 1& k\notin I\end{array}$$

$$d_I(y_{\sigma })=\sum _{\tau }(-1{)}^{n(\tau )}{y}_{\tau }$$

$$C_K=\underset{I\subset [n]}{⨁}C(K_I)$$

$$s_i⌣s_i=0,t_i⌣t_i=t_i,s_i⌣t_i=s_i,t_i⌣s_i=0.$$

Full text

1. Introduction

Fixing a pair of topological spaces $(X,A)$, polyhedral product spaces $Z_{K}({X},{A})$ give a family of spaces where $K$ is a simplicial complex (see Definition 2.1). Examples include moment-angle complexes, complements of complex coordinate subspace arrangements, and intersections of quadrics among others. In certain cases, polyhedral products provide geometric realizations of right-angled Artin groups and the Stanley-Reisner ring (see Definition 2.5).

The real moment-angle complex, $Z_{K}(D^{1},S^{0})$, and its complex analog (arising from the pair of spaces, the unit disc $D^{2}$ and the circle $S^{1}$) feature in toric topology, as they have been key in showing applications in combinatorics and algebraic geometry, among others [5]. The cohomology ring of the moment-angle complex is shown to be isomorphic to the Tor-algebra $Tor_{{\mathbb{Z}}[v_{1},\ldots,v_{m}]}({\mathbb{Z}}[K],{\mathbb{Z}})$ in [4], where ${\mathbb{Z}}[K]$ is the Stanley-Reisner (or face ring) of $K$ and the indeterminates $v_{i}$ are of degree two (see Section 2). The generators correspond to certain subsets of integers and the product of two generators corresponding to non-disjoint subsets is trivial, forming a symplectic basis.

On the other hand, the cohomology ring of the real moment-angle complex is not completely understood. The group structure is known to be given by $Tor_{{\mathbb{Z}}[v_{1},\ldots,v_{m}]}({\mathbb{Z}}[K],{\mathbb{Z}})$ with indeterminates $v_{i}$ of degree one. Theorem 3.5 gives the ring structure for real moment-angle complexes over certain simplicial complexes. A consequence of Theorem 3.5 is that the multiplicative structure does not have the same nice closed form as that of moment-angle complexes. In other words, generators corresponding to non-disjoint subsets do not necessarily have trivial product.

This set of combinatorially defined generators can be identified using Bahri-Bendersky-Cohen-Gitler’s Splitting Theorem [1] and Welker-Ziegler-Z̆ivaljević’s wedge lemma [13]. In this paper, we consider the case when the simplicial complex $K$ is the boundary of an $n$-gon, and describe the ring structure in terms of the combinatorial generators in Theorem 3.5.

In full generality, for a simplicial complex $K$ on $m$ vertices, polyhedral product spaces $Z_{K}(\underline{X},\underline{A})$ are defined in terms of a collection of pairs of spaces $(\underline{X},\underline{A})=\{X_{i},A_{i}\}_{i=1}^{m}$. The ring structure for the real moment-angle complex $Z_{K}(D^{1},S^{0})$ is particularly useful in that the cohomology ring of the more general polyhedral product $Z_{K}(\underline{CA},\underline{A})$ when $CA_{i}$ is the cone on $A_{i}$, can be described in terms of the ring structure of $H^{*}(Z_{K}(D^{1},S^{0}))$ and $H^{*}(A)$ [3].

Moreover, this problem of understanding the cohomology ring of a real moment-angle complex has connections to studying the topology of intersections of quadrics and real coordinate subspace arrangements, specifically the case when $K$ is the pentagon is discussed in [10]. The cohomology of real moment-angle complexes and related spaces has also been studied in [7], in the case of rational coefficients.

In Section 3.2, we will illustrate the main theorem with some examples. As a corollary we will see that, even though real moment-angle complexes over an $n$-gon are orientable surfaces, the combinatorial generators do not form a symplectic basis.

Acknowledgements. This work is part of the author’s doctoral dissertation at the City University of New York Graduate Center. The author would like to thank Martin Bendersky for his guidance throughout this research.

2. Polyhedral Product Spaces

In this section, we will give a brief introduction to polyhedral products, moment-angle complexes, and real moment-angle complexes, with an emphasis on the multiplicative structure of their respective cohomology rings.

Let $[m]=\{1,2,\ldots,m\}$ denote the set of integers from $1$ to $m$. An abstract simplicial complex, $K$, on $[m]$ is a subset of the power set of $[m]$, such that:

1. (1)

$\emptyset\in K$. 2. (2)

If $\sigma\in K$ with $\tau\subset\sigma$, then $\tau\in K$.

An $n$-simplex is the full power set of $[n+1]$ and is denoted $\Delta^{n}$. Associated to an abstract simplicial complex is its geometric realization, denoted $\mathcal{K}$ or $|K|$ (also called a geometric simplicial complex). A (geometric) $n$-simplex, $\Delta^{n}$, is the convex hull of $n+1$ points.

We do not assume $m$ is minimal, i.e. there may exist $[n]\subsetneq[m]$ such that $K$ is contained in the power set of $[n]$.

Let $I$ be a subset of $[m]$. The full subcomplex of $K$ in $I$ is denoted $K_{I}$. It is a simplicial complex on the set $I$ and defined

[TABLE]

It is often called the restriction of $K$ to $I$ in the literature.

Given an abstract simplicial complex $K$, let $\mathcal{S}_{K}$ be the category with simplices of $K$ as the objects and inclusions as the morphisms. In particular, for $\sigma,\tau\in ob(\mathcal{S}_{K})$, there is a morphism $\sigma\rightarrow\tau$ whenever $\sigma\subset\tau$. Define $\mathcal{CW}$ to be the category of CW-complexes and continuous maps. Define $(\underline{X},\underline{A})$ to be a collection of pairs of CW-complexes $\{(X_{i},A_{i})\}_{i=1}^{m}$, where $A_{i}$ is a subspace of $X_{i}$ for all $i$.

Definition 2.1**.**

Given an abstract simplicial complex $K$ on $[m]$, simplices $\sigma,\tau$ of $K$ and a collection of pairs of CW-complexes $(\underline{X},\underline{A})$, define a diagram $D:\mathcal{S}_{K}\rightarrow\mathcal{CW}$ given by

[TABLE]

For a morphism $f:\sigma\rightarrow\tau$, the functor $D$ maps $f$ to $\iota:D(\sigma)\rightarrow D(\tau)$ where $\iota$ is the canonical injection.

The polyhedral product space is defined as

[TABLE]

and is topologized as a subspace of $\displaystyle\prod_{i\in[m]}X_{i}$.

Notice that it suffices to take the colimit over the maximal simplices of $K$. In fact, simplicial complexes can be defined by their maximal simplices and this description will be used throughout. In the case where $(X_{i},A_{i})=(X,A)$ for all $i$, we write $Z_{K}(X,A)$.

Some examples of polyhedral products are moment-angle complexes $Z_{K}(D^{2},S^{1})$, which have the homotopy type of the complement of a complex coordinate subspace arrangement, and Davis-Januszkiewicz spaces $Z_{K}(\mathbb{C}P^{\infty},*)$, which have the Stanley-Reisner ring as cohomology ring. For a simple example, consider the following.

Example 1*.*

Let $K$ be the boundary of a $2$-simplex with vertices labelled $1,2,3$.

[TABLE]

In general, $Z_{\partial\Delta^{m}}(D^{1},S^{0})\cong S^{m}$ (see examples in [1]).

Next we will define the polyhedral smash product, a space analogous to the polyhedral product with the smash product operation in place of the Cartesian product. Define $\mathcal{CW}_{*}$ to be the category of based CW-complexes and based continuous maps.

Definition 2.2**.**

Let the CW-pairs $(\underline{X},\underline{A})$ be pointed. Likewise, define a functor $\widehat{D}(\sigma):\mathcal{S}_{K}\rightarrow\mathcal{CW}_{*}$ by

[TABLE]

Then the polyhedral smash product is

[TABLE]

For the remainder of the paper, we will assume that $(X_{i},A_{i})$ is a pair of pointed CW-complexes where $A_{i}$ is a subspace of $X_{i}$.

The following theorem of Bahri, Bendersky, Cohen and Gitler (BBCG) gives a stable decomposition of a polyhedral product.

Theorem 2.3** (Splitting Theorem, [1]).**

Let $(\underline{X_{I}},\underline{A_{I}})=\{(X_{i},A_{i})\}_{i\in I}$. Then

[TABLE]

where $\Sigma$ denotes the reduced suspension.

In [1], the authors apply the wedge lemma from [13] to polyhedral smash products and obtain the following:

Theorem 2.4** (Wedge Lemma, [13]).**

If $X_{i}$ is contractible for all $i$, then

[TABLE]

where $A^{\wedge[m]}=A_{1}\wedge\ldots\wedge A_{m}$

Since $S^{0}$ serves as an identity for the smash product operation, computing the cohomology groups of real moment-angle complexes becomes a combinatorial process that involves examining only the simplicial complex. This follows from the previous two theorems.

[TABLE]

The generators of the cohomology ring are given by the subsets of $[m]$ that yield a noncontractible full subcomplex of $K$ after suspension, which we call the combinatorial generators.

To describe the ring structure of the cohomology of the moment-angle complex, we will introduce some notation. The graded ring $\mathbb{Z}[m]$ is the polynomial ring on $m$ variables $\mathbb{Z}[v_{1},v_{2},\ldots,v_{m}]$ with $\deg(v_{i})=2$.

Definition 2.5**.**

The Stanley-Reisner ring (or face ring) of the simplicial complex $K$ is the quotient of $\mathbb{Z}[m]$ by the ideal generated by square-free monomials associated to nonfaces of $K$

[TABLE]

The following was first proved by Franz in [9] and stated in terms of smooth toric varieties. Another proof was later given by Baskakov, Buchstaber, Panov in [4].

Theorem 2.6** (Franz, [9]).**

The cohomology ring of the moment-angle complex $Z_{K}(D^{2},S^{1})$ is given by

[TABLE]

A description of the multiplicative structure in terms of full subcomplexes comes from Hochster’s theorem in commutative algebra on the $Tor$-module [11]. We obtain the following analogous formula

[TABLE]

For the multiplicative structure, take classes $\alpha\in{H}^{i}(Z_{K}(D^{2},S^{1}))$ and $\beta\in\widetilde{H}^{k}(Z_{K}(D^{2},S^{1}))$. Then $\alpha$ corresponds to some class in $\widetilde{H}^{i-|J|-1}(K_{J})$ for some subset $J\subset[m]$ , and similarly $\beta$ to some class in $\widetilde{H}^{k-|L|-1}(K_{L})$ for some $L\subset[m]$. Their product is

[TABLE]

for some $\gamma$ coming from $\widetilde{H}^{i+k-|L|-|J|-1}(K_{J\cup L})$. See [12] for more details.

2.1. The BBCG spectral sequence

We will use a spectral sequence developed by BBCG [3]. It gives a Künneth-like formula for the cohomology of a polyhedral product as long as the pairs $(\underline{X},\underline{A})$ satisfy the following freeness condition.

Definition 2.7**.**

Given the pair $(X_{i},A_{i})$, the associated long exact sequence is given by

[TABLE]

The pair is said to satisfy the strong $h^{*}$ freeness condition if there are free $h^{*}-$modules $E_{i},B_{i},C_{i}$ and $W_{i}$ satisfying

[TABLE]

where $W_{i}$ is $sE_{i}$, the suspension of $E_{i}$. Additionally, assume $1\in B_{i}$, and for $b\in B_{i},c\in C_{i},e\in E_{i},w\in W_{i}=sE_{i}$, we have

$b\overset{f}{\mapsto}b\overset{\delta}{\mapsto}0,\qquad c\overset{g}{\mapsto}c\overset{f}{\mapsto}0,\qquad e\overset{\delta}{\mapsto}w\overset{g}{\mapsto}0$.

Before defining the spectral sequence, we will give some notation and recall the definition of a half smash product:

1. (1)

for $\sigma=\{i_{1},\ldots,i_{k}\}$, define $\widehat{X}^{\sigma}:=X_{i_{1}}\wedge\ldots\wedge X_{i_{k}}$ and ${A}^{\sigma}=A_{i_{1}}\times\ldots\times A_{i_{k}}$ 2. (2)

the complement of a set $\sigma\subset[m]$ is $\sigma^{c}=[m]\backslash\sigma$ 3. (3)

given a basepoint $x_{0}\in X$, the right half smash product $X\rtimes Y=(X\times Y)/(x_{0}\times Y)$ 4. (4)

for a subset $I$ and a simplex $\sigma$ such that $\sigma\subset I$, define

[TABLE]

Choosing a lexicographical ordering for the simplices of $K$ gives a filtration of the associated polyhedral product space and polyhedral smash product, which in turn leads to a spectral sequence converging to the reduced cohomology of $Z_{K}(\underline{X},\underline{A})$ and a spectral sequence converging to the reduced cohomology of $\widehat{Z}_{K}(\underline{X},\underline{A})$. The $E_{1}^{s,t}$ term for $Z_{K}(\underline{X},\underline{A})$ has the following description.

Theorem 2.8** (Bahri, Bendersky, Cohen and Gitler [3]).**

There exist spectral sequences

[TABLE]

[TABLE]

with $E^{s,t}_{1}=\widetilde{H}^{t}((\widehat{X/A})^{\sigma}\rtimes{A}^{\sigma^{c}})$ and $\widehat{E}^{s,t}_{1}=\widetilde{H}^{t}((\widehat{X/A})^{\sigma}\wedge\widehat{A}^{\sigma^{c}})$ where $s$ is the index of $\sigma$ in the lexicographical ordering and the differential $d_{r}:E_{r}^{s,t}\rightarrow E_{r}^{s+r,t+1}$ is induced by the coboundary map $\delta:E\rightarrow W=sE$. Moreover, the spectral sequence is natural for embeddings of simplicial maps with the same number of vertices and with respect to maps of pairs. The natural quotient map

[TABLE]

induces a morphism of spectral sequences and the Splitting Theorem (2.3) induces a morphism of spectral sequences.

Following [3], Definition 2.7 and the Künneth Theorem imply that the entries $\widetilde{H}^{t}((\widehat{X/A})^{\sigma}\wedge\widehat{A}^{\sigma^{c}})$ in the first page of the spectral sequence for $\widehat{Z}_{K}(\underline{X},\underline{A})$ decompose as a direct sum of spaces $W^{N}\otimes C^{S}\otimes B^{T}\otimes E^{J}$ such that $N\cup S=\sigma$, $T\cup J=\sigma^{c}$ and $N,S,J,T$ are disjoint. We have that $S$ is a simplex in $K$ as $N\cup S$ is a simplex in $K$. Since the differential is induced by the coboundary $\delta:E\rightarrow W$, consider all the possible summands $W^{N}\otimes C^{S}\otimes B^{T}\otimes E^{J}$ for $S$ and $T$ fixed. It must be the case that $N$ is a simplex in $K$ and that $N$ is a subset of $[m]\backslash(S\cup T)$. Therefore all such $N$ correspond to simplices in the link of $S$ in $K$ restricted to the vertex set $[m]\backslash(S\cup T)$.

Theorem 2.9** (Bahri, Bendersky, Cohen and Gitler [3]).**

Let $(\underline{X},\underline{A})$ satisfy the decomposition described in Definition 2.7

[TABLE]

Then

[TABLE]

where:

1. (1)

$\sigma$* is a simplex in $K$,* 2. (2)

$lk(\sigma)_{I^{c}}=\{\tau\subset[m]\backslash I\;|\;\tau\cup\sigma\in K\}$* is the link of $\sigma$ in $K$ restricted to the set $[m]\backslash I$,* 3. (3)

$Y^{I,\sigma}=\bigotimes\limits_{i\in\sigma}C_{i}\otimes\bigotimes\limits_{i\in I-\sigma}B_{i}$, and 4. (4)

$\widetilde{H}^{*}(\Sigma\emptyset)=1$.

Theorem 2.10** (Bahri, Bendersky, Cohen and Gitler [3]).**

Let

[TABLE]

Then

[TABLE]

where:

1. (1)

$\sigma$* is a simplex in $K$,* 2. (2)

$lk(\sigma)_{I^{c}}=\{\tau\subset[m]\backslash I\;|\;\tau\cup\sigma\in K\}$* is the link of $\sigma$ in $K$ restricted to the set $[m]\backslash I$,* 3. (3)

$Y^{I,\sigma}=\bigotimes\limits_{i\in\sigma}C_{i}\otimes\bigotimes\limits_{i\in I-\sigma}\widetilde{B}_{i}$* where $\widetilde{B}_{i}=B_{i}\backslash\{1\}$,* 4. (4)

$\widetilde{H}^{*}(\Sigma\emptyset)=1$.

A description of the ring structure in $H^{*}(Z_{K}(\underline{X},\underline{A}))$ is given using the decomposition from Theorems 2.9 and 2.10. It is induced by a pairing involving links

[TABLE]

defined in terms of the $*$-product, introduced in [2], where $I_{3}$ and $\sigma_{3}$ are defined in terms of $\sigma_{1},\sigma_{2},I_{1}$ and $I_{2}$.

Theorem 2.11** (Theorem 6.1 in [3]).**

Two classes

[TABLE]

are of the form

[TABLE]

where $n_{\alpha}\in\widetilde{H}^{*}(\Sigma\text{lk}(\sigma)_{I^{c}})$ and $n_{\beta}\in\widetilde{H}^{*}(\Sigma\text{lk}(\tau)_{J^{c}})$.

The cup product of $\alpha$ and $\beta$ is given in terms of the $*$-product and a componentwise product induced by the multiplicative structure of $H^{*}(X_{i})$ and $H^{*}(A_{i})$.

For the pair of spaces $(CA_{i},A_{i})$, where $CA_{i}$ is the cone on $A_{i}$, the modules are given by $B_{i}=1$, $C_{i}=0$ and $E_{i}=\widetilde{H}^{*}(A)$. The links are all of the form $K_{I}$ for $I\subset[m]$. Therefore, it can be seen from Theorem 2.9 that the product structure in $H^{*}(Z_{K}(\underline{CA},\underline{A}))$ can be described in terms of the product structure in $H^{*}(A)$ and $\widetilde{H}^{*}(\Sigma\mathcal{K}_{I})$.

Due to the decomposition in Equation 2.1 and work in [2], the ring structure in $H^{*}(Z_{K}(\underline{CA},\underline{A}))$ can be described in terms of the ring structure in $H^{*}(A)$ and $H^{*}(Z_{K}(D^{1},S^{0}))$.

Theorem 2.12** (Theorem 1.9 in [2]).**

Assume that any finite product of $A_{i}$ with $Z_{K_{I}}(D^{1},S^{0})$ for all $I$ satisfies the strong form of the Künneth Theorem. Then the cup product structure for the cohomology algebra $H^{*}(Z_{K}(\underline{CA},\underline{A}))$ is a functor of the cohomology algebras of $A_{i}$, and $Z_{K_{I}}(D^{1},S^{0})$ for all $I$.

3. Multiplicative structure of $H^{*}(Z_{K}(D^{1},S^{0}))$

Recall from Equation 2.1 that each subset $I$ of $[m]$ such that the full subcomplex $\mathcal{K}_{I}$ is not contractible corresponds to a generator of $H^{*}(Z_{K}(D^{1},S^{0}))$.

To compute the cohomology of a real moment-angle complex, we will use a filtered chain complex induced by the long exact sequence of the pair $(D^{1},S^{0})$, denoted $C_{K}$ and constructed in [3]. For $(X_{i},A_{i})=(D^{1},S^{0})$, let $\widetilde{H}^{*}(A_{k})=\widetilde{H}^{*}(S^{0})$ be generated by $t_{k}$ and $\widetilde{H}^{*}(X_{k}/A_{k})=\widetilde{H}^{*}(S^{1})$ be generated by $s_{k}$.

Definition 3.1**.**

The chain complex $C(K_{I})$ is generated by $y_{\sigma}:=\otimes y_{i}$ where $\sigma\in K_{I}$ and

[TABLE]

The differential is defined by

[TABLE]

where $\sigma\subset\tau\in K_{I}$ and $\tau=\sigma\cup v$ for some vertex $v\in I$. The integer $n(\tau)$ is defined by the usual sign convention of a graded derivation. In particular, the coboundary $\delta$ acts on each factor of $y_{\sigma}$ by $\delta(s_{i})=0$ and $\delta(t_{i})=s_{i}$, and every time it passes an $s_{i}$ a factor of $(-1)$ is introduced.

Then

[TABLE]

and $H^{*}(C_{K})=H^{*}(Z_{K}(D^{1},S^{0}))$.

It follows from work of Li Cai in [6] that the chain level cup product of two generators is induced by the following

[TABLE]

3.1. Boundary of a polygon

We will consider the case of $K$ the boundary of a polygon. By Theorem 2.1, we need to consider all subsets of $[n]$ to find the cohomology groups. By convention, when $I$ is the empty set, $H^{*}(\Sigma\mathcal{K}_{I})=1$. The suspension of the whole complex $K$ is a degree two generator. The following lemma gives the generators of degree one.

Lemma 3.2**.**

Suppose $K$ is the boundary of an $n$-gon. Let $I=I_{1}\sqcup I_{2}\sqcup\ldots\sqcup I_{p}$ be a subset of $[n]$ such that $\mathcal{K}_{I}$ has exactly $p$ maximal connected components, $\mathcal{K}_{I}\simeq\bigvee_{p-1}S^{0}$. Then

[TABLE]

Proof.

Let $I_{1}=\{i_{1},\ldots,i_{c}\}$. If $i_{1}=i_{c}$, then $d(y_{\{i_{1}\}})=0$ and $y_{\{i_{1}\}}$ is clearly a cocycle. If $i_{1}\neq i_{c}$, then the differential will not be trivial. If $+y_{\{e\}}$ for some edge $e\in K_{I_{1}}$ appears as a summand in the image of $d(y_{\{v\}})$ for some vertex $v\in I_{1}$, then $e=\{v-1,v\}$ or $e=\{1,n\}$ (since $y_{\{e\}}$ was positive, we could not have passed an $s$). Additionally, since $y_{\{e\}}$ was in the image of $y_{\{v\}}$, it must be the case that $v-1\in I_{1}$, so $-y_{\{e\}}$ is a term in the image of $y_{\{v-1\}}$ under $d$. If it had been the case that $e=\{1,n\}$, then $\{1\}\in K_{I}$ and $-y_{\{e\}}$ would be in the image of $y_{\{1\}}$. Since it is only possible for $y_{\{e\}}$ to be in the image of $y_{\{v-1\}}$ or $y_{\{v\}}$, the terms $y_{\{e\}}$ cancel. This means that $d(y_{\{i_{1}\}}+\ldots+y_{\{i_{c}\}})=0$ since $i_{1},\ldots,i_{c}$ are all the vertices in a connected component of $\mathcal{K}_{I}$. Without loss of generality, the same is true for the other connected components. Lastly, $d(\emptyset)$ is the sum of $y_{\{i\}}$ for $i\in I$. ∎

Corollary 3.3**.**

If $I=I_{1}\sqcup I_{2}\sqcup\ldots\sqcup I_{p}$ is a subset of $[n]$ such that $\mathcal{K}_{I}$ has exactly $p$ maximal connected components with $p>1$, then $H^{1}(\Sigma|\mathcal{K}_{I}|)$ has rank $p-1$ and a basis of generators can be chosen by picking any $p-1$ of the $p$ disjoint subsets $I_{1},I_{2},\ldots,I_{p}$.

Next, the following lemma will show how generators coming from different subsets of $[n]$ multiply. Consider subsets $I,J\subset[n]$ such that $I\cup J=[n]$. We will employ a slight change in notation: replacing $y$’s associated to $I$ with $a$’s and $y$’s associated to $J$ with $b$’s to differentiate between generators in $C(K_{I})$ and generators in $C(K_{J})$. Recall that $a_{\{i\}}$ is the generator associated to the vertex $i$, whereas $a_{i}$ is the $i$th factor of a generator.

Lemma 3.4**.**

Suppose $a_{\{i\}}\in C(K_{I})$ and $b_{\{j\}}\in C(K_{J})$. Then $a_{\{i\}}=a_{1}\otimes a_{2}\otimes\ldots\otimes\ a_{n}$ where

[TABLE]

Define $b_{\{j\}}$ similarly. Then

[TABLE]

Proof.

If $|i-j|\neq 1$, then $\{i,j\}$ is not a simplex in $K$ and $a_{\{i\}}\smile b_{\{j\}}=0$. Therefore, we will now consider cases where $|i-j|=1$.

Recall that $s\smile t=s\smile 1=s$ and $t\smile s=0$.

Suppose $j\in I$. Since $j\in I$, $a_{j}=t_{j}$. In particular, in the $j$th coordinate of $a_{\{i\}}\smile b_{\{j\}}$, we will have $a_{j}\smile b_{j}=t_{j}\smile s_{j}=0$ so $a_{\{i\}}\smile b_{\{j\}}=0$.

Next suppose $j\notin I$. Then $a_{j}=1$ and $a_{j}\smile b_{j}=s_{j}$. If $j=i+1$, then

$a_{\{i\}}\smile b_{\{j\}}$

\begin{array}[]{cccccccc}=&(a_{1}\smile b_{1})&\otimes\ldots\otimes&(a_{i}\smile b_{i})&\otimes&(a_{j}\smile b_{j})&\otimes\ldots\otimes&(a_{n}\smile b_{n})\\ =&t_{1}&\otimes\ldots\otimes&s_{i}\smile b_{i}&\otimes&1\smile s_{j}&\otimes\ldots\otimes&t_{n}\\ =&t_{1}&\otimes\ldots\otimes&s_{i}&\otimes&s_{j}&\otimes\ldots\otimes&t_{n}\\ =&y_{\{i,j\}}\end{array}

since the only coordinate of $b_{\{j\}}$ that is an $s$ is $b_{j}$ and all other coordinates are $t$ or $1$.

If $j=i-1$, since $a_{i}\smile b_{i-1}=(-1)^{|b_{i-i}||a_{i}|}(b_{i-1}\smile a_{i})$, we have

$a_{\{i\}}\smile b_{\{j\}}$

\begin{array}[]{cccccccc}=&(a_{1}\smile b_{1})&\otimes\ldots\otimes&(-1)^{|b_{j}||a_{i}|}(a_{j}\smile b_{j})&\otimes&(a_{i}\smile b_{i})&\otimes\ldots\otimes&(a_{n}\smile b_{n})\\ =&t_{1}&\otimes\ldots\otimes&(-1)(1\smile s_{j})&\otimes&s_{i}\smile b_{i}&\otimes\ldots\otimes&t_{n}\\ =&t_{1}&\otimes\ldots\otimes&(-1)s_{j}&\otimes&s_{i}&\otimes\ldots\otimes&t_{n}\\ =&-y_{\{j,i\}}\end{array} ∎

The following theorem uses the previous lemmas to show what the only non-trivial products in $H^{*}(Z_{K}(D^{1},S^{0}))$ are.

Theorem 3.5**.**

Let $K$ be the boundary of an $n$-gon. Let $I=I_{1}\sqcup I_{2}\sqcup\ldots\sqcup I_{p}$ and $J=J_{1}\sqcup J_{2}\sqcup\ldots\sqcup J_{q}$ be nonempty subsets of $[n]$ such that $I\cup J=[n]$ and ${K}_{I}\simeq\bigvee_{1}^{p-1}S^{0}$ and ${K}_{J}\simeq\bigvee_{1}^{q-1}S^{0}$. Given generators $\alpha$ and $\beta$ of $H^{*}(Z_{K}(D^{1},S^{0}))$ such that one is associated to some $I_{g}$ for $1\leq g\leq p$ and the other is associated to some $J_{h}$ for some $1\leq h\leq q$. If $\gamma$ is the second degree generator of $H^{*}(Z_{K}(D^{1},S^{0}))$, then $\alpha\smile\beta=\pm\gamma$ if and only if the following conditions are met

• •

$I_{g}\nsubseteq J_{h}$**

• •

$J_{h}\nsubseteq I_{g}$**

• •

${K}_{I_{g}\cup J_{h}}$* is contractible*

Proof.

We will compute $H^{*}({Z}_{K}(D^{1},S^{0}))$ using the chain complex described previously.

[TABLE]

Therefore, all the classes in $H^{*}({Z}(K;(D^{1},S^{0})))$ represented by an edge are cohomologous, except $y_{\{1,n\}}$, which is the negative. Note that if $1$ and $n$ are in $I$, then $1$ and $n$ are in $I_{i}$ for some $1\leq i\leq p$ (since $\mathcal{K}_{I}\simeq\bigvee_{1}^{p-1}S^{0}$, it cannot be that $1$ and $n$ are in different subsets of $I$). By Corollary 3.3, we only need to consider all but one of the disjoint subset of $I$ and all but one of the disjoint subsets of $J$. Therefore, it suffices to only consider when $1,n\notin I_{g}\cup J_{h}$. As a consequence, the class cannot $y_{\{1,n\}}$ occur in the product $\alpha\smile\beta$.

Let $I_{g}=\{i_{1},\ldots,i_{c}\}$ and $J_{h}=\{j_{1},\ldots,j_{d}\}$. By Lemma 3.2, we have that

[TABLE]

Since $I\cup J=[n]$, we have that $i_{1}\neq j_{1}$ and $i_{c}\neq j_{d}$.

First, suppose $I_{g}\cap J_{h}=\emptyset$. In the case that $i_{c}<j_{1}$, we must have $i_{c}=j_{1}-1$ so that there is at least one edge after expanding the product. Then there is only one nonzero term

[TABLE]

by Lemma 3.4. Similarly, if $j_{d}=i_{1}-1$, then

[TABLE]

Secondly, suppose $I_{g}\cap J_{h}\neq\emptyset$ and that neither set is contained in the other. If $j_{1}\leq i_{c}$, then there exists $j\in J_{h}$ such that $j=i_{c}$. Note that $j+1\in J_{h}$ (because $i_{c}\neq j_{d}$ and so $j\neq j_{d}$). Since $\{i_{c},j+1\}$ is an edge and $j+1=i_{c}+1\notin I$, by Lemma 3.4 we have only one nonzero term

[TABLE]

Similarly, if $i_{1}\leq j_{d}$ and $j=i_{1}$ for some $j\in J_{h}$, then $j-1\notin I$ and $i_{1}-(j-1)=1$. Then

[TABLE]

If $J_{h}\subset I_{g}$, then $\alpha\smile\beta=0$ by Lemma 3.4. If $I_{g}\subset J_{h}$, then there are only two possible nonzero products between the summands of $\alpha$ and $\beta$. There exists $j\in J_{h}$ such that $j=i_{1}$. Then

[TABLE]

∎

3.2. Example and related consequences

To illustrate an application of the theorem and some important consequences, we will consider the case when $K$ is the boundary of the pentagon, denoted $K_{5}$. Let the $1$-simplices of $K_{5}$ be labeled $\{1,2\},\{2,3\},\{3,4\},\{4,5\},\{1,5\}$.

It follows from [8] that the real moment-angle complex over the boundary of an $n$-gon is a closed orientable surface of genus $1+(n-4)2^{n-3}$, which means that in this example the associated real moment-angle complex has genus five. For the combinatorial generators, there are ten subsets $I$ of $[5]$ that yield a full subcomplex $K_{I}$ equivalent to a wedge of [math]-spheres. The cohomology of $Z_{K_{I}}(D^{1},S^{0})$ has an identity, ten degree one generators $x_{0},\ldots,x_{4},w_{0},\ldots,w_{4}$, and a degree two generator $z$, subject to a graded commutative product. The identity corresponds to the empty set. The generators $x_{i}$ correspond to the subsets that yield a full subcomplex of $K$ of an edge and the opposite vertex, such as the subset $I=\{1,2,4\}$. The generators $w_{i}$ correspond to the subsets that produce a full subcomplex of two disjoint vertices. Lastly, $z$ corresponds to the full vertex set $[5]=\{1,2,3,4,5\}$.

[TABLE]

where $\delta$ is the Dirac delta function and the subscripts $i,j$ are integers modulo $4$.

Notice that for subsets $J=\{2,4,5\}$ and $I=\{1,3,4\},$ $\alpha_{I}\smile\alpha_{J}=\gamma$. This is an example where generators coming from non-disjoint subsets have a nontrivial product, unlike the ring for moment-angle complexes. The cohomology ring of $Z_{K}(D^{1},S^{0})$ is not isomorphic to the Tor-module as rings. Moreover, this application of Theorem 3.5 also shows that the basis of combinatorial generators is not symplectic.

Lastly, recall that the multiplicative structure of the cohomology of real moment-angle complexes plays an important role in the product structure for more general polyhedral product spaces [2]. Theorem 2.12 gives the algebra $H^{*}(Z_{K}(\underline{CA},\underline{A}))$ in terms of the cohomology algebras of $A_{i}$ and $H^{*}(Z_{K_{I}}(D^{1},S^{0}))$.