\begin{document}

\title [\ ] {On a class of quotient spaces of moment-angle complexes}

\author{Li Yu}
\address{Department of Mathematics and IMS, Nanjing University, Nanjing, 210093, P.R.China
 }
 \email{yuli@nju.edu.cn}
 \thanks{2010 \textit{Mathematics Subject Classification}. 57S15, 14M25, \\
 The author is partially supported by
 Natural Science Foundation of China (grant no.11371188) and the
 PAPD (priority academic program development) of Jiangsu higher education institutions.}


\keywords{Moment-angle complex, Hochster's formula, Tor algebra, balanced simple polytope}


\begin{abstract}
 For a simplicial complex $\mathcal{K}$ and a partition $\upalpha$ 
 of the vertex set of $\mathcal{K}$, we define
  a quotient space of the (real) moment-angle complex of $\K$ by some 
  (not necessarily free) torus action determined by $\upalpha$.
  We obtain a Hochster type formula to compute the cohomology groups 
  of such spaces. Meanwhile, we can derive the formula
   from some stable decompositions of these spaces. 
  Moreover,
  we show that 
   their cohomology rings with $\Z_2$-coefficients 
 are isomorphic as multigraded $\Z_2$-modules (or algebras) to 
 the cohomology of some multigraded differential algebras determined by 
 $\K$ and $\upalpha$. 
  \end{abstract}

\maketitle

 \section{Introduction} \label{Sec:Penal-Struct}
 
  An \emph{abstract simplicial complex} on a set $[m]=\{ v_1,\cdots, v_m\}$ 
  is a collection $\mathcal{K}$ of subsets $\sigma\subseteq [m]$ such that
  if $\sigma\in\mathcal{K}$, then any subset of $\sigma$ also belongs to $\mathcal{K}$.
  We always assume that the empty set belongs to $\mathcal{K}$ and refer
  to $\sigma\in \mathcal{K}$ as an abstract \emph{simplex} of $\mathcal{K}$.
  The simplex corresponding to the empty set is denoted by $\hat{\mathbf{0}}$.
  In particular, any element of $[m]$ that belongs to $\mathcal{K}$ is 
  called a \emph{vertex} of $\mathcal{K}$.  
  We call the number of
  vertices of a simplex $\sigma$ the \emph{rank} of $\sigma$, denoted by $\mathrm{rank}(\sigma)$.
  Let $\dim(\sigma)$ denote the dimension of a simplex $\sigma$.
   So $\mathrm{rank}(\sigma)=\dim(\sigma)+1$.\n
  
   Any finite abstract simplicial complex $\K$ admits a
  \emph{geometric realization} $|\K|$ in some Euclidean space. 
  If not particularly indicated, we do not
  distinguish $\K$ and its geometric realization $|\K|$ in this paper. \n
  
  Given a finite abstract simplicial complex $\mathcal{K}$ on a set $[m]$ and 
  a pair of spaces $(X,A)$ with $A\subset X$, we can construct of a topological space 
  $(X,A)^{\mathcal{K}}$ by:
    \begin{equation}\label{Equ:Construction}
     (X,A)^{\mathcal{K}}= \bigcup_{\sigma\in \mathcal{K}} (X,A)^{\sigma}, \ \text{where}\
      (X,A)^{\sigma}= \prod_{v_i\in \sigma} X \times \prod_{v_i\notin \sigma} A. 
    \end{equation}  
  The symbol $\prod$ here and in the rest of this paper means Cartesian product.\n
  
   By definition $(X,A)^{\mathcal{K}}$ is a subspace of the Cartesian product of
  $m$ copies of $X$. It is called the \emph{polyhedral product}
   or the \emph{generalized moment-angle complex}
   of $\mathcal{K}$ and $(X,A)$. 
  In particular, $\mathcal{Z}_{\mathcal{K}}
    =(D^2,S^1)^{\mathcal{K}}$ and 
    $\R\mathcal{Z}_{\mathcal{K}}=(D^1,S^0)^{\mathcal{K}}$ are called the 
  \emph{moment-angle complex of $\mathcal{K}$} and 
  \emph{real moment-angle complex of $\mathcal{K}$}, respectively (see~\cite{BP02}).
  Moreover, we can define the polyhedral product 
  $(\mathbb{X},\mathbb{A})^{\K}$ of $\K$ with a set of $m$ pairs of spaces
  $(\mathbb{X},\mathbb{A}) = \{ (X_1,A_1),\cdots, (X_m,A_m) \}$ 
  (see~\cite{BBCG10} or~\cite[Sec 4.2]{BP13}).\n
  
  
  Originally, $\mathcal{Z}_{\mathcal{K}}$ and  
    $\R\mathcal{Z}_{\mathcal{K}}$ are introduced by Davis and Januszkiewicz~\cite{DaJan91} 
    in a completely different way.
   Let $\K'$ denote the barycentric subdivision of $\K$. We can consider
    $\K'$
   as the set of chains of simplices in $\K$ ordered by inclusions. 
     For each simplex
   $\sigma\in \K$, let $F_{\sigma}$ denote the geometric realization of the poset
   $\K_{\geq \sigma}=\{ \tau\in \K\,|\, \sigma\subseteq \tau \}$. Thus,  
    $F_{\sigma}$
   is the subcomplex of $\K'$ consisting of all simplices of the form
   $\sigma=\sigma_0 < \sigma_1 < \cdots < \sigma_l$.    
    Let $P_{\K}$ denote the cone on $\K'$.   
    If $\sigma$ is a $(k-1)$-simplex, then we say that $F_{\sigma}\subset P_{\K}$ is a face 
    of codimension $k$ in $P_{\K}$. 
      The polyhedron $P_{\K}$ together
    with its decomposition into ``faces'' $\{ F_{\sigma} \}_{\sigma\in \K}$ 
    is called a \emph{simple polyhedral complex} in~\cite[p.428]{DaJan91}.
     \n

   Let $V(\K)$ denote the vertex set of $\K$.
    Any map $\lambda: V(\K)\rightarrow (\Z_2)^r$ is
    called a \emph{$(\Z_2)^r$-coloring of $\K$}, and any element of $(\Z_2)^r$ is called a 
    \emph{color}.
     For any
   $\sigma\in \K$, let $V(\sigma)$ denote the vertex set of 
   $\sigma$ and let $G_{\lambda}(\sigma)$ 
    denote the subgroup of $(\Z_2)^r$ generated by
   $\{ \lambda(v) \, |\, v\in V(\sigma) \}$. 
  Define
   \begin{equation}\label{Def:Real-Quotient}
     X(\K,\lambda):= P_{\K}\times (\Z_2)^r \slash \sim 
   \end{equation}
  where $(p,g)\sim (p',g')$ whenever $p'=p\in F_{\sigma}$ and $g'-g\in G_{\lambda}(\sigma)$
  for some $\sigma\in \K$. So
  when $p$ lies in the relative interior of $F_{\sigma}$, $(p,g)\sim (p',g')$ if and only if
  $p'=p$ and $g'-g\in G_{\lambda}(\sigma)$.   \n
  
  We call $\lambda$ \emph{non-degenerate} if
   $\mathrm{rank}_{\Z_2}(G_{\lambda}(\sigma))= \mathrm{rank}(\sigma)$ for any simplex
   $\sigma$ in $\K$. 
  In particular, when $r=|V(\K)|=m$ and $\{\lambda(v_i)\,; \, 1\leq i \leq m\}$ 
  is a basis of $(\Z_2)^m$,
 the space $X(\K,\lambda)$ is homeomorphic to $\R\mathcal{Z}_{\K}$.
 Let $\pi_{\K}: P_{\K}\times (\Z_2)^m \rightarrow \R\mathcal{Z}_{\K}$ be the corresponding 
 quotient map in~\eqref{Def:Real-Quotient}. 
 There is a canonical action of $(\Z_2)^m$ on $\R\mathcal{Z}_{\K}$
  defined by:
  \[  g'\cdot \pi_{\K}(p,g) = \pi_{\K} (p,g+g'),\ p\in P_{\K},\, g,g'\in (\Z_2)^m.\]
   Then any subgroup of $(\Z_2)^m$ acts canonically on $\R\mathcal{Z}_{\K}$ through this action.
      \n
      
    Let $\{e_1,\cdots, e_m\}$ be a basis of $(\Z_2)^m$.
   For a map $\lambda: V(\K)\rightarrow (\Z_2)^r$ whose image spans the whole $(\Z_2)^r$, 
   we have a short exact
   sequence of groups
   \[ 0\longrightarrow (\Z_2)^{m-r} \longrightarrow (\Z_2)^m 
             \overset{\bar{\lambda}}{\longrightarrow} (\Z_2)^r \longrightarrow 0   \] 
     where $\bar{\lambda}$ is the group homomorphism defined by 
     $\bar{\lambda}(e_i) = \lambda(v_i)$, 
     $1\leq i \leq m$. The kernel of $\bar{\lambda}$ is a subgroup of $(\Z_2)^m$, denoted by
     $H_{\lambda}$, which is isomorphic to $(\Z_2)^{m-r}$.
      It 
      is easy to see that $ X(\K,\lambda)$ is homeomorphic to the quotient space $\R\mathcal{Z}_{\K}\slash H_{\lambda}$, where $H_{\lambda}$ acts on $\R\mathcal{Z}_{\K}$ 
          via the canonical action.\n
          
       The cohomology groups of $\R\mathcal{Z}_{\K}$ 
       can be computed via a Hochster type
       formula as follows (see~\cite{Cai-13} or~\cite{LuZhi09}). 
       For any subset $J\subset [m]$, let $\K_J$ denote the full 
       subcomplex of $\K$ obtained by restricting to $J$.
       Let $\mathbf{k}$ denote a field or $\Z$ in the rest of the paper. We have      
       \begin{equation}\label{Equ:Hoch-real}
          H^q(\R\mathcal{Z}_{\K};\mathbf{k}) \cong \bigoplus_{J\subset [m]} 
       \widetilde{H}^{q-1}(\K_J;\mathbf{k}), \ q\geq 0 
       \end{equation}
      where $\widetilde{H}^{q-1}(\K_J;\mathbf{k})$ is the reduced cohomology groups of $\K_J$. 
      Here and in the rest
      we adopt the convention $\widetilde{H}^{-1}(\K_{\varnothing};\mathbf{k})=\mathbf{k}$.
     \nn
            
      There is a completely parallel
       version of the construction~\eqref{Def:Real-Quotient} for 
       any $\Z^r$-coloring $\Lambda : V(\K)\rightarrow \Z^r$ (see~\cite{DaJan91}).
       Indeed, let $\widehat{G}_{\Lambda}(\sigma)$ 
    denote the toral subgroup of $T^r$ corresponding to the subgroup of $\Z^r$
   generated by $\{ \Lambda(v) \, |\, v\in V(\sigma) \}$. 
  Define
   \begin{equation}\label{Def:Complex-Quotient}
     X(\K,\Lambda):= P_{\K}\times T^r \slash \sim 
   \end{equation}
  where $(p,g)\sim (p',g')$ whenever $p'=p\in F_{\sigma}$ and 
  $g'g^{-1}\in \widehat{G}_{\Lambda}(\sigma)$
  for some $\sigma\in \K$.   \n
  
   We call $\Lambda$ \emph{non-degenerate} if for any $\sigma\in\K$,
   the subgroup of $\Z^r$ generated by
    $\{ \Lambda(v) \, |\, v\in V(\sigma) \}$ is a direct summand of $\Z^r$ with rank equal to
    $\mathrm{rank}(\sigma)$. This implies that   
   $\dim(\widehat{G}_{\Lambda}(\sigma))= \mathrm{rank}(\sigma)$ for any simplex
   $\sigma$ in $\K$. 
   In particular, when $r=|V(\K)|=m$ and $\{\Lambda(v_i)\,; \, 1\leq i \leq m\}$ 
  is a basis of $\Z^m$,
  $X(\K,\Lambda)$ is homeomorphic to $\mathcal{Z}_{\K}$.  
  Let $\widehat{\pi}_{\K}: P_{\K}\times T^m \rightarrow \mathcal{Z}_{\K}$ be the corresponding 
 quotient map in~\eqref{Def:Complex-Quotient}. 
 There is a canonical action of $T^m$ on $\mathcal{Z}_{\K}$
  defined by:
  \[  g'\cdot \widehat{\pi}_{\K}(p,g) = \widehat{\pi}_{\K} 
        (p,gg'),\ p\in P_{\K},\, g,g'\in T^m.\]
   Then any subgroup of $T^m$ acts canonically on $\mathcal{Z}_{\K}$ through this action.\n
   
    Let $\{\hat{e}_1,\cdots, \hat{e}_m\}$ be a basis of $\Z^m$.
   For a map $\Lambda: V(\K)\rightarrow \Z^r$ whose image spans the whole $\Z^r$, 
   we have a short exact sequence
    \[ 0\longrightarrow \Z^{m-r} \longrightarrow \Z^m 
             \overset{\bar{\Lambda}}{\longrightarrow} \Z^r \longrightarrow 0   \] 
  where $\bar{\Lambda}$ is the group homomorphism defined by 
  $\bar{\Lambda}(\hat{e}_i) = \Lambda(v_i)$, $1\leq i \leq m$. 
  The kernel of $\bar{\Lambda}$ is a subgroup of $\Z^m$ which determines an $(m-r)$-dimensional 
  toral subgroup $S_{\Lambda} \subset T^m$.
          Then it is easy to see that $ X(\K,\Lambda)$ is homeomorphic to the quotient space 
          $\mathcal{Z}_{\K}\slash S_{\Lambda}$, where $S_{\Lambda}$ acts on 
          $\mathcal{Z}_{\K}$ via the canonical action.\n

 
       Similarly, there is a Hochster type formula for the cohomology 
       groups for $\mathcal{Z}_{\K}$ (see~\cite{BasKov02} or~\cite{BP02}).
       \begin{equation}\label{Equ:Hoch-complex}
          H^q(\mathcal{Z}_{\K};\mathbf{k}) = \bigoplus_{J\subset [m]} 
       \widetilde{H}^{q-|J|-1}(\K_J;\mathbf{k}), \ q\geq 0. 
       \end{equation}
       
     It is shown in~\cite{BP02} that there is a natural bigrading on
     $H^*(\mathcal{Z}_{\K};\mathbf{k})$ so that it is
    isomorphic to  $\mathrm{Tor}_{\mathbf{k}[v_1,\cdots, v_m]}(\mathbf{k}[\K];\mathbf{k})$
        as bigraded algebras where $\mathbf{k}[\K]$ is the face ring of $\K$ over $\mathbf{k}$.
     \n
      
    
     Recall that
       the \emph{face ring} (or the \emph{Stanley-Reisner ring}) of a simplicial complex $\K$
      on the set $[m]$ with $\mathbf{k}$-coefficients is the quotient ring
      \begin{equation} \label{Equ:Face-Ring-Def-1}
         \mathbf{k}(\K)=\mathbf{k}[v_1,\cdots, v_m]\slash \mathcal{I}_{\K}
       \end{equation}  
      where $\mathcal{I}_{\K}$ 
      is the ideal generated by the monomials $v_{i_1}\cdots v_{i_s}$ for which 
      $\{v_{i_1},\cdots, v_{i_s}\}$ does not span a
       simplex of $\K$. The ideal $\mathcal{I}_{\K}$ is 
      called the \emph{Stanley-Reisner ideal} of $\K$.     
     The face ring $\mathbf{k}[\K]$ can be equivalently defined as the quotient algebra 
        \begin{equation}\label{Equ:Face-Ring-Def-2}
           \mathbf{k}[\K]:= \mathbf{k}[\,\mathbf{v}_{\sigma} :\sigma\in \K\,] \slash 
          \mathcal{I}'_{\K}  
         \end{equation}  
        where $\mathcal{I}'_{\K}$ is the ideal generated by all the elements of 
        the form
        \[ \mathbf{v}_{\hat{\mathbf{0}}}-1, \ \ \mathbf{v}_{\sigma}\mathbf{v}_{\tau} - \mathbf{v}_{\sigma\wedge \tau} \cdot 
                         \mathbf{v}_{\sigma\vee\tau}, \]
       where $\sigma\wedge\tau$ denotes the maximal common face of $\sigma$ and $\tau$, 
       and $\sigma\vee\tau$ denotes the minimal face of $\K$ that contains
       both $\sigma$ and $\tau$. If there is no simplex contains both $\sigma$ and $\tau$,
      we let $\mathbf{v}_{\sigma\vee\tau}=0$. 
      This way of defining face rings is introduced by R.~Stanley in~\cite{St91} 
      in the context of simplicial posets. The definition~\eqref{Equ:Face-Ring-Def-2}
      of $\mathbf{k}[\K]$ is related to our previous definition~\eqref{Equ:Face-Ring-Def-1}
      of $\mathbf{k}[\K]$ by
      mapping $\mathbf{v}_{\sigma}$ to $\prod_{v_j\in V(\sigma)} v_j$.
      In particular, we use $\mathbf{v}_{\{j\}}$ to denote the generator corresponding to 
      the vertex $v_j$.
      We will use either definition of $\mathbf{k}[\K]$ in our argument depending on which
      one is more convenient.\n
    
   
        
     The topology of $\R\mathcal{Z}_{\K}$ and 
     $\mathcal{Z}_{\K}$ for a general finite simplicial complex $\K$
      has been extensively studied in the recent years.  
     But the study of the quotient spaces of $\R\mathcal{Z}_{\K}$ or $\mathcal{Z}_{\K}$
      by the toral subgroups of $(\Z_2)^m$ or $T^m$ are mainly done for
      $\K$ being a simplicial sphere only.
      \n
     Let $\K^P$ be a simplicial sphere that is dual to an $n$-dimensional simple
            convex polytope $P$ with $m$ facets. 
     For brevity, we denote $\mathcal{Z}_{\K^P}$ by $\mathcal{Z}_P$ and 
     $\R\mathcal{Z}_{\K^P}$ by 
     $\R\mathcal{Z}_P$. Then $\mathcal{Z}_P$ and $\R\mathcal{Z}_P$ 
      are $(m+n)$-dimensional and $n$-dimensional closed manifolds, respectively.  
     We call $\mathcal{Z}_P$ the \emph{moment-angle manifold} of $P$ 
     and $\R\mathcal{Z}_P$ the \emph{real moment-angle manifold}
      of $P$. \n
      
      $\bullet$ Suppose $H\cong (\Z_2)^{m-n}$ is a subgroup of $(\Z_2)^m$ that acts freely on
      $\R\mathcal{Z}_P$ through the canonical action, the quotient space $\R\mathcal{Z}_P\slash H$
      is called a \emph{small cover} over $P$.\n
      
      $\bullet$ Suppose $S\cong T^{m-n}$ is a subgroup of $T^m$ that acts freely on
      $\mathcal{Z}_P$ through the canonical action, the quotient space $\mathcal{Z}_P\slash S$
      is called a \emph{quasitoric manifold} over $P$.\n
      
       The topology of small covers and quasitoric manifolds are thoroughly
      studied by Davis and Januszkiewicz in~\cite{DaJan91}. Some other discussions
      on partial quotients of $\mathcal{Z}_P$ can be found in~\cite[Sec 7.5]{BP02}.
     \n      
      
      
     $\bullet$ Let $S\subset T^m$ be a $k$-dimensional subtorus which acts freely 
      $\mathcal{Z}_P$ through the canonical action. Then by choosing a basis of $\Z^n$,
      we can write $S$ in the form:
      \[ S = \{ (e^{2\pi i(a_{11}s_1+\cdots + a_{1k}s_k)}),\cdots, 
       e^{2\pi i(a_{m1}s_1+\cdots + a_{mk}s_k)}) \in T^m \} ,\]
      where $s_1, \cdots, s_k\in \R$ and $A=(a_{ij})$
      is an $m\times k$ integer matrix which defines a monomorphism $\Z^k\rightarrow \Z^m$
      whose image is a direct summand of $\Z^m$. Moreover,
       we can choose an $(m-k)\times m$ integer
      matrix $U=(u_{ij})$ of rank $(m-k)$ satisfying 
      \[ \text{$U\cdot A =0$ 
      (see~\cite[Proposition 7.34]{BP02})}.\]   
       It is shown by~\cite[Theorem 7.37]{BP02}
      that $H^*(\mathcal{Z}_P\slash S; \mathbf{k})$ is isomorphic as a graded algebra
      to $\mathrm{Tor}_{\mathbf{k}[u_1,\cdots, u_{m-k}]}(\mathbf{k}[\K^P];\mathbf{k})$, 
      where the $\mathbf{k}[u_1,\cdots, u_{m-k}]$-module structure on 
       $\mathbf{k}[\K^P]=\mathbf{k}[v_1,\cdots, v_m]\slash \mathcal{I}_{\K^P}$ is given by
      \begin{align*} 
        \mathbf{k}[u_1,\cdots, u_{m-k}] &\longrightarrow \mathbf{k}[v_1,\cdots, v_m]\\
                     u_i    \   &\mapsto    \  u_{i1}v_1+\cdots + u_{im} v_m.
      \end{align*}
     It is easy to see that
      $\mathcal{Z}_P\slash S$ is homeomorphic to the space $X(\K^P,\Lambda_S)$ 
      (see~\eqref{Def:Complex-Quotient})
      where $\Lambda_{S}(v_j)\in \Z^{m-k}$ is the $j$-th column vector of $U$
      for any $1\leq j \leq m$. \nn
      
       The proof of~\cite[Theorem 7.37]{BP02} uses the Eilenberg-Moore spectral
      sequence of fiberations over simply connected spaces. It is not 
       clear whether the argument can be used to compute the cohomology rings of
       the free quotient
      spaces of $\mathcal{Z}_{\K}$ or $\R\mathcal{Z}_{\K}$ for a general 
       simplicial complex $\K$.
      Besides, it is not known whether there exists a Hochster type formula for 
      the cohomology groups $H^*(\mathcal{Z}_P\slash S;\mathbf{k})$ as~\eqref{Equ:Hoch-complex}.\n
      
      In this paper, we will study a special class of quotient spaces of 
      $\R\mathcal{Z}_{\K}$ and $\mathcal{Z}_{\K}$ by some (not necessarily free)
      action of toral
      subgroups of $(\Z_2)^m$ and $T^m$. We will show that the
       cohomology groups of these spaces
       can be computed by Hochster type formulae which 
       generalize~\eqref{Equ:Hoch-real} and~\eqref{Equ:Hoch-complex}. These 
        spaces are defined as follows.
       \n
       
       Let $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ 
       be a \emph{partition} of the vertex set $V(\K)$
       of a simplicial complex $\K$, i.e. $\alpha_i$'s are disjoint subsets of $V(\K)$
       with $\alpha_1 \cup\cdots \cup \alpha_k=V(\K)$.\n
       
          Let $\{ e_1,\cdots,e_k\}$ be a basis of $(\Z_2)^k$ and
          $\{ \hat{e}_1,\cdots, \hat{e}_k\}$ a basis of $\Z^k$. 
          Let $\lambda_{\upalpha}$ be a $(\Z_2)^k$-coloring of $\K$ 
       defined by 
       $\lambda_{\upalpha}(\alpha_i)= e_i$, $1\leq i \leq k$. Similarly,  
       let $\Lambda_{\upalpha}$ be a $\Z^k$-coloring of $\K$ 
       defined by 
       $\Lambda_{\upalpha}(\alpha_i)= \hat{e}_i$, $1\leq i \leq k$.
       Then we obtain two spaces $X(\K,\lambda_{\upalpha})$ and 
       $X(\K,\Lambda_{\upalpha})$ which are essentially determined by
       $\K$ and $\upalpha$. We will see that many  
       properties of $X(\K,\lambda_{\upalpha})$ and 
       $X(\K,\Lambda_{\upalpha})$ are similar to $\R\mathcal{Z}_{\K}$
       and $\mathcal{Z}_{\K}$.\n
       
        For any subset
       $\mathrm{J}\subset [k]=\{ 1,\cdots, k\}$, let 
       $$G_{\mathrm{J}} := \text{the subgroup of $(\Z_2)^k$ generated by}\ \{ e_j \, |\, 
         j\in \mathrm{J}\}. $$ 
         $$\widehat{G}_{\mathrm{J}} := \text{the toral 
         subgroup of $T^k$ corresponding to the subgroup}$$
         $$ \text{of $\Z^k$ spanned by}\ \{ \hat{e}_j \, |\, 
         j\in \mathrm{J}\}.\qquad\qquad\qquad\qquad\ \, $$

       
         For any simplex $\sigma\in \K$, let
        $$ \I_{\upalpha}(\sigma) := \{ i\in [k] \,
                  ;\, V(\sigma)\cap \alpha_i \neq \varnothing \}
     \subset [k],\ \text{and let}\ k_{\upalpha}(\sigma) = |\I_{\upalpha}(\sigma)|.$$
     It is clear that $0\leq k_{\upalpha}(\sigma) \leq \mathrm{rank}(\sigma)$.
     Notice that by our notations,
      $$G_{\mathrm{I}_{\upalpha}(\sigma)} = G_{\lambda_{\upalpha}}(\sigma),\ \
       \widehat{G}_{\mathrm{I}_{\upalpha}(\sigma)} = \widehat{G}_{\Lambda_{\upalpha}}(\sigma).
        $$
    \n

     
     For any subset $\mathrm{L}\subset [k]=\{ 1,\cdots, k \}$, define
    \begin{equation} \label{Equ:K-Subcomplexes}
        \K_{\upalpha, \mathrm{L}}
      = \text{the subcomplex of $\K$ consisting of $\{ \sigma\in \K\,;\, 
     \I_{\upalpha}(\sigma)\subset \mathrm{L} \}$}.
    \end{equation} 
      
  
      The main theorems of this paper are the following.
      
      \begin{thm}\label{Thm:Main-1}
       Let $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ 
        be a partition of the vertex set of a finite simplicial complex $\K$. Then we have
       \begin{itemize}
       \item[(i)] $H^q(X(\K, \lambda_{\upalpha});\mathbf{k}) \cong \underset{\mathrm{L}
           \subset [k]}{\bigoplus} 
                   \widetilde{H}^{q-1}(\K_{\upalpha, \mathrm{L}};\mathbf{k})$ for any $q\geq 0$. 
       \item[(ii)] $H^q(X(\K, \Lambda_{\upalpha});\mathbf{k}) \cong 
          \underset{\mathrm{L}\subset [k]}{\bigoplus} \widetilde{H}^{q-|\mathrm{L}|-1} 
          (\K_{\upalpha, \mathrm{L}};\mathbf{k})$ for any $q\geq 0$. 
      \end{itemize}
      \end{thm}  
      Let $\upalpha^*$ denote the \emph{trivial partition}
       of $V(\K)$, i.e. $\upalpha^*=(\alpha_1,\cdots, \alpha_m)$
      where each $\alpha_j=\{ v_j\}$ consists of only one vertex of $\K$. Then by definition
      $X(\K,\lambda_{\upalpha^*}) = \R\mathcal{Z}_{\K}$,
       $X(\K,\Lambda_{\upalpha^*}) = \mathcal{Z}_{\K}$,
        and the formulae in Theorem~\ref{Thm:Main-1} for $\upalpha^*$
    agree with~\eqref{Equ:Hoch-real} and~\eqref{Equ:Hoch-complex}.
      \nn
      In addition, it is shown in~\cite[Corollary 2.23]{BBCG10}
       that the Hochster's formula for the
      cohomology groups of $\mathcal{Z}_{\K}$ follows from a 
      stable decomposition of $\mathcal{Z}_{\K}$. We have completely parallel 
      results for $X(\K,\lambda_{\upalpha})$
   and $X(\K,\Lambda_{\upalpha})$ below.\n
      
      
      
       \begin{thm}\label{Thm:Stable-Decomp}
      Let $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ 
        be a partition of the vertex set of a finite simplicial complex $\K$.
          There are homotopy equivalences:
         $$ \mathbf{\Sigma}( X(\K,\lambda_{\upalpha}) ) \simeq \bigvee_{\mathrm{L}\subset [k]} 
           \mathbf{\Sigma}^2( \K_{\upalpha,\mathrm{L}}),\ \ 
           \mathbf{\Sigma}( X(\K,\Lambda_{\upalpha}) ) \simeq \bigvee_{\mathrm{L}\subset [k]} 
           \mathbf{\Sigma}^{|\mathrm{L}|+2}( \K_{\upalpha,\mathrm{L}})$$
          where the bold $\mathbf{\Sigma}$ denotes the suspension. 
       \end{thm}
       
       It is easy to see that Theorem~\ref{Thm:Main-1} follows from
   the stable decompositions of $X(\K,\lambda_{\upalpha})$
   and $X(\K,\Lambda_{\upalpha})$ in Theorem~\ref{Thm:Stable-Decomp}. But 
   we can not obtain the cohomology ring structures of 
   $X(\K,\lambda_{\upalpha})$ and 
  $X(\K,\Lambda_{\upalpha})$ from Theorem~\ref{Thm:Stable-Decomp}.   \nn
      
       We call a partition $\upalpha$ of $V(\K)$ 
       \emph{non-degenerate} if for any simplex $\sigma$ of $\K$,
         $$ |\I_{\upalpha}(\sigma)|=\mathrm{rank}(\sigma).$$
     It is easy to see that the following statements are equivalent.
     \begin{itemize}
       \item $\upalpha=\{ \alpha_1,\cdots, \alpha_k\}$ is a non-degenerate partition of $V(\K)$.\n
       \item the two vertices of any $1$-simplex of $\K$ belong to different $\alpha_i$.\n
       \item The coloring $\lambda_{\upalpha}$ or $\Lambda_{\upalpha}$ on $\K$ 
               is non-degenerate.\n
      \end{itemize} 
      Note that the trivial partition $\upalpha^*$ of $V(\K)$ is always non-degenerate.  \n
      
         
     We define
       a $\mathbf{k}[u_1,\cdots,u_k]$-module structure on 
       $\mathbf{k}[\K]=\mathbf{k}[\,\mathbf{v}_{\sigma} :\sigma\in \K\,] \slash 
          \mathcal{I}'_{\K} $
       by
     \begin{equation} \label{Equ:K-alpha-def}
        \mathbf{k}[u_1,\cdots, u_k] \longrightarrow 
       \mathbf{k}[\,\mathbf{v}_{\sigma} :\sigma\in \K\,],\ \ 
       u_i \mapsto  \underset{v_j\in \alpha_i}{\sum} \mathbf{v}_{\{j\}}.
     \end{equation}  
    
     \begin{thm}\label{Thm:Main-2}
     Let $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ be a non-degenerate partition of
      the vertex set of a finite simplicial complex $\K$. Then 
      with respect to some properly defined multi-gradings, \n
          \begin{itemize}
           \item[(i)] there is an isomorphism of multigraded $\Z_2$-modules from 
               $H^*(X(\K,\lambda_{\upalpha});\Z_2)$ to 
               $\mathrm{Tor}_{\Z_2[u_1,\cdots,u_k]}(\Z_2[\K];\Z_2)$.\n
              
          \item[(ii)] there is an isomorphism of multigraded $\Z_2$-algebras from 
                   $H^*(X(\K,\Lambda_{\upalpha});\Z_2)$ to
              $\mathrm{Tor}_{\Z_2[u_1,\cdots,u_k]}(\Z_2[\K];\Z_2)$.
           \end{itemize}    
        \end{thm} 
        
   \begin{rem}
       For the trivial partition $\upalpha^*$ of $V(\K)$, the space
      $X(\K,\lambda_{\upalpha^*})= \R\mathcal{Z}_{\K}$ and 
      $X(\K,\Lambda_{\upalpha^*})= \mathcal{Z}_{\K}$, and the
    assertions in Theorem~\ref{Thm:Main-2} actually hold for any coefficients $\mathbf{k}$
    (see~\cite[Proposition 7.12]{BP02} and~\cite[Theorem 4.2]{LuZhi09}).
   \end{rem}
     
       
    
   If $\upalpha$ is a general partition of $V(\K)$ which is not necessarily non-degenerate, 
   we can describe 
   $H^*(X(\K,\lambda_{\upalpha});\Z_2)$ and $H^*(X(\K,\Lambda_{\upalpha});\Z_2)$
   in terms of some differential graded algebra determined by $\K$ and $\upalpha$ as follows. 
   Let $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]$ denote the exterior algebra
      over $\mathbf{k}$ with $k$ generators. For any partition $\upalpha$ of $V(\K)$,
    we define a differential $d_{\upalpha}$ on the tensor product
     $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K]$ 
      by:
     \begin{equation}\label{Equ:R-boundary-1}
        d_{\upalpha}(t_i) := \sum_{v_j\in \alpha_i} \mathbf{v}_{\{j\}} ,\ 1\leq i \leq n,\
           \ \text{and}
      \ \ d_{\upalpha}(\mathbf{v}_{\sigma}):= \underset{\dim(\omega) = \dim(\sigma)+1}
       {\sum_{\sigma\subset \omega,\, 
        \mathrm{I}_{\upalpha}(\omega) =\mathrm{I}_{\upalpha}(\sigma)}} 
        \varepsilon(\sigma,\omega)\cdot \mathbf{v}_{\omega}. 
     \end{equation}                 
       where $\varepsilon(\sigma,\omega)$ is the sign of $\sigma$  
    relative to the boundary of $\omega$ with respect to some chosen orientations of the 
    simplices of $\K$ (see~\eqref{Equ:Sign-Boundary}).
     Note that here and in the rest of this paper we use 
     $\otimes$ for $\otimes_{\mathbf{k}}$. 
      It is easy to see the following fact.\nn
      
    \textbf{Fact:}
       $d_{\upalpha}$ is zero on $\mathbf{k}[\K]$ if and only if $\upalpha$ is non-degenerate.
  \n
     \begin{thm}\label{Thm:Main-3}
         Let $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ be a partition of
      the vertex set of a finite simplicial complex $\K$. Then 
      with respect to some properly defined multi-gradings, \n
          \begin{itemize}
           \item[(i)] there is an isomorphism of multigraded $\Z_2$-modules from 
               $H^*(X(\K,\lambda_{\upalpha});\Z_2)$ to 
               $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K], d_{\upalpha})$.
               \n
              
          \item[(ii)] there is an isomorphism of multigraded $\Z_2$-algebras from 
                   $H^*(X(\K,\Lambda_{\upalpha});\Z_2)$ to
              $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K], d_{\upalpha})$.
           \end{itemize}    
     \end{thm}
  Note that when $\upalpha$ is non-degenerate,  
  $H^*(\Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K], d_{\upalpha})$ is isomorphic 
  to $\mathrm{Tor}_{\mathbf{k}[u_1,\cdots,u_k]}(\mathbf{k}[\K];\mathbf{k})$
  with respect to the $\mathbf{k}[u_1,\cdots,u_k]$-module structure on 
       $\mathbf{k}[\K]$ defined in~\eqref{Equ:K-alpha-def}. Indeed, we can
  compute $\mathrm{Tor}_{\mathbf{k}[u_1,\cdots,u_k]}(\mathbf{k}[\K];\mathbf{k})$ 
  via the \emph{Koszul resolution} of $\mathbf{k}$ (see~\cite[Sec 3.4]{BP02}).
  Recall that    
     the \emph{Koszul resolution} of $\mathbf{k}$ is:
 \[ 0\longrightarrow  \Lambda^k_{\mathbf{k}}[t_1,\cdots, t_k]\otimes\mathbf{k}[u_1,\cdots, u_k]
 \longrightarrow \cdots \qquad\qquad\qquad\qquad\qquad\qquad
  \]
  \[\qquad\qquad\qquad \ \longrightarrow \Lambda^1_{\mathbf{k}}[t_1,\cdots, t_k]\otimes\mathbf{k}[u_1,\cdots, u_k]
  \longrightarrow \mathbf{k}[u_1,\cdots, u_k] \longrightarrow \mathbf{k} \longrightarrow 0, \]
  where $d (t_i)=u_i$ and $d (u_i)=0$ for any $1\leq i \leq k$.
  Then we obtain
   \begin{align*}
    \mathrm{Tor}_{\mathbf{k}[u_1,\cdots, u_k]}(\mathbf{k}[\K],\mathbf{k})
     & \cong \mathrm{Tor}_{\mathbf{k}[u_1,\cdots, u_k]}(\mathbf{k}, \mathbf{k}[\K])\\
     &\cong 
       H^*\big( \big(\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]\otimes\mathbf{k}[u_1,\cdots, u_k]\big)
      \otimes_{\mathbf{k}[u_1,\cdots, u_k]}  \mathbf{k}[\K] \big)\\
     & \cong H^*\big(\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]\otimes \mathbf{k}[\K] \big)
   \end{align*}
  When $\upalpha$ is non-degenerate, the
    differential on $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]\otimes \mathbf{k}[\K]$  
  induced from 
  $ \mathrm{Tor}_{\mathbf{k}[u_1,\cdots, u_k]}(\mathbf{k}, \mathbf{k}[\K])$ is exactly 
  given by $d_{\upalpha}$ (which is zero on $\mathbf{k}[\K]$ in this case). 
 So Theorem~\ref{Thm:Main-2} is actually a corollary of Theorem~\ref{Thm:Main-3}.
  \n
  
       
      Our discussion of the cohomology rings of
         $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$
        generalizes the discussion of (real) moment-angle complexes
        in~\cite{BBCG10, BasKov02, BP02, Cai-13, LuZhi09, LuPanov11, Panov08} in several aspects.
        The reader is referred to~\cite{BP02}, \cite{BP13} and~\cite{Panov08} for a
         comprehensive introduction to moment-angle complexes and related topics. 
               \n
      
      The paper is organized as follows. In section 2, we construct some
      natural cell decompositions of $X(\K,\lambda_{\upalpha})$ and 
       $X(\K, \Lambda_{\upalpha})$. Then in section 3 we use these cell decompositions to
       compute the the cohomology groups of $X(\K,\lambda_{\upalpha})$ and 
       $X(\K, \Lambda_{\upalpha})$ and the give a proof of
        Theorem~\ref{Thm:Main-1}. In section 4, we use
        the strategies in~\cite{BBCG10} to study the stable decompositions of
         $X(\K,\lambda_{\upalpha})$ and 
       $X(\K, \Lambda_{\upalpha})$, which leads to a proof of Theorem~\ref{Thm:Stable-Decomp}.
         In section 5, we 
       build some connections between the differential algebra
       $(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K], d_{\upalpha})$ 
       and the cellular cochain complexes of $X(\K,\lambda_{\upalpha})$ and
       $X(\K, \Lambda_{\upalpha})$, which imply some
        multigraded additive isomorphisms from 
        $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K], d_{\upalpha})$ to
         $H^*(X(\K,\lambda_{\upalpha});\Z_2)$ and  $H^*(X(\K,\Lambda_{\upalpha});\Z_2)$. 
       In section 6, we study the cohomology ring structures of 
       $X(\K,\lambda_{\upalpha})$ and
       $X(\K, \Lambda_{\upalpha})$ and finish the proof of Theorem~\ref{Thm:Main-3}. 
       In addition, we show that the moment-angle complex of any finite 
       simplicial poset is homotopy
       equivalent to $X(\K, \Lambda_{\upalpha})$ for some simplicial complex $\K$ and 
       partition $\upalpha$ of $V(\K)$.
       In the finial section 7, we generalize the main results of this paper
       to a wider range of spaces.\\
       
     
   \section{Cell decompositions of $X(\K,\lambda_{\upalpha})$ and 
       $X(\K, \Lambda_{\upalpha})$} 
             
     For any partition $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ 
       of the vertex set of a simplicial complex $\K$, 
       we construct some cell decompositions of $X(\K,\lambda_{\upalpha})$ and 
       $X(\K, \Lambda_{\upalpha})$ in this section which will be used
       for the calculation of their cohomology groups later.
      Although the cell decompositions of
         $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$ 
       are quite similar in taste, there are still some differences 
       which make it hard to treat them in a uniform way. 
       So we construct the cell decompositions of
         $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$
          in two parallel subsections below, which helps the reader to see where the difference
          lies. \n
     
     \subsection{A cell decomposition of $X(\K,\lambda_{\upalpha})$}
    \ \n
      First of all, let 
      $\pi_{\upalpha}: P_{\K} \times (\Z_2)^k \rightarrow  
     X_{(\K,\lambda_{\upalpha})}$ 
     be the quotient map  
     in~\eqref{Def:Real-Quotient}. For any $p\in P_{\K}$ and $g\in (\Z_2)^k$, let
     $[(p,g)]$ denote the equivalence class of $(p,g)$ with respect to the $\sim$ 
     in~\eqref{Def:Real-Quotient}. \n
     
          By~\cite[Construction 4.9]{BP02}, $P_{\K}$ has a natural cubical 
      cell decomposition
      $$ P_{\K} = \bigcup_{\sigma\in\K} C_{\sigma}$$
      where $C_{\sigma}$ is the cone of the barycentric subdivision of 
      $\sigma$, and
        $\dim(C_{\sigma})=\mathrm{rank}(\sigma)$. 
        The boundary of $C_{\sigma}$
     consists of $\sigma$
     and $\{ C_{\tau}\,|\, \tau\leq \sigma, \mathrm{rank}(\tau)
       =\mathrm{rank}(\sigma)-1 \}$. In particular, 
       $C_{\hat{\mathbf{0}}}$ be the cone point
       of $P_{\K}$. In addition, we let $\mathrm{Cone}(\K)$
       be the cone of $\K$ as a simplicial complex where the cone point is denoted by $v_0$. 
       Then
       \[  \mathrm{Cone}(\K) = \bigcup_{\sigma\in\K} \mathrm{Cone}(\sigma), \]
      where $\mathrm{Cone}(\sigma)$ is the simplex with vertex set $V(\sigma)\cup \{v_0\}$.
      It is clear that $\mathrm{Cone}(\sigma)$ agrees with $C_{\sigma}$ 
      as a set, but they have different
      combinatorial structures.\n
      
       Let $b_{\sigma}$ be the barycenter  of $\sigma$ which lies in
     the relative interior of $F_{\sigma}$. For any $g\in (\Z_2)^k$,
      the equivalence class of $[(b_{\sigma},g)]$ consists of 
      $2^{k_{\upalpha}(\sigma)}$ elements in $P_{\K} \times (\Z_2)^k$.
        More specifically,
       $[(b_{\sigma},g)]=\{ (b_{\sigma},g+h) \,|\,
        h\in  G_{\mathrm{I}_{\upalpha}(\sigma)}\} \subset P_{\K} \times (\Z_2)^k$.
      So $\pi_{\upalpha}( b_{\sigma} \times (\Z_2)^k )$
         consists of 
       exactly $2^{k-k_{\upalpha}(\sigma)}$ points which form
       an orbit of the canonical $(\Z_2)^k$-action on $X(\K,\lambda_{\upalpha})$.
        Indeed, the points in $\pi_{\upalpha}( b_{\sigma} \times (\Z_2)^k )$
        are indexed by all the elements of 
       $G_{[k]\backslash \mathrm{I}_{\upalpha}(\sigma)}$ where 
     $g\in G_{[k]\backslash \mathrm{I}_{\upalpha}(\sigma)}$ corresponds to
        $$ b_{(\sigma,g)} := g\cdot \pi_{\upalpha}(b_{\sigma}\times \{ 0\}). $$
    For any $\sigma\in \K$ and $g\in G_{[k]\backslash \mathrm{I}_{\upalpha}(\sigma)}$,
    we define a subset $\mathbf{B}_{(\sigma,g)} \subset X(\K,\lambda_{\upalpha})$ by
   \begin{equation}\label{Equ:B-cells}
       \mathbf{B}_{(\sigma,g)} =
       \bigcup_{h\in G_{\mathrm{I}_{\upalpha}(\sigma)}} 
           \pi_{\upalpha}(C_{\sigma}\times \{ g+h\})
       = \bigcup_{h\in G_{\mathrm{I}_{\upalpha}(\sigma)}} 
           g\cdot\pi_{\upalpha}(C_{\sigma}\times \{h\}). 
    \end{equation}
   So $\mathbf{B}_{(\sigma,g)}$ is a union of $2^{k_{\upalpha}(\sigma)}$
    cubes of dimension $\mathrm{rank}(\sigma)$.\nn
     
   \noindent \textbf{Claim:}
       $\mathbf{B}_{(\sigma,g)}$ is homeomorphic to
    a closed ball of dimension $\mathrm{rank}(\sigma)$ 
     which contains $b_{(\sigma,g)}$ in its interior.  \nn
      
   Note that
    $\mathbf{B}_{(\sigma,g)} = g\cdot \mathbf{B}_{(\sigma,0)}$ for any 
    $\sigma \in \K$, $g\in G_{[k]\backslash \mathrm{I}_{\upalpha}(\sigma)}$. So
    we only need to examine $\mathbf{B}_{(\sigma,0)}$.     
    Note that when $\sigma=\hat{\mathbf{0}}$, $\mathbf{B}_{(\hat{\mathbf{0}},0)}$ is just 
    the cone point of
     $P_{\K}\times\{ 0\}$.
    Now assume $V(\sigma)=\{ v_1,\cdots, v_n\} \subset V(\K)$.
     For each $1\leq j \leq n$, let $f_j$ be the unique cubic
     facet (codimension-one face) of $C_{\sigma}$ containing both $b_{\sigma}$ and $v_j$.
    The $(\Z_2)^k$-coloring $\lambda_{\upalpha}$ of $\K$ induces a map
       $$ \lambda^{\sigma}_{\upalpha}: \{ f_1,\cdots, f_n \} 
       \longrightarrow 
        G_{\mathrm{I}_{\upalpha}(\sigma)} \subset (\Z_2)^k,\
        \text{where $ \lambda^{\sigma}_{\upalpha}(f_j) = 
          \lambda_{\upalpha}(v_j)$, $1\leq j \leq n$.}$$      
     Then similarly to the construction~\eqref{Def:Real-Quotient},
     we define a space $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$ 
    by              
     \begin{equation} \label{Equ:M-Construc}
       M(C_{\sigma}, \lambda^{\sigma}_{\upalpha}) := 
     C_{\sigma}\times G_{\mathrm{I}_{\upalpha}(\sigma)} \slash \sim 
    \end{equation}  
     where $(y,h)\sim (y',h')$ whenever $y'=y\in f_j$ and 
     $h'-h=\lambda^{\sigma}_{\upalpha}(f_j)$ for some 
     $1\leq j \leq n$. Note that the facets of $C_{\sigma}$ other than
     $f_1,\cdots, f_n$ are 
     not glued under the relation $\sim$ in~\eqref{Equ:M-Construc}.
     Let $[(y,h)]$ denote the equivalence class of 
      $(y,h)$ in~\eqref{Equ:M-Construc}. There is 
      a natural action of $G_{\mathrm{I}_{\upalpha}(\sigma)}$
      on $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$ by:
      \[ h'\cdot [(y,h)] = [(y, h+h')], \ y\in C_{\sigma},\, 
      h, h'\in G_{\mathrm{I}_{\upalpha}(\sigma)}.  \]
    Then we can define a natural equivariant homeomorphism
      $$ \Theta_{\sigma}: M(C_{\sigma}, \lambda^{\sigma}_{\upalpha}) \longrightarrow  
     \mathbf{B}_{(\sigma,0)} = \bigcup_{h\in G_{\mathrm{I}_{\upalpha}(\sigma)}} 
     \pi_{\upalpha}(C_{\sigma}\times \{ h\}) $$ 
    with respect to the natural actions of $G_{\mathrm{I}_{\upalpha}(\sigma)}$
    on $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$ and $\mathbf{B}_{(\sigma,0)}$ 
     where $$ \Theta_{\sigma}([(y,h)])=  \pi_{\upalpha}(y , h), \ 
     y\in C_{\sigma},\, h\in G_{\mathrm{I}_{\upalpha}(\sigma)}. $$
     So it amounts to show that $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$
      is homeomorphic to a closed ball.\n
      
      In the rest we will use the following notations.
     \begin{itemize}
       \item Let $\sigma_{(i)}$ be the face of $\sigma$ spanned by the vertices 
          $V(\sigma)\cap \alpha_i$. So we have
           $$ \sum_{i\in\I_{\upalpha}(\sigma)} \mathrm{rank}(\sigma_{(i)}) = \mathrm{rank}(\sigma). $$
          
      \item For any $r\geq 0$, let $\Delta^r$ denote the standard $r$-simplex in $\R^r$.\n    
      \end{itemize} 
      
       Now we define 
      $$ \mathbf{f}^{\sigma}_i = \bigcup_{v_j\in V(\sigma)\cap \alpha_i} f_j, \ 
                                  \forall   i\in \mathrm{I}_{\upalpha}(\sigma).$$
        Notice that all the components in 
      $\mathbf{f}^{\sigma}_i$ are assigned the same
      value by $\lambda^{\sigma}_{\upalpha}$. 
      We readily think of each $\mathbf{f}^{\sigma}_i$ as a ``big''
      facet of $C_{\sigma}$. It is shown in~\cite[Theorem 3.7]{Yu2011} that $C_{\sigma}$
      with these
       ``big'' facets $\{\mathbf{f}^{\sigma}_i;\, i \in \mathrm{I}_{\upalpha}(\sigma)\}$ 
       is still a nice manifold with
      corners, denoted by $Q^{\sigma}_{\upalpha}$, 
      which is combinatorially equivalent to a product
      of simplices. More precisely, we have
            \[ Q^{\sigma}_{\upalpha} \cong \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} 
        \mathrm{Cone}(\sigma_{(i)}). \]
        \n       
       
      The barycenter $b_{\sigma}$ of $\sigma$ lies in the relative interior of 
      the face $\underset{i\in \mathrm{I}_{\upalpha}(\sigma)}{\bigcap}  \mathbf{f}^{\sigma}_i$
      of $Q^{\sigma}_{\upalpha}$.  
      Moreover, the map $\lambda^{\sigma}_{\upalpha}$
       induces a map
       $$ \tilde{\lambda}^{\sigma}_{\upalpha}: \{ \mathbf{f}^{\sigma}_i\, ;\, 
               i\in \mathrm{I}_{\upalpha}(\sigma) \} 
       \longrightarrow 
        G_{\mathrm{I}_{\upalpha}(\sigma)}  \subset (\Z_2)^k,\ 
        \text{where $\tilde{\lambda}^{\sigma}_{\upalpha}(\mathbf{f}^{\sigma}_i) 
             = \lambda_{\upalpha}(\alpha_i)= e_i$, $i\in \mathrm{I}_{\upalpha}(\sigma)$.}$$   
     Then we can construct a space $M(Q^{\sigma}_{\upalpha}, \tilde{\lambda}^{\sigma}_{\upalpha})$ 
     in the same way as~\eqref{Equ:M-Construc} by
     \begin{equation} \label{Equ:M-Construct-red}
         M(Q^{\sigma}_{\upalpha}, \tilde{\lambda}^{\sigma}_{\upalpha}) = 
           Q^{\sigma}_{\upalpha}\times G_{\mathrm{I}_{\upalpha}(\sigma)} \slash \sim 
     \end{equation}    
     where $(y,h)\sim (y',h')$ whenever $y'=y\in \mathbf{f}^{\sigma}_i$ and 
     $y'-y=\tilde{\lambda}^{\sigma}_{\upalpha}(\mathbf{f}^{\sigma}_i)$ for some 
     $i\in \mathrm{I}_{\upalpha}(\sigma)$. Then
     $M(Q^{\sigma}_{\upalpha}, \tilde{\lambda}^{\sigma}_{\upalpha})$ is obviously homeomorphic to
     $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$.
     Moreover, it is easy
      to see that $M(Q^{\sigma}_{\upalpha}, \tilde{\lambda}^{\sigma}_{\upalpha})$
      is homeomorphic to a product of closed balls
      \begin{equation} \label{Equ:Cells-real}
            M(Q^{\sigma}_{\upalpha}, \tilde{\lambda}^{\sigma}_{\upalpha}) \cong
      \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^0*\sigma_{(i)}, 
        \end{equation}
       where $S^0*\sigma_{(i)}$ is the the join of $S^0$ with 
       $\sigma_{(i)}$ (i.e. the suspension of $\sigma_{(i)}$).  Indeed,
       $S^0*\sigma_{(i)}$ is obtained by applying the rule in~\eqref{Equ:M-Construct-red}
        to $\mathrm{Cone}(\sigma_{(i)})$ with $\sigma_{(i)}$ 
       colored by $\lambda(\alpha_i)$.
       For example, Figure~\ref{p:Smoothing-Faces} 
       shows a $2$-simplex $\sigma$ whose vertices $v_1,v_2,v_3$ are colored
       by $e_1$ and $e_2$, and we have $Q^{\sigma}_{\upalpha}\cong \Delta^1\times\Delta^2$,
       $ M(Q^{\sigma}_{\upalpha}, \tilde{\lambda}^{\sigma}_{\upalpha}) \cong 
       (S^0*\Delta^1) \times (S^0*\Delta^2)$. \n
             
      Since $\dim( S^0*\sigma_{(i)})=\mathrm{rank}(\sigma_{(i)})$,  the space    
       $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$ is homeomorphic to
      a closed ball of dimension $\mathrm{rank}(\sigma)$. The claim is proved. \qquad $\qed$\n
      \begin{figure}
         \includegraphics[width=0.9\textwidth]{Smoothing-Faces.eps}\\
          \caption{} 
          \label{p:Smoothing-Faces}
      \end{figure}
    
     Let $B_{(\sigma,g)}$ be the relative interior of $\mathbf{B}_{(\sigma,g)}$
     in $X(\K,\lambda_{\upalpha})$. Define
   \begin{equation} \label{Equ:Cell-Decomp-1}
        \mathscr{B}_{\upalpha}(\K):=\{ B_{(\sigma,g)} \, |\, \sigma\in \K,\
       g\in G_{[k]\backslash \mathrm{I}_{\upalpha}(\sigma)}\}.  
   \end{equation}  
      By our construction, 
   \[ X(\K,\lambda_{\upalpha})=
    \underset{g \in G_{[k]\backslash \mathrm{I}_{\upalpha}(\sigma)}}{\bigcup_{\sigma\in \K}}  
      B_{(\sigma,g)},\ \ \dim(B_{(\sigma,g)})=\mathrm{rank}(\sigma). \]
   
    So $\mathscr{B}_{\upalpha}(\K)$ 
     is a cell decomposition of $X(\K,\lambda_{\upalpha})$. 
    Let $C_*(X(\K,\lambda_{\upalpha});\mathbf{k})$ be the cellular chain complex corresponding to
    $\mathscr{B}_{\upalpha}(\K)$.
    For any $\J\subset [k]\backslash \mathrm{I}_{\upalpha}(\sigma)$, let
    \begin{equation}\label{Equ:Cell-Decomp-1-2}
    \mathbf{B}_{(\sigma,\J)}:= \bigcup_{g\in G_{\J}} \mathbf{B}_{(\sigma,g)} ,\ \ 
      B_{(\sigma,\J)}:= \bigcup_{g\in G_{\J}} B_{(\sigma,g)}.  
    \end{equation}  
    For any fixed $\sigma\in \K$, 
     $\{ B_{(\sigma,\J)} \, |\, \J \subset [k]\backslash \mathrm{I}_{\upalpha}(\sigma)\}$
    is a basis of the $\mathbf{k}$-submodule of
    $C_*(X(\K,\lambda_{\upalpha});\mathbf{k})$ spanned by
    $\{ B_{(\sigma,g)} \, |\, g\in G_{[k]\backslash \mathrm{I}_{\upalpha}(\sigma)}\}$.
    In the following sections, we use
     $\{ B_{(\sigma,\J)} \, |\, \sigma\in\K,\
        \J \subset [k]\backslash \mathrm{I}_{\upalpha}(\sigma)\}$ as the basis of
       $C_*(X(\K,\lambda_{\upalpha});\mathbf{k})$.\nn
       
       Notice that $\mathbf{B}_{(\sigma,\J)}\subset  
       \mathbf{B}_{(\sigma,\J')}$ and $B_{(\sigma,\J)}\subset  B_{(\sigma,\J')}$ for any
        $\J \subset \J' \subset [k]\backslash \mathrm{I}_{\upalpha}(\sigma) $. So we have
        the decomposition of $X(\K,\lambda_{\upalpha})$ (as a set)
         \[ X(\K,\lambda_{\upalpha})= 
         \bigcup_{\sigma\in \K} 
         \mathbf{B}_{(\sigma, [k]\backslash \mathrm{I}_{\upalpha}(\sigma))}=
    \bigcup_{\sigma\in \K} B_{(\sigma, [k]\backslash \mathrm{I}_{\upalpha}(\sigma))}.\]
    \n   
     
          
    \subsection{A cell decomposition of 
     $X(\K, \Lambda_{\upalpha})$}\ \n
     
   We will see that the construction of the cell decomposition of 
     $X(\K, \Lambda_{\upalpha})$ is very similar to $X(\K, \lambda_{\upalpha})$ but with
     an extra ingredient. First of all, we put a standard cell decomposition
     on $T^k$ for any $k\geq 0$. The circle $S^1=\{ z\in\mathbb{C}\,; |z|=1 \}$ has a natural
     cell decomposition $\{ e^0,e^1\}$ where $e^0=\{1\}\in S^1$ and $e^1=S^1\backslash e^0$.
     Then consider $T^k$ as a product of $k$ copies of $S^1$ and
     equip $T^k$ with the product cell structure (see~\cite[3.B]{Hatcher}).
     All the cells in $T^k$ can be indexed by subsets of $[k]=\{ 1,\cdots, k\}$. 
     Indeed, any $\J\subset [k]$ determines a unique cell $U_{\J}$ in $T^k$ where
     \[ U_{\J} =  \prod_{i\in \J} e^1_i \times \prod_{i\in [k]\backslash \J} e^0_i,\ 
     \dim(U_{\J})=|\J|. \]
     Here $\{ e^0_i, e^1_i\}$ denote the cells in the $i$-th $S^1$-factor of $T^k$, 
     $1\leq i \leq k$.  Let 
      $$\widehat{\pi}_{\upalpha}: P_{\K} \times T^k \rightarrow  
     X_{(\K, \Lambda_{\upalpha})}$$ 
     be the quotient map in~\eqref{Def:Real-Quotient}.
     For any simplex $\sigma\in \K$ and 
     $\J\subset [k]\backslash \mathrm{I}_{\upalpha}(\sigma)$,
    let
   \begin{equation}\label{Equ:B-cells-2}
       \widehat{\mathbf{B}}_{(\sigma,\J)} =
           \widehat{\pi}_{\upalpha}\big(C_{\sigma}\times 
           (\widehat{G}_{\I_{\upalpha}(\sigma)}\times U_{\J})\big)
   \subset X(\K,\Lambda_{\upalpha}).
    \end{equation}
  From the definition of $X_{(\K, \Lambda_{\upalpha})}$,
  we can easily see that $\widehat{\mathbf{B}}_{(\sigma,\J)}$ is homeomorphic to
  $U_{\J}\times \widehat{\mathbf{B}}_{(\sigma,\varnothing)}$. So to understand 
  $\widehat{\mathbf{B}}_{(\sigma,\J)}$, it amounts to understand 
  $\widehat{\mathbf{B}}_{(\sigma,\varnothing)}$.
  Note that when $\sigma=\hat{\mathbf{0}}$, the space 
  $\widehat{\mathbf{B}}_{(\hat{\mathbf{0}},\varnothing)}$ is just a point.
     Assume $V(\sigma)=\{ v_1,\cdots, v_n \}\subset V(\K)$ and use the previous notations 
     for the faces of the cube $C_{\sigma}$.
      The $\Z^k$-coloring $\Lambda_{\upalpha}$ of $\K$ induces a map
       $$ \Lambda^{\sigma}_{\upalpha}: \{ f_1,\cdots, f_n \} 
       \longrightarrow  \Z^k,\
        \text{where $ \Lambda^{\sigma}_{\upalpha}(f_j) 
        = \Lambda_{\upalpha}(v_j)$, $1\leq j \leq n$.}$$   
     Then parallel to the space $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$,
     we obtain a space $M(C_{\sigma}, \Lambda^{\sigma}_{\upalpha})$ by replacing
     $G_{\mathrm{I}_{\upalpha}(\sigma)}$ by $\widehat{G}_{\mathrm{I}_{\upalpha}(\sigma)}$
     and $\lambda^{\sigma}_{\upalpha}$ by $\Lambda^{\sigma}_{\upalpha}$ 
     in~\eqref{Equ:M-Construc}. 
     It is clear that $M(C_{\sigma}, \Lambda^{\sigma}_{\upalpha})$ is homeomorphic to
     $\widehat{\mathbf{B}}_{(\sigma,\varnothing)}$. Moreover, on $Q^{\sigma}_{\upalpha}$ the map 
     $\Lambda^{\sigma}_{\upalpha}$
     induces a map 
      $$ \tilde{\Lambda}^{\sigma}_{\upalpha}: \{ \mathbf{f}^{\sigma}_i\, ;\, 
               i\in \mathrm{I}_{\upalpha}(\sigma) \} 
       \longrightarrow  \Z^k,\ 
        \text{where $\tilde{\Lambda}^{\sigma}_{\upalpha}(\mathbf{f}^{\sigma}_i) 
             = \Lambda_{\upalpha}(\alpha_i)= \hat{e}_i$, 
              $i\in \mathrm{I}_{\upalpha}(\sigma)$.}$$   
     Then parallel to the space $M(Q^{\sigma}_{\upalpha}, \tilde{\lambda}^{\sigma}_{\upalpha})$, 
     we obtain a space 
     $M(Q^{\sigma}_{\upalpha}, \tilde{\Lambda}^{\sigma}_{\upalpha})$ 
     by replacing $G_{\mathrm{I}_{\upalpha}(\sigma)}$ by 
      $\widehat{G}_{\mathrm{I}_{\upalpha}(\sigma)}$
     and $\tilde{\lambda}^{\sigma}_{\upalpha}$ by $\tilde{\Lambda}^{\sigma}_{\upalpha}$
    in~\eqref{Equ:M-Construct-red}. 
    Similarly, we can show that 
     \begin{equation} \label{Equ:Cells-Complex}
         M(Q^{\sigma}_{\upalpha}, \tilde{\Lambda}^{\sigma}_{\upalpha}) 
       \cong \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^1* \sigma_{(i)}. 
     \end{equation}  
     Since
        $\dim(S^1*\sigma_{(i)})=\mathrm{rank}(\sigma_{(i)})+1$,
       $\widehat{\mathbf{B}}_{(\sigma,\varnothing)}$ is homeomorphic to
      a closed ball of dimension $\mathrm{rank}(\sigma)+|\I_{\upalpha}(\sigma)|$.
     \n
      
    For any subset $\J\subset 
      [k]\backslash \I_{\upalpha}(\sigma)$, let 
      $\widehat{B}_{(\sigma,\J)}$ be the relative interior of 
      $\widehat{\mathbf{B}}_{(\sigma,\J)}$.
      So
      $\widehat{B}_{(\sigma,\J)} \cong U_{\J}\times \widehat{B}_{(\sigma,\varnothing)}$
      is an open ball of dimension
    $\mathrm{rank}(\sigma)+|\I_{\upalpha}(\sigma)|+|\J|$. Define
       \begin{equation} \label{Equ:Cell-Decomp-Complex}
        \widehat{\mathscr{B}}_{\upalpha}(\K):=\{ \widehat{B}_{(\sigma,\J)} \, |\, \sigma\in \K,\
    \J \subset [k]\backslash \mathrm{I}_{\upalpha}(\sigma)\}    
   \end{equation} 
   From the definition of $\widehat{\mathbf{B}}_{(\sigma,\J)}$, it is clear that
       \[ X(\K,\Lambda_{\upalpha})=
    \underset{\J \subset [k]\backslash \mathrm{I}_{\upalpha}(\sigma)}{\bigcup_{\sigma\in \K}}  
    \widehat{B}_{(\sigma,\J)}. \]
    So $\widehat{\mathscr{B}}_{\upalpha}(\K)$ is a cell decomposition of 
    $X(\K,\Lambda_{\upalpha})$.
    \\
     
         
     
  \section{Cohomology groups of $X(\K,\lambda_{\upalpha})$ and 
     $X(\K, \Lambda_{\upalpha})$}
    For any coefficients $\mathbf{k}$,
    let $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ and $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$
     be the 
     cellular cochain complexes corresponding to the
      cell decompositions $\mathscr{B}_{\upalpha}(\K)$ 
     and $\widehat{\mathscr{B}}_{\upalpha}(\K)$ defined in~\eqref{Equ:Cell-Decomp-1}
     and~\eqref{Equ:Cell-Decomp-Complex}, respectively. 
     In this section we compute 
     $H^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ and $H^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$. 
     First, we give a new description of $X(\K,\lambda_{\upalpha})$ and 
     $X(\K,\Lambda_{\upalpha})$. \n
     
     Suppose the vertex set of $\K$ is $[m]=\{ v_1,\cdots, v_m\}$. Let 
     $\Delta^{[m]}$ be the $m$-simplex with vertex set $[m]$. For a partition
     $\upalpha=\{ \alpha_1,\cdots, \alpha_k \}$ of $[m]$, let
     $\Delta^{\alpha_i}$ denote the face of $\Delta^{[m]}$ whose 
     vertex set is $\alpha_i$. Then $\K$ 
     is a simplicial subcomplex of $\Delta^{[m]}$. 
      Let $b_{\Delta^{[m]}}$ be the barycenter of $\Delta^{[m]}$ and
     $C_{\Delta^{[m]}}=P_{\Delta^{[m]}}$ be the cone of the barycentric subdivision of $\Delta^{[m]}$ which
     is an $m$-dimensional cube.  For each $1\leq j \leq m$,   
     let $f_j$ be the unique cubic facet of $C_{\Delta^{[m]}}$ containing both 
     $b_{\Delta^{[m]}}$ and $v_j$. 
     For any partition $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ of 
     $\{ v_1,\cdots, v_m\}$, 
     \[ \text{define}\ \, \mathbf{f}^{\Delta^{[m]}}_i := 
        \bigcup_{v_j\in \alpha_i} f_j, \  i\in [k].
        \]
      Then by the same argument as in the previous section,
      $C_{\Delta^{[m]}}$ with these
       ``big'' facets $\{\mathbf{f}^{\Delta^{[m]}}_i;\, i \in [k]\}$ 
       is a nice manifold with
      corners, denoted by $Q^{\Delta^{[m]}}_{\upalpha}$, which
      is combinatorially equivalent to a product of simplices
      $\prod_{i\in [k]} \Delta^{\alpha_i}$.\n
      
      Moreover, we can also consider
     $\lambda_{\upalpha}$ and $\Lambda_{\upalpha}$ as a $(\Z_2)^k$-coloring and $\Z^k$-coloring
     of $\Delta^{[m]}$, respectively.
     Then we have the following maps
       $$ \lambda^{\Delta^{[m]}}_{\upalpha}: \{ f_1,\cdots, f_n \} 
       \longrightarrow  (\Z_2)^k,\
        \text{where $ \lambda^{\Delta^{[m]}}_{\upalpha}(f_j) = \lambda_{\upalpha}(v_j)$, 
           $1\leq j \leq m$.}$$  
       $$ \Lambda^{\Delta^{[m]}}_{\upalpha}: \{ f_1,\cdots, f_n \} 
       \longrightarrow  \Z^k,\
        \text{where $ \Lambda^{\Delta^{[m]}}_{\upalpha}(f_j) = \Lambda_{\upalpha}(v_j)$, 
           $1\leq j \leq m$.}$$               
     Then parallel to $M(C_{\sigma}, \lambda^{\sigma}_{\upalpha})$ and 
     $M(C_{\sigma}, \Lambda^{\sigma}_{\upalpha})$, we obtain homeomorphisms:
     \begin{equation} \label{Equ:Universal-Space}
       M(C_{\Delta^{[m]}}, \lambda^{\Delta^{[m]}}_{\upalpha})\cong \prod_{i\in [k]} 
           S^0*\Delta^{\alpha_i},\ \ M(C_{\Delta^{[m]}}, \Lambda^{\Delta^{[m]}}_{\upalpha})\cong 
     \prod_{i\in [k]} S^1*\Delta^{\alpha_i}.
     \end{equation}
    By definition, we clearly have homeomorphisms
     $$X(\Delta^{[m]},\lambda_{\upalpha}) \cong  M(C_{\Delta^{[m]}}, 
            \lambda^{\Delta^{[m]}}_{\upalpha}),\ \
     X(\Delta^{[m]},\Lambda_{\upalpha}) \cong  M(C_{\Delta^{[m]}}, 
         \Lambda^{\Delta^{[m]}}_{\upalpha}).$$
         Note 
          $\dim\big(X(\Delta^{[m]},\lambda_{\upalpha})\big) =
     |\alpha_1| + \cdots + |\alpha_k| = m$ and $\dim\big(
     X(\Delta^{[m]},\Lambda_{\upalpha})\big)= m+k$.\n
     
      Since $\K$ is a subcomplex of $\Delta^{[m]}$, 
     $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$ are
     subspaces of $M(C_{\Delta^{[m]}}, \lambda^{\Delta^{[m]}}_{\upalpha})$
     and $M(C_{\Delta^{[m]}}, \Lambda^{\Delta^{[m]}}_{\upalpha})$, respectively.
     Indeed, the cubical complex $P_{\K}$ naturally embeds into the cube $P_{\Delta^{[m]}}$ 
     (see~\cite[4.2]{BP02}), which gives the following commutative diagrams     
     \begin{equation}\label{Equ:Pull-Back}
        \xymatrix{
            X(\K,\lambda_{\upalpha}) \ar[r] \ar[d] &
             X(\Delta^{[m]},\lambda_{\upalpha})
            \ar[d] \\
            P_{\K} \ar[r] &  P_{\Delta^{[m]}}
        } \quad \ \
         \xymatrix{
            X(\K,\Lambda_{\upalpha}) \ar[r] \ar[d] &
             X(\Delta^{[m]},\Lambda_{\upalpha})
            \ar[d] \\
            P_{\K} \ar[r] &  P_{\Delta^{[m]}}
        }
     \end{equation}
  where the horizontal arrows are embeddings, while the vertical ones are orbit maps
  for the $(\Z_2)^k$ and $T^k$ actions, respectively.\n
  
   Note that for any simplex
  $\sigma\in \K$, $\sigma_{(i)} =\sigma\cap \Delta^{\alpha_i}$. So we can write
  $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$ as:
  \begin{equation} \label{Equ:Decomp-Real-form}
       X(\K,\lambda_{\upalpha}) = \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^0_{(i)}*(\sigma\cap \Delta^{\alpha_i}) 
       \times \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} S^0_{(i)} \Big),
     \end{equation}    
  \begin{equation} \label{Equ:Decomp-Complex-form}
      X(\K,\Lambda_{\upalpha}) = \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^1_{(i)}*(\sigma\cap \Delta^{\alpha_i}) 
        \times \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} S^1_{(i)} \Big).
   \end{equation} 
   Here $S^0_{(i)}$ and $S^1_{(i)}$ denote a copy of $S^0$ and $S^1$ associated to the index
   $i\in [k]$. We can easily
    recover the cell decomposition $\mathscr{B}_{\upalpha}(\K)$ of $X(\K,\lambda_{\upalpha})$ 
    and
    $\widehat{\mathscr{B}}_{\upalpha}(\K)$ 
     of $X(\K,\Lambda_{\upalpha})$ from~\eqref{Equ:Decomp-Real-form} 
     and~\eqref{Equ:Decomp-Complex-form}, respectively. Indeed, for any simplex $\sigma\in\K$, let
     \begin{equation} \label{Equ:D-sigma-real}
      \mathbf{D}_{\upalpha}(\sigma) = \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} 
       S^0_{(i)}*(\sigma\cap \Delta^{\alpha_i}) \times \prod_{i\in [k]\backslash 
       \mathrm{I}_{\upalpha}(\sigma)} S^0_{(i)},
    \end{equation} 
    \begin{equation}\label{Equ:D-sigma-complex}
     \widehat{\mathbf{D}}_{\upalpha}(\sigma) = 
      \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^1_{(i)}*(\sigma\cap \Delta^{\alpha_i}) 
        \times \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} S^1_{(i)}.
    \end{equation}    
 It is clear that 
 $\mathbf{D}_{\upalpha}(\sigma)$ and $\widehat{\mathbf{D}}_{\upalpha}(\sigma)$ are the closure of 
 $B_{(\sigma, [k]\backslash \mathrm{I}_{\upalpha}(\sigma))}$ and
 $\widehat{B}_{(\sigma, [k]\backslash \mathrm{I}_{\upalpha}(\sigma))}$ in
 $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$, respectively.\nn
 
   Note that $S^0$ can be naturally identified with $\Z_2$. Then
    the caonical action of $(\Z_2)^k$ (or $T^k$) on 
   $X(\K,\lambda_{\upalpha})$ (or $X(\K,\Lambda_{\upalpha})$) can be equivalently defined 
  by:
   for any $(g_1,\cdots, g_k)\in (\Z_2)^k$ (or $T^k$), 
   let $g_i\in \Z_2$ (or $S^1$) act on the $S^0_{(i)}$ (or $S^1_{(i)}$) through
   left translations.\n
  
   
 Moreover, a simplex $\sigma\in \K$ with $V(\sigma)=\{ v_{i_1},\cdots, v_{i_s}\}$ 
   can be written as a join 
   $$ \sigma = \underset{v\in V(\sigma)}{\Asterisk} v = v_{i_1} *\cdots * v_{i_s}. $$
  So we can write
  $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$ equivalently as:
   \begin{equation} \label{Equ:Decomp-Real-form-star}
       X(\K,\lambda_{\upalpha}) = \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} 
       \big(S^0_{(i)} * \underset{v\in V(\sigma)\cap \alpha_i}{\Asterisk} v \big)\times
        \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} S^0_{(i)} \Big),
     \end{equation}    
  \begin{equation} \label{Equ:Decomp-Complex-form-star}
      X(\K,\Lambda_{\upalpha}) = \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} 
       \big(S^1_{(i)} * \underset{v\in V(\sigma)\cap \alpha_i}{\Asterisk} v \big)\times
        \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} S^1_{(i)} \Big).
   \end{equation}  
  \n
  
  \begin{rem}
  The building blocks of 
    $X(\K,\lambda_{\upalpha}) $ and $ X(\K,\Lambda_{\upalpha})$
    are spaces obtained by mixtures of
     Cartesian products and joins of some simple spaces 
    (points, $S^0$ and $S^1$). 
    The building blocks of polyhedral products $(X,A)^{\mathcal{K}}$, however, only involve
    Cartesian products of spaces. 
    In addition, we have \emph{polyhedral join} (see~\cite{Anton13}) and
    \emph{polyhedral smash product} (see~\cite{BBCG10})
     whose building blocks only involve joins and smash products of spaces.
   It should be interesting to explore 
    spaces whose building blocks involve mixtures of 
     Cartesian products, joins and smash products. 
 \end{rem}
  
       When we write the boundary maps in the cochain
       complexes of $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$,
       we need to put orientations on
        their basis elements.  To do this, we need to first assign
         orientations to all the simplicies of $\K$. 
       For convenience, we put a total 
      ordering $\prec$ on the vertex set 
      $\{v_1,\cdots, v_m\}$ of $\Delta^{[m]}$ so that they appear in the increasing 
      order in $\alpha_1$ until $\alpha_k$.
      In other words, for any $1\leq i < k$ all the vertices of $\Delta^{\alpha_i}$ have less
      order than the vertices in $\Delta^{\alpha_{i+1}}$. 
      Moreover, the vertex-ordering of $\Delta^{[m]}$ induces a vertex-ordering
      of any simplex $\omega\in \Delta^{[m]}$, 
       which determines an orientation of $\omega$. Then the boundary of
       $\omega$ is
     \begin{equation}\label{Equ:Sign-Boundary}
        \partial \omega =\underset{\dim(\sigma)=\dim(\omega)-1}{\sum_{\sigma\subset \omega}}
          \varepsilon(\sigma,\omega)\, \sigma . 
      \end{equation}    
      Here if $V(\omega)=V(\sigma)\cup \{ v\}$, then
    $\varepsilon(\sigma,\omega)$ is equal to 
    $(-1)^{l(v,\omega)}$ where $l(v,\omega)$ is the number 
    vertices of $\omega$ that are less than $v$ with respect to the vertex-ordering $\prec$.\n
      
         
    \begin{defi}[Simplex with a ghost face] \label{Def:Simplex-Ghost-Face}
    For any $r\geq 0$, the $r$-simplex $\Delta^r$ has an empty face $\hat{0}$ by our convention (note here we do not use bold font since we want to distinguish $\hat{0}$ from the empty simplex
    $\hat{\mathbf{0}}$ of $\K$).
     In addition, we attach a
      \emph{ghost face} $-\hat{1}$ to $\Delta^r$ with 
     the following conventions.
      \begin{itemize}        
        \item $\dim(\hat{0})=\dim(-\hat{1})= -1$, $\mathrm{rank}(\hat{0})=
      \mathrm{rank}(-\hat{1}) = 0$. \n
      
        \item The interiors of $\hat{0}$ and $-\hat{1}$ are themselves. \n
        
        \item The boundary of any vertex of $\Delta^r$ is $\hat{0}$.\n
        
        \item The boundaries of $\hat{0}$ and $-\hat{1}$ are empty, \n
      \end{itemize}
     We denote $\Delta^r$ with the face $\hat{0}$ and $-\hat{1}$ by $\widehat{\Delta}^r$.
     This construction is very useful for us to simplify our argument later.
     \end{defi} 
    

    
   \subsection{Cochain complex of $X(\K,\lambda_{\upalpha})$}
   \ \n
    Let $S^0=\{ a^0, a^1\}$ where we assign $-1$ to $a^0$ and $+1$ to $a^1$ as the 
    orientation of $S^0$. Given a total ordering
   of all the vertices of $\Delta^r$, we can determine an orientation of each face 
   of $\Delta^r$.  Let $\sigma^{\circ}$ denote the \emph{relative interior}
     of $\sigma$ in $\Delta^r$.
     Then a cell decomposition of $S^0*\Delta^r$ is given by
     $$\{a^0\},\ \{a^1\}, 
      \ \{ S^0 * \sigma^{\circ}\,|\, \sigma \ \text{is a nonempty simplex in}\ \Delta^r 
       \},$$
       where the orientation of $S^0 *\sigma^{\circ}$ 
       is canonically determined by the orientations
       of $S^0$ and $\sigma$.
     We formally define 
     $$S^0*\hat{0} := a^1-a^0, \ \  
     S^0* -\hat{1} := a^0.$$
      Then we can write a basis of 
     the cellular chain complex $C_*(S^0*\Delta^r;\mathbf{k})$
    as
     $$ \{ S^0*\sigma^{\circ} \, |\, \sigma \in \widehat{\Delta}^r \}.$$
   
     
     Let $\{ x^{\sigma}\, |\, \sigma \in \widehat{\Delta}^r \}$ denote a basis 
     for the cellular cochain complex $C^*(S^0*\Delta^r;\mathbf{k})$  
     where $x^{\sigma}$ is the dual of $S^0*\sigma^{\circ}$. So $x^{\sigma}$
      is a cochain in dimension $\mathrm{rank}(\sigma)$.\n
      
     For a nonempty simplex $\sigma\in \Delta^r$, 
     the differential $d(x^{\sigma})$ is: 
     \begin{equation} \label{Equ:Diff-Real-0}
         d(x^{\sigma}) = \underset{\dim(\tau)=\dim(\sigma)+1}{\sum_{\sigma\subset \tau}} 
               \varepsilon{(\sigma,\tau})\cdot x^{\tau},
      \end{equation}         
       where 
       $\varepsilon{(\sigma,\tau})\in \{-1,1\}$ is defined as in~\eqref{Equ:Sign-Boundary}.
      In addition, our convention implies 
      $$ 
        d(x^{-\hat{1}})=0,\ \ d(x^{\hat{0}}) = \sum_{v\in V(\Delta^r)} x^{v}. $$
      \begin{rem} \label{Rem:S0-Cochains}
        Let $\{(a^0)^*, (a^1)^*\}\in C^*(S^0;\mathbf{k})$ be the dual basis of $\{a_0, a_1\}$.
        Then we can write $x^{\hat{0}}=(a^1)^*$ and $x^{-\hat{1}} =(a^0)^* + (a^1)^*$. The same
         convention is used in~\cite{Cai-13} to study the cohomology rings 
         of real moment-angle complexes.
      \end{rem}
      
      By the
      homeomorphism in~\eqref{Equ:Universal-Space}, 
      let $X(\Delta^{[m]},\lambda_{\upalpha})
      \cong \prod_{i\in [k]} S^0*\Delta^{\alpha_i}$ be equipped with the 
      product cell structure of each
      of $S^0*\Delta^{\alpha_i}$. Then the corresponding
       cellular cochain complex $C^*(X(\Delta^{[m]},\lambda_{\upalpha});\Z_2)$
       has a basis
       $$\{ \mathbf{x}^{\Phi} = x^{\sigma_1}\times \cdots \times x^{\sigma_k}; \   
       \Phi =(\sigma_1,\cdots, \sigma_k), \ \text{where}\ 
       \sigma_i\in \widehat{\Delta}^{\alpha_i},\, 1\leq i \leq k \}.$$
      Here the ``$\times$'' should be understood as the cross product of cochains.
       By the rule of
    taking differentials of cross products of cochains (see~\cite[3.B]{Hatcher}),
    we obtain
       \begin{equation} \label{Equ:Diff-Real-1}
        d(\mathbf{x}^{\Phi}) := \sum_{1\leq i \leq k}  \iota(\Phi,\sigma_i)\, 
         x^{\sigma_1}\times \cdots \times d x^{\sigma_i} \times \cdots \times x^{\sigma_k},  
        \end{equation}    
   where $\iota(\Phi,\sigma_i) =(-1)^{\sum^{i-1}_{l=1}\dim(x^{\sigma_l})}$.
   For a simplex $\sigma\in \Delta^{[m]}$ and $\J\subset [k]\backslash \I_{\upalpha}(\sigma)$,
     let 
     $$\Phi^{\J}_{\sigma}=(\sigma^{\J}_1,\cdots, \sigma^{\J}_k)$$
     \begin{equation} \label{Equ:Phi-sigma-J}
       \text{where}\ \,   \sigma^{\J}_i = \begin{cases}
          \sigma\cap \Delta^{\alpha_i}=\sigma_{(i)},   &  
                        \text{ $i\in \I_{\upalpha}(\sigma)$; } \\
          \ \hat{0}\in \widehat{\Delta}^{\alpha_i},     &   \text{ $i\in \J$; } \\
          -\hat{1}\in \widehat{\Delta}^{\alpha_i},   &   \text{ $i\in  [k]\backslash 
              (\I_{\upalpha}(\sigma)\cup \J)$;} 
              \end{cases} \ 
               \dim(x^{\sigma^{\J}_i})= \mathrm{rank}(\sigma^{\J}_i).
     \end{equation} 
    So for $\Phi=\Phi^{\J}_{\sigma}$ the equation~\eqref{Equ:Diff-Real-1} reads:
        \begin{align*} \label{Equ:Diff-Real-2}
           d (\mathbf{x}^{\Phi^{\J}_{\sigma}})
             & \ = \sum_{1\leq i \leq k}  \iota(\Phi^{\J}_{\sigma},\sigma^{\J}_i)\, 
                x^{\sigma^{\J}_1}\times \cdots \times d x^{\sigma^{\J}_i} \times \cdots \times 
                x^{\sigma^{\J}_k}\\
            & \overset{\eqref{Equ:Diff-Real-1}}{=}  
             \sum_{1\leq i \leq k}  \iota(\Phi^{\J}_{\sigma},\sigma^{\J}_i)\, 
                x^{\sigma^{\J}_1}\times \cdots \times \Big( 
                  \underset{\dim(\tau)=\dim(\sigma)+1}{\sum_{\sigma^{\J}_i\subset \tau \subset
                  \Delta^{\alpha_i}}} 
               \varepsilon{(\sigma^{\J}_i,\tau})\cdot x^{\tau} \Big) \times \cdots \times 
                x^{\sigma^{\J}_k} .
   \end{align*}
   Note that if $V(\omega)=V(\sigma)\cup \{ v\}$ where
 $v\in \alpha_i$, then $\omega^{\J}_i=\sigma^{\J}_i\cup \{ v\}$ and we have
 $$\varepsilon(\sigma,\omega)=\iota(\Phi^{\J}_{\sigma},\sigma^{\J}_i)\,
        \cdot \varepsilon(\sigma^{\J}_i,\tau).$$ 
  This follows from our convention on the vertex-ordering of $\Delta^{[m]}$. So we obtain
       \begin{equation} \label{Equ:Diff-Real-2}
       d (\mathbf{x}^{\Phi^{\J}_{\sigma}})   =  \underset{\dim(\omega) 
         =\dim(\sigma)+1} {\sum_{\sigma\subset \omega,\, \mathrm{I}_{\upalpha}(\omega)
                  \subset\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}}} 
             \varepsilon(\sigma,\omega)\,     
                \mathbf{x}^{\Phi_{\omega}^{\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)}}.  
       \end{equation}  
         
  For the simplicial complex $\K \subset \Delta^{[m]}$, let 
  $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ be the cellular 
   cochain complex corresponding to $\mathscr{B}_{\upalpha}(\K)$.
   Then for any $\sigma\in \K$, it is easy to see that 
   $\mathbf{x}^{\Phi^{\J}_{\sigma}}$ is
   a cochain of dimension $\mathrm{rank}(\sigma)$ in $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$
  which is the dual of the chain $B_{(\sigma,\J)} \in 
  C_*(X(\K,\lambda_{\upalpha});\mathbf{k})$ (see~\eqref{Equ:Cells-real} 
  and~\eqref{Equ:Cell-Decomp-1-2}). So we can choose a basis of
  $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ as
      $$\{  \mathbf{x}^{\Phi^{\J}_{\sigma}}\,;\, \sigma\in\K,\ 
       \J\subset [k]\backslash \I_{\upalpha}(\sigma)\}.$$
     Note that for any $\sigma\in \K$, the differential
     $d (\mathbf{x}^{\Phi^{\J}_{\sigma}})$ in $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$
     is only the sum of those terms 
     $\mathbf{x}^{\Phi_{\omega}^{\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)}}$ on the right side 
                      of~\eqref{Equ:Diff-Real-2} with $\omega\in \K$.\nn
  
  \subsection{Cochain complex of $X(\K,\Lambda_{\upalpha})$}
   \ \n

 Let $\{ e^0, e^1\}$ be the cell decomposition of $S^1$ where
 $\dim(e^0)=0$, $\dim(e^1)=1$, and $e^0$, $e^1$ are both oriented.
  Then given an orientation of each face of $\Delta^r$,
     we obtain an oriented cell decomposition of $S^1*\Delta^r$ by
     $$\{e^0\},\ \{e^1\}, 
      \ \{ S^1 * \sigma^{\circ}\,|\, \sigma \ \text{is a nonempty simplex in}\ \Delta^r 
       \}.$$ 
        If we formally define $S^1*\hat{0} = e^1$ and
     $S^1* -\hat{1} = e^0$, we can write a basis of 
     the cellular chain complex $C_*(S^1*\Delta^r;\mathbf{k})$
    as $ \{ S^1*\sigma^{\circ} \, |\, \sigma \in \widehat{\Delta}^r \}$ where
    the orientation of $S^1*\sigma^{\circ}$ 
    is determined canonically by the orientations of $S^1$ and
    $\sigma$. \n
       
    Let $\{ y^{\sigma}\, |\, \sigma \in \widehat{\Delta}^r \}$ denote a basis 
     for the cellular cochain complex $C^*(S^1*\Delta^r;\Z)$, 
     where $y^{\sigma}$ is the dual of $S^1*\sigma^{\circ}$. 
      For any nonempty simplex $\sigma$ in $\Delta^r$, 
     \begin{equation} \label{Equ:Diff-Complex-0}
         d(y^{\sigma}) = \underset{\dim(\tau)=\dim(\sigma)+1}{\sum_{\sigma\subset \tau}} 
               \varepsilon{(\sigma,\tau})\cdot y^{\tau}.
      \end{equation}        
       In addition, we have
     \begin{equation} \label{Equ:Diff-Complex-0-0}
        d(y^{-\hat{1}})=0,\ \ d(y^{\hat{0}}) = \sum_{v\in V(\Delta^r)} y^{v}. 
      \end{equation}  
      Note that $y^{-\hat{1}}$ and $y^{\hat{0}}$ are cochains in dimension $0$ and $1$, 
      respectively.\n
      By the
      homeomorphism in~\eqref{Equ:Universal-Space}, 
      let $X(\Delta^{[m]},\Lambda_{\upalpha})
      \cong \prod_{i\in [k]} S^1*\Delta^{\alpha_i}$ be equipped with the 
      product cell structure of each
      $S^1*\Delta^{\alpha_i}$.       
      The corresponding cellular cochain complex $C^*(X(\Delta^{[m]},\Lambda_{\upalpha});
      \mathbf{k})$ has a basis
       $$\{ \mathbf{y}^{\Phi} = y^{\sigma_1}\times \cdots \times y^{\sigma_k}; \   
       \Phi =(\sigma_1,\cdots, \sigma_k), \ \text{where}\ 
       \sigma_i\in \widehat{\Delta}^{\alpha_i},\, 1\leq i \leq k \}.$$ 
       
    We need two more notations for our argument below.               
    \begin{itemize}
    \item For any vertex $v$ of $\K$ and a partition $\upalpha=\{ \alpha_1,\cdots,\alpha_k\}$
     of $V(\K)$,
     let $i_{\upalpha}(v) \in [k]$ denote the index so that $v\in \alpha_{i_{\upalpha}(v)}$.\n 
    
   \item  For any $i\in \mathrm{L} \subset [k]$, define
     $\kappa(i,\mathrm{L})= (-1)^{r(i,\mathrm{L})}$, where
     $r(i,\mathrm{L})$ is the number of elements in $\mathrm{L}$ less than $i$. 
    Moreover, for any simplex $\sigma \in \K_{\upalpha, \mathrm{L}}$ we define               
    \begin{equation}\label{Equ:Kappa}
       \kappa(\sigma, \mathrm{L}) := \prod_{v\in V(\sigma)}
     \kappa(i_{\upalpha}(v), \mathrm{L}). 
   \end{equation}  
    \begin{equation} \label{Equ:Sign-Relation-1}
      \text{ So if $V(\omega)=V(\sigma)\cup \{v \}$, we have
    $\kappa(\omega, \mathrm{L})=\kappa(\sigma, \mathrm{L}) \cdot
      \kappa(i_{\upalpha}(v), \mathrm{L})$.}
    \end{equation}  
 \end{itemize}
     \n
     
  The differential of $\mathbf{y}^{\Phi}$ in $C^*(X(\Delta^{[m]},\Lambda_{\upalpha});
      \mathbf{k})$ is given by:   
       \begin{equation} \label{Equ:Diff-complex-1}
      d(\mathbf{y}^{\Phi}) := \sum_{1\leq i \leq k} 
       \hat{\iota}(\Phi,\sigma_i)\, y^{\sigma_1}\times \cdots \times d y^{\sigma_i} 
        \times \cdots \times y^{\sigma_k}, 
        \end{equation}
       where $\hat{\iota}(\Phi,\sigma_i) =
        (-1)^{\sum^{i-1}_{l=1}\dim(y^{\sigma_l})}$.
    Notice that when $\Phi=\Phi^{\J}_{\sigma}=(\sigma^{\J}_1,\cdots, \sigma^{\J}_k)$, 
         $$\dim(y^{\sigma^{\J}_i})=
              \begin{cases}
          \mathrm{rank}(\sigma^{\J}_i)+1,   &   \text{ $i\in \I_{\upalpha}(\sigma)$; } \\
            \ 1,     &   \text{ $i\in \J$; } \\
          \  0,   &   \text{ $i\in  [k]\backslash 
              (\I_{\upalpha}(\sigma)\cup \J)$.} 
              \end{cases} \qquad
         $$

     \begin{align} 
         d (\mathbf{y}^{\Phi^{\J}_{\sigma}})
           & \ = \sum_{1\leq i \leq k}  \hat{\iota}(\Phi^{\J}_{\sigma},\sigma^{\J}_i)\, 
                y^{\sigma^{\J}_1}\times \cdots \times d y^{\sigma^{\J}_i} \times \cdots \times 
                y^{\sigma^{\J}_k} \notag \\ 
           & \overset{\eqref{Equ:Diff-Complex-0}}{=} 
           \sum_{1\leq i \leq k}  \hat{\iota}(\Phi^{\J}_{\sigma},\sigma^{\J}_i)\, 
                y^{\sigma^{\J}_1}\times \cdots \times \Big( 
                  \underset{\dim(\tau)=\dim(\sigma)+1}{\sum_{\sigma^{\J}_i\subset \tau \subset
                  \Delta^{\alpha_i}}} 
               \varepsilon{(\sigma^{\J}_i,\tau})\cdot y^{\tau} \Big) \times \cdots \times 
                y^{\sigma^{\J}_k} . \notag 
         \end{align}   
 Note that if $V(\omega)=V(\sigma)\cup \{ v\}$ where
 $v\in \alpha_i$, then $\omega^{\J}_i=\sigma^{\J}_i\cup \{ v\}$ and we have
  $$\kappa(i, \I_{\upalpha}(\sigma)\cup\J) \cdot 
    \varepsilon(\sigma,\omega)= \hat{\iota}(\Phi^{\J}_{\sigma},\sigma^{\J}_i)\cdot
      \varepsilon(\sigma^{\J}_i,\tau).
  $$   
  This is because for each $i\in  [k]\backslash 
              (\I_{\upalpha}(\sigma)\cup \J)$, $\dim(y^{\sigma^{\J}_i}) = 0$. So we obtain
 \begin{equation} \label{Equ:Diff-complex-2}
      d (\mathbf{y}^{\Phi^{\J}_{\sigma}}) =        
            \underset{\dim(\omega)=\dim(\sigma)+1} 
            {\sum_{\sigma\subset \omega,\, \mathrm{I}_{\upalpha}(\omega)
                  \subset \mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}}} 
             \kappa\big( i_{\upalpha}(\omega\backslash \sigma), \I_{\upalpha}(\sigma)\cup\J \big) 
             \cdot  \varepsilon(\sigma,\omega) \,
                 \mathbf{y}^{\Phi_{\omega}^{\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)}}  
   \end{equation}                   
  where $\omega\backslash \sigma$ denote the vertex of $\omega$ that is 
  not in $\sigma$.\nn
  
          For the simplicial complex $\K \subset \Delta^{[m]}$, let 
  $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$ be the cellular 
   cochain complex corresponding to $\widehat{\mathscr{B}}_{\upalpha}(\K)$.
   Then for any simplex $\sigma\in \K$, $\mathbf{y}^{\Phi^{\J}_{\sigma}}$ is
   the cochain of dimension $\mathrm{rank}(\sigma)+|\I_{\upalpha}(\sigma)|+|\J|$
    in $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$
  that is dual to the cell $\widehat{B}_{(\sigma,\J)}$
  (see~\eqref{Equ:Cells-Complex} and~\eqref{Equ:Cell-Decomp-Complex}). 
     So $C^*(X(\K,\Lambda_{\upalpha});\Z)$ has a basis 
      $$\{  \mathbf{y}^{\Phi^{\J}_{\sigma}}\,;\, \sigma\in\K,\ 
       \J\subset [k]\backslash \I_{\upalpha}(\sigma)\}.$$                  
    Note that for any $\sigma\in \K$, the differential $d (\mathbf{y}^{\Phi^{\J}_{\sigma}})$ 
    in $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$
     is only the sum of those terms 
     $\mathbf{y}^{\Phi_{\omega}^{\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)}}$ on the right side 
                      of~\eqref{Equ:Diff-complex-2} with $\omega\in \K$.
                      \nn
                      
    
    
   
   \begin{proof}[\textbf{Proof of Theorem~\ref{Thm:Main-1}}]\ \n
   
      For any subset $\mathrm{L}\subset [k]$, let 
      $C^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\mathbf{k})$ denote the linear subspace of
      $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ generated by the following set  
          $$\{ B_{(\sigma, \mathrm{J})} \,|\, \mathrm{I}_{\upalpha}(\sigma)\cup 
            \mathrm{J} =\mathrm{L},\, \sigma\in\K,\, 
       \J\subset [k]\backslash \I_{\upalpha}(\sigma)\}.$$
      Similarly, let $C^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\mathbf{k})$ 
      denote the submodule of
      $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$ generated by the set
          $\{ \widehat{B}_{(\sigma, \mathrm{J})} \,|\, 
          \mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J} =\mathrm{L},\, \sigma\in\K,\, 
       \J\subset [k]\backslash \I_{\upalpha}(\sigma)\}$.\n
       
    From the differential of $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ and 
    $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$ shown in~\eqref{Equ:Diff-Real-2} 
    and~\eqref{Equ:Diff-complex-2},
    we see that $C^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\mathbf{k})$ and
    $C^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\mathbf{k})$ are actually cochain subcomplexes of
     $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ and $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$, respectively.
     We denote their cohomology groups by
     $H^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\mathbf{k})$ and  
       $H^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\mathbf{k})$.      
      Then we have the following decompositions:
        \begin{equation} \label{Equ:Decomp-Real}  
          H^*(X(\K,\lambda_{\upalpha});\mathbf{k})=\bigoplus_{\mathrm{L}\subset [k]}
        H^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\mathbf{k}),
       \end{equation}
   \begin{equation} \label{Equ:Decomp-Complex}  
          H^*(X(\K,\Lambda_{\upalpha});\mathbf{k})=\bigoplus_{\mathrm{L}\subset [k]}
        H^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\mathbf{k}).
       \end{equation}
   For any subset $\mathrm{L}\subset [k]$, let $C^*(\K_{\upalpha, \mathrm{L}};\mathbf{k})$ 
   denote the simplicial cochain complex of $\K_{\upalpha, \mathrm{L}}$. 
   For any simplex $\sigma\in \K$, let
   $\sigma^*$ be the basis cochain dual to $\sigma$ in 
   $C^*(\K_{\upalpha, \mathrm{L}};\mathbf{k})$.
   Then we have the following isomorphisms of $\mathbf{k}$-modules:
   \[ \varphi^{\mathrm{L}}_{\upalpha}: C^*(\K_{\upalpha, \mathrm{L}};\mathbf{k}) \rightarrow  
                                C^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\mathbf{k}),\ \ 
     \widehat{\varphi}^{\mathrm{L}}_{\upalpha}: C^*(\K_{\upalpha, \mathrm{L}};\mathbf{k}) \rightarrow  
                                     C^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\mathbf{k}) \]
   where 
   $\varphi^{\mathrm{L}}_{\upalpha}(\sigma^*)= 
    \mathbf{x}^{\Phi_{\sigma}^{\mathrm{L}\backslash \I_{\upalpha}(\sigma)}}$ and 
    $\widehat{\varphi}^{\mathrm{L}}_{\upalpha}(\sigma^*)= 
    \kappa(\sigma, \mathrm{L})
     \, \mathbf{y}^{\Phi_{\sigma}^{\mathrm{L}\backslash \I_{\upalpha}(\sigma)}}$ for any $\sigma\in \K_{\upalpha, \mathrm{L}}$. Note 
  $$ \dim(\mathbf{x}^{\Phi_{\sigma}^{\mathrm{L}\backslash \I_{\upalpha}(\sigma)}})
    =\mathrm{rank}(\sigma),\ \ 
    \dim(\mathbf{y}^{\Phi_{\sigma}^{\mathrm{L}\backslash \I_{\upalpha}(\sigma)}})=
     \mathrm{rank}(\sigma) + |\mathrm{L}|.$$\n
   
  Moreover, by the differential of $C^*(X(\K,\lambda_{\upalpha});\mathbf{k})$ and 
    $C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$ shown in~\eqref{Equ:Diff-Real-2} 
    and~\eqref{Equ:Diff-complex-2}, we can prove that $\varphi^{\mathrm{L}}_{\upalpha}$ and 
    $\widehat{\varphi}^{\mathrm{L}}_{\upalpha}$
    are chain complex isomorphisms. Indeed, the proof for 
    $\varphi^{\mathrm{L}}_{\upalpha}$ is trivial. For $\widehat{\varphi}^{\mathrm{L}}_{\upalpha}$ we have:
    \begin{align*}
     d (\widehat{\varphi}^{\mathrm{L}}_{\upalpha}(\sigma^*)) & = 
    d\big( \kappa(\sigma, \mathrm{L})
     \, \mathbf{y}^{\Phi_{\sigma}^{\mathrm{L}\backslash \I_{\upalpha}(\sigma)}} \big) \\
     & =  \kappa(\sigma, \mathrm{L})    \underset{\dim(\omega)=\dim(\sigma)+1} 
            {\sum_{\sigma\subset \omega\in \K,\, \mathrm{I}_{\upalpha}(\omega)
                  \subset \mathrm{L}}} 
              \kappa( i_{\upalpha}(\omega\backslash \sigma), \mathrm{L}) 
              \cdot  \varepsilon(\sigma,\omega) \,
                 \mathbf{y}^{\Phi_{\omega}^{\mathrm{L}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)}}  \\
     & =    \underset{\dim(\omega)=\dim(\sigma)+1} 
            {\sum_{\sigma\subset \omega\in \K,\, \mathrm{I}_{\upalpha}(\omega)
                  \subset \mathrm{L}}} 
             \kappa(\omega, \mathrm{L})  \cdot  \varepsilon(\sigma,\omega) \,
                 \mathbf{y}^{\Phi_{\omega}^{\mathrm{L}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)}} = \widehat{\varphi}^{\mathrm{L}}_{\upalpha}(d\sigma^*).        
   \end{align*}            
    The third ``$=$'' uses the relation
     $\kappa(\omega, \mathrm{L})=\kappa(\sigma, \mathrm{L})
      \cdot \kappa( i_{\upalpha}(\omega\backslash \sigma), \mathrm{L})$ 
      (see~\eqref{Equ:Sign-Relation-1}).   
    So we have additive isomorphisms of cohomology groups
    \[ \widetilde{H}^q(\K_{\upalpha, \mathrm{L}};\mathbf{k}) \cong 
                                H^{q+1,\mathrm{L}}(X(\K,\lambda_{\upalpha});\mathbf{k}),\ \ 
     \widetilde{H}^q(\K_{\upalpha, \mathrm{L}};\mathbf{k}) \cong  
                                     H^{q+|\mathrm{L}|+1,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\mathbf{k}).\]
  Let  $$\;\, \varphi_{\upalpha} = \underset{\mathrm{L}\subset [k]}{\bigoplus}
      \varphi^{\mathrm{L}}_{\upalpha} :\
      \underset{\mathrm{L}\subset [k]}{\bigoplus} 
                   \widetilde{H}^{q-1}(\K_{\upalpha, \mathrm{L}};\mathbf{k}) \longrightarrow
                    H^q(X(\K,\lambda_{\upalpha});\mathbf{k}),$$ 
     $$\qquad \widehat{\varphi}_{\upalpha} = \underset{\mathrm{L}\subset [k]}{\bigoplus}
      \widehat{\varphi}^{\mathrm{L}}_{\upalpha} :\
      \underset{\mathrm{L}\subset [k]}{\bigoplus} 
                   \widetilde{H}^{q-|\mathrm{L}|-1}(\K_{\upalpha, \mathrm{L}};\mathbf{k}) \longrightarrow
                    H^q(X(\K,\Lambda_{\upalpha});\mathbf{k}).$$      
   Then $\varphi_{\upalpha}$ and $\widehat{\varphi}_{\upalpha}$
   are the isomorphisms that satisfy our requirement.                               
   \end{proof}                   
  
  In the following, 
  we use Theorem~\ref{Thm:Main-1} to compute the cohomology groups of some examples
  explicitly.
  
      \begin{exam} \label{Example:Pentagon}
        Let $e_1,\cdots, e_5$ be a basis of $(\Z_2)^5$. Let $P$ be a pentagon.
        Then the real moment-angle
        complex $M=\R\mathcal{Z}_P$ is an orientable closed surface of genus five.
        Note that $M$ can be obtained either from $P$ with
        the $(\Z_2)^5$-coloring shown in the left picture of Figure~\ref{p:Pentagon-Poset}
        or from an octagon $P'$ with the $(\Z_2)^3$-coloring shown in the middle picture of
        Figure~\ref{p:Pentagon-Poset}. Indeed, the octagon $P'$ is obtained from 
        gluing four copies of $P$ along their edges colored by $e_4$ and $e_5$.
       \n
        The simplical complex $\K^{P'}$ dual to $P'$ is
        a simplicial circle with eight vertices that are partitioned 
        into three subsets labelled by $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$. 
        By Theorem~\ref{Thm:Main-1}, the 
        cohomology groups of
        $M$ can be computed from the cohomology groups of the simplicial 
       subcomplexes $\{ \K^{P'}_{\upalpha,\mathrm{L}}; \, \mathrm{L}\subset [3] \}$ 
       which are shown
       in Figure~\ref{p:Pentagon-Subcomplex} except 
       $\K^{P'}_{\upalpha,\varnothing}=\hat{\mathbf{0}}$ 
       and $\K^{P'}_{\upalpha,[3]} = \K^{P'}$. 
     \begin{figure}
         \includegraphics[width=0.9\textwidth]{Pentagon-Poset.eps}\\
          \caption{} 
          \label{p:Pentagon-Poset}
      \end{figure}
      
      \begin{figure}
         \includegraphics[width=0.95\textwidth]{Pentagon-Subcomplex-2.eps}\\
          \caption{} 
          \label{p:Pentagon-Subcomplex}
      \end{figure}
  \end{exam}

  \begin{rem}
     The above example implies that for any simplicial complex $\K$, 
   $\R\mathcal{Z}_{\K}$ (or $\mathcal{Z}_{\K}$)
    can be thought of as the quotient space of 
    $\R\mathcal{Z}_{\widehat{\K}}$ (or $\mathcal{Z}_{\widehat{\K}}$)
  where $\widehat{\K}$ is some larger simplicial complex constructed from $\K$.
  So we can also compute the cohomology groups of $\R\mathcal{Z}_{\K}$ (or $\mathcal{Z}_{\K}$)  
  from some $(\Z_2)^k$-coloring (or $\Z^k$-coloring) on $\widehat{\K}$ via Theorem~\ref{Thm:Main-1}.\n
  \end{rem} 
  
  
   
 \begin{exam}[Balanced Simplicial Complex and Pullbacks from the Linear Model] 
     \ \ \
      An $(n-1)$-dimensional simplicial complex $\K$ is called \emph{balanced} if 
    there exists a map $\phi : V(\K) \rightarrow [n]=\{1,\cdots, n\}$ such that
    if $\{ v,v'\}$ is an edge of $\K$, then $\phi(x)\neq \phi(y)$ (see~\cite{St96}). 
    We call $\phi$
    an \emph{$n$-coloring} on $\K$. 
     It is easy to see that $\K$ is balanced if and only if $\K$ admits 
     a non-degenerate simplicial map onto $\Delta^{n-1}$. In fact, if we identify
     the vertex set of $\Delta^{n-1}$ with $[n]$, any $n$-coloring $\phi$ on $\K$
     induces a non-degenerate simplicial map from $\K$ to $\Delta^{n-1}$ which
     sends a simplex $\sigma\in \K$ with $V(\sigma)=\{ v_{i_1},\cdots, v_{i_s}\}$
     to the face of $\Delta^{n-1}$ spanned by $\{ \phi(v_{i_1}),\cdots, \phi(v_{i_s}) \}$.
       \n
    
    Suppose
    $\phi:V(\K)\rightarrow [n]$ is an $n$-coloring of an $(n-1)$-dimensional simplicial complex
     $\K$.
    Let $\{e_1,\cdots, e_n\}$ be a basis of $(\Z_2)^n$.
    Then $\phi$ uniquely determines a $(\Z_2)^n$-coloring 
    $\lambda^{\phi}: V(\K)\rightarrow (\Z_2)^n$ where $\lambda^{\phi}(v)= e_{\phi(v)}$.
    It is clear that $\lambda^{\phi}$ is a non-degenerate $(\Z_2)^n$-coloring on $\K$.
   The space $X(\K,\lambda^{\phi})$ is also called a \emph{pullback from the linear model} 
    in~\cite[Example 1.15]{DaJan91}.      
          Notice that
     $$ \upalpha_{\phi} := \{ \phi^{-1}(1),\cdots \phi^{-1}(n) \}$$ 
     is a partition of $V(\K)$. So by our notation in section 1, 
     $X(\K,\lambda^{\phi}) =X(\K,\lambda_{\upalpha_{\phi}})$. 
  Then by Theorem~\ref{Thm:Main-1}, the cohomology groups of $X(\K,\lambda^{\phi})$ 
   can be computed by
     \[  H^q(X(\K,\lambda^{\phi});\mathbf{k}) = \sum_{\mathrm{L}
           \subset [n]}  \widetilde{H}^{q-1}
                (\K_{\upalpha_{\phi}, \mathrm{L}};\mathbf{k}),\ \forall q\geq 0. \]
   \end{exam}
   
   \vskip .7cm
   
   \section{Stable Decompositions of $X(\K,\lambda_{\upalpha})$ and
   $X(\K,\Lambda_{\upalpha})$}
   
   It is shown in~\cite{BBCG10} that the stable homotopy type of a polyhedral
   product $(\mathbb{X},\mathbb{A})^{\K}$  is a wedge of
   spaces, which provides corresponding homological decompositions of
   $(\mathbb{X},\mathbb{A})^{\K}$. In this section, we prove parallel results of
   for $X(\K,\lambda_{\upalpha})$ and
   $X(\K,\Lambda_{\upalpha})$. Our argument proceeds along the same line as~\cite{BBCG10}.\n
   
     Let
    $\upalpha=\{ \alpha_1,\cdots,\alpha_k \}$ be a
     partition of the vertex set $V(\K)$ of a simplicial complex $\K$.
     For any subset $\mathrm{L} \subset [k]$
     and any simplex $\sigma\in\K$, define
    \[ \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma) := 
    \bigwedge_{i\in \mathrm{I}_{\upalpha}(\sigma)\cap \mathrm{L}} 
    S^0_{(i)}*(\sigma\cap \Delta^{\alpha_i}) 
    \wedge \bigwedge_{i\in \mathrm{L} \backslash 
    (\mathrm{I}_{\upalpha}(\sigma)\cap \mathrm{L})} S^0_{(i)} \]
    \[ \mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma) := 
    \bigwedge_{i\in \mathrm{I}_{\upalpha}(\sigma)\cap \mathrm{L}} 
    S^1_{(i)}*(\sigma\cap \Delta^{\alpha_i}) 
    \wedge \bigwedge_{i\in \mathrm{L} \backslash 
    (\mathrm{I}_{\upalpha}(\sigma)\cap \mathrm{L})} S^1_{(i)} \]
   where $\wedge$ and $\bigwedge$ denote the smash product.
   We adopt the convention that the smash product of a space with the empty space is empty.
    Then the following lemma is
    immediate from the definitions of $ \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma)$
    and $\mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma)$.\n
   
    \begin{lem}\label{Lem:W-Simplex-Equiv} 
      For any subset $\mathrm{L} \subset [k]$
     and any simplex $\sigma\in\K$,
     \begin{itemize}
      \item[(i)] $\mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma) =
     \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma\cap \K_{\upalpha,\mathrm{L}})$,
     $\mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma) =
     \mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma\cap \K_{\upalpha,\mathrm{L}})
       $.\n
       \item[(ii)] $\mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma)$ and 
     $\mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma)$ are  
     contractible whenever $\mathrm{I}_{\upalpha}(\sigma)\cap \mathrm{L}\neq \varnothing$.\n
     \item[(iii)] $\mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\hat{\mathbf{0}}) = 
     \bigwedge_{i\in \mathrm{L}} S^0_{(i)}\cong S^0$ while $\mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\hat{\mathbf{0}}) = 
     \bigwedge_{i\in \mathrm{L}} S^1_{(i)}\cong S^{|\mathrm{L}|}$.
    \end{itemize}
    \end{lem}
   
 
  Next, we assign a base-point to each of $S^0_{(i)}$ and $S^1_{(i)}$, which then
   determines the base-points of $\mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma)$
   and $\mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma)$.
 By~\cite[Theorem 2.8]{BBCG10}, there are natural pointed homotopy equivalences
    (see~\eqref{Equ:D-sigma-real} and~\eqref{Equ:D-sigma-complex})
    \[ \mathbf{\Sigma} (\mathbf{D}_{\upalpha}(\sigma)) \simeq  \mathbf{\Sigma} \Big( 
      \bigvee_{\mathrm{L}\subset [k]} \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}
          (\sigma) \Big),\ \
     \mathbf{\Sigma} (\widehat{\mathbf{D}}_{\upalpha}(\sigma)) \simeq 
    \mathbf{\Sigma} \Big( 
      \bigvee_{\mathrm{L}\subset [k]} \mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}
        (\sigma) \Big)  \]
       where $\mathbf{\Sigma}$ is the reduced suspension. Define 
        $$\mathbf{E}_{\upalpha}(\sigma):= \bigvee_{\mathrm{L}\subset [k]}
            \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}
          (\sigma),\ \ \
          \widehat{\mathbf{E}}_{\upalpha}
          (\sigma) := \bigvee_{\mathrm{L}\subset [k]} \mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}
          (\sigma).$$
          
       Let $\mathrm{Cat}(\K)$ denote the \emph{face category} of $\K$ whose objects are
       simplices $\sigma\in\K$ and there is a morphism from $\sigma$ to $\tau$ whenever
       $\sigma\subseteq \tau$. Then   
     we can consider $\mathbf{D}_{\upalpha}$, $\widehat{\mathbf{D}}_{\upalpha}$,
      $\mathbf{E}_{\upalpha}$, $\widehat{\mathbf{E}}_{\upalpha}$ as functors from 
      $\mathrm{Cat}(\K)$ to the category $\mathrm{CW}_*$ of connected, 
      based CW-complexes and based continuous maps. It is clear that      
      \[ X(\K,\lambda_{\upalpha}) = \bigcup_{\sigma\in \K} \mathbf{D}_{\upalpha}(\sigma) =
      \mathrm{colim}(\mathbf{D}_{\upalpha}(\sigma)),  \ \ 
      X(\K,\Lambda_{\upalpha}) = \bigcup_{\sigma\in \K} \widehat{\mathbf{D}}_{\upalpha}(\sigma) =
      \mathrm{colim}(\widehat{\mathbf{D}}_{\upalpha}(\sigma)).   \] 
       For any subset $\mathrm{L} =\{ l_1,\cdots, l_s\}\subset [k]$, define
       $$ X^{\wedge}(\K_{\upalpha, \mathrm{L}}, \lambda_{\upalpha}) 
       := \bigcup_{\sigma\in\K} \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma)
       =\bigcup_{\sigma\in\K_{\upalpha,\mathrm{L}}} 
       \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma)$$
       $$ X^{\wedge}(\K_{\upalpha, \mathrm{L}}, \Lambda_{\upalpha}) :=
         \bigcup_{\sigma\in\K} \mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma)
         =\bigcup_{\sigma\in\K_{\upalpha,\mathrm{L}}} 
         \mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma).$$
      So $X^{\wedge}(\K,\lambda_{\upalpha})$ and 
      $X^{\wedge}(\K_{\upalpha, \mathrm{L}}, \Lambda_{\upalpha})$ are
       the colimits of $ \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma)$ and
     $\mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma)$ over face category $\mathrm{Cat}(\K_{\upalpha, \mathrm{L}})$ of $\K_{\upalpha, \mathrm{L}}$, respectively.
       In addition, we have
        $$ 
        \mathrm{colim}(\mathbf{E}_{\upalpha}(\sigma))=
           \bigvee_{\mathrm{L}\subset [k]} X^{\wedge}(\K_{\upalpha, \mathrm{L}}, 
           \lambda_{\upalpha}),\ \ 
        \mathrm{colim}(\widehat{\mathbf{E}}_{\upalpha}(\sigma))=
        \bigvee_{\mathrm{L}\subset [k]} X^{\wedge}(\K_{\upalpha, \mathrm{L}},\Lambda_{\upalpha}).$$
        
       
       
      Since the suspension commutes with the colimits up to homotopy
      equivalence (see~\cite[Theorem 4.3]{BBCG10}), 
      we obtain homotopy equivalences
      \begin{equation*}
       \mathbf{\Sigma}( X(\K,\lambda_{\upalpha}) ) \simeq 
        \mathrm{colim}(\mathbf{\Sigma}(\mathbf{D}_{\upalpha}(\sigma)))
      \simeq \mathrm{colim}(\mathbf{\Sigma} (\mathbf{E}_{\upalpha}(\sigma))) \simeq 
      \mathbf{\Sigma} \big( \bigvee_{\mathrm{L}\subset [k]} 
           X^{\wedge}(\K_{\upalpha, \mathrm{L}},\lambda_{\upalpha}) \big),
      \end{equation*}
       \begin{equation*}
        \mathbf{\Sigma}( X(\K,\Lambda_{\upalpha}) ) \simeq 
      \mathrm{colim}(\mathbf{\Sigma} (\widehat{\mathbf{D}}_{\upalpha}(\sigma)))
      \simeq \mathrm{colim}(\mathbf{\Sigma}
          (\widehat{\mathbf{E}}_{\upalpha}(\sigma))) \simeq \mathbf{\Sigma}
       \big( \bigvee_{\mathrm{L}\subset [k]} 
        X^{\wedge}(\K_{\upalpha, \mathrm{L}},\Lambda_{\upalpha})  \big). 
      \end{equation*}  
      
    \begin{lem} \label{Lem:Wedge-Equiv}
     For any $\mathrm{L}\subset [k]$, there are homotopy equivalences:
      $$X^{\wedge}(\K_{\upalpha, \mathrm{L}},\lambda_{\upalpha}) \simeq
      \bigvee_{\sigma\in \K_{\upalpha,\mathrm{L}}} 
      |\Delta((\K_{\upalpha, \mathrm{L}})_{>\sigma})| *
      \mathbf{W}^{S^0}_{\upalpha, \mathrm{L}}(\sigma), $$
      $$X^{\wedge}(\K_{\upalpha, \mathrm{L}},\Lambda_{\upalpha}) \simeq
      \bigvee_{\sigma\in \K_{\upalpha,\mathrm{L}}} 
      |\Delta((\K_{\upalpha, \mathrm{L}})_{>\sigma})| *
      \mathbf{W}^{S^1}_{\upalpha, \mathrm{L}}(\sigma).$$
      where $\Delta((\K_{\upalpha, \mathrm{L}})_{>\sigma})$ is the order complex
      of the poset $\{ \tau\in \K_{\upalpha, \mathrm{L}}\,|\, \tau\supsetneq \sigma \}$ 
      with respect to
      the inclusion and $|\Delta((\K_{\upalpha, \mathrm{L}})_{>\sigma})|$ is 
      the geometric realization of $\Delta((\K_{\upalpha, \mathrm{L}})_{>\sigma})$.
    \end{lem}
    \begin{proof}
        Note that the natural inclusion
      $S^0\hookrightarrow S^0 * \Delta^r$ is
      null-homotopic for any $r\geq 0$. Then parallel to the proof 
      of~\cite[Theorem 2.12]{BBCG10}, we have a homotopy equivalence
       $H_{\mathrm{L}}(\sigma): \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma) \rightarrow 
       \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma)$ 
      for each simplex $\sigma\in \K_{\upalpha,\mathrm{L}}$ so that the following 
      diagram commutes for any
      $\tau\subseteq \sigma\in \K_{\upalpha,\mathrm{L}}$,
     \begin{equation}\label{Equ:Wedge-Equiv}
        \xymatrix{
          \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\tau) 
              \ar[r]^{H_{\mathrm{L}}(\tau)} \ar[d]_{\zeta_{\sigma,\tau}} &
              \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\tau)
            \ar[d]^{c_{\sigma,\tau}} \\
            \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma)
             \ar[r]^{H_{\mathrm{L}}(\sigma)} & 
             \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma)        
           } 
      \end{equation}  
  where $\zeta_{\sigma,\tau}$ is the natural inclusion and
  $c_{\sigma,\tau}$ is the constant map
  to the base-point. Then by~\cite[Theorem 4.1]{BBCG10} 
  and~\cite[Theorem 4.2]{BBCG10}, there is a homotopy equivalence
  $$
     X^{\wedge}(\K_{\upalpha,\mathrm{L}}, \lambda_{\upalpha})
        =\mathrm{colim} (\mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma)) \ 
  \ \simeq \ 
  \bigvee_{\sigma\in \K_{\upalpha,\mathrm{L}}} |\Delta((\K_{\upalpha,\mathrm{L}})_{>\sigma})| *
      \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma).$$
  The argument for 
      $X^{\wedge}_{\mathrm{L}}(\K,\Lambda_{\upalpha})$ is completely parallel (by replacing
      $S^0$ by $S^1$).
   \end{proof}

 From the above discussions, we obtain homotopy equivalences:
  \[   \mathbf{\Sigma}( X(\K,\lambda_{\upalpha}) ) \simeq 
       \mathbf{\Sigma} \Big( \bigvee_{\mathrm{L}\subset [k]} 
           \bigvee_{\sigma\in \K_{\upalpha,\mathrm{L}}}
            |\Delta((\K_{\upalpha,\mathrm{L}})_{>\sigma})| *
      \mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma) \Big), \]
 \[   \mathbf{\Sigma}( X(\K,\Lambda_{\upalpha}) ) \simeq 
     \mathbf{\Sigma} \Big( \bigvee_{\mathrm{L}\subset [k]} 
           \bigvee_{\sigma\in \K_{\upalpha,\mathrm{L}}}
            |\Delta((\K_{\upalpha,\mathrm{L}})_{>\sigma})| *
      \mathbf{W}^{S^1}_{\upalpha,\mathrm{L}}(\sigma) \Big). \]  
         
  Moreover, the above spaces can be simplified by the following facts.\nn
  \begin{itemize}
  \item
   $\mathbf{W}^{S^0}_{\upalpha,\mathrm{L}}(\sigma)$ and
  $\mathbf{W}^{S^1}_{\upalpha,\mathrm{L}}(\sigma)$ are contractible whenever $\sigma\neq 
  \hat{\mathbf{0}}\in \K_{\upalpha,\mathrm{L}}$ (see Lemma~\ref{Lem:W-Simplex-Equiv}(ii)). \n
  \item
  $\Delta((\K_{\upalpha,\mathrm{L}})_{>\hat{\mathbf{0}}})$
  is isomorphic to the barycentric subdivision $\K'_{\upalpha,\mathrm{L}}$ of
   $\K_{\upalpha,\mathrm{L}}$
  whose geometric realization is homeomorphic to that of $\K_{\upalpha,\mathrm{L}}$.\n
  \end{itemize}
  
  Then we have the following homotopy equivalences which proves Theorem~\ref{Thm:Stable-Decomp}.
  \[  \mathbf{\Sigma}( X(\K,\lambda_{\upalpha}) ) \simeq 
  \mathbf{\Sigma} \Big( \bigvee_{\mathrm{L}\subset [k]} 
            |\K_{\upalpha,\mathrm{L}}| * S^0
     \Big) \simeq  \bigvee_{\mathrm{L}\subset [k]} 
           \mathbf{\Sigma}^2( |\K_{\upalpha,\mathrm{L}}|), \]
     \[  \mathbf{\Sigma}( X(\K,\Lambda_{\upalpha}) ) \simeq 
         \mathbf{\Sigma} \Big( \bigvee_{\mathrm{L}\subset [k]} 
            |\K_{\upalpha,\mathrm{L}}| * S^{|\mathrm{L}|}
     \Big) \simeq  \bigvee_{\mathrm{L}\subset [k]} 
           \mathbf{\Sigma}^{|\mathrm{L}|+2}( |\K_{\upalpha,\mathrm{L}}|). \] 
  
                
  \ \\
   
  \section{Another interpretation of $H^*(X(\K,\lambda_{\upalpha});\Z_2)$ and 
  $H^*(X(\K,\Lambda_{\upalpha});\Z_2)$}      
        Let
    $\upalpha=\{ \alpha_1,\cdots,\alpha_k \}$ be a
     partition of the vertex set $V(\K)$ of a simplicial complex $\K$.
    We have defined a differential $d_{\upalpha}$ on 
     $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K]$ 
     in~\eqref{Equ:R-boundary-1}.\nn
             
  \textbf{Claim:}
     $d_{\upalpha}\circ d_{\upalpha}=0$.   
     \[ \text{Indeed, we have}\ d_{\upalpha}\circ d_{\upalpha}(t_i)
      = d_{\upalpha}\Big(\sum_{v_j\in \alpha_i} \mathbf{v}_{\{j\}} \Big) 
      =\sum_{v_j\in \alpha_i} \Big( \underset{\dim(\omega) =1}
       {\sum_{v_j\in \omega,\, \mathrm{I}_{\upalpha}(\omega) =\{i\}}} 
        \varepsilon(\omega,v_j) \cdot \mathbf{v}_{\omega} \Big) =0 \]
  since any $\mathbf{v}_{\omega}$  with $\dim(\omega) =1$ and 
           $\mathrm{I}_{\upalpha}(\omega) =\{i\}$ appears twice with opposite signs
           in the above sum. Similarly, we obtain
    $d_{\upalpha}\circ d_{\upalpha}(\mathbf{v}_{\sigma})=0$ from the face that
    $\partial \circ \partial=0$ in $C_*(\K;\mathbf{k})$.
    The claim is proved.  \ \ $\qed$
    \n
    
     For any subset $\mathrm{J}=\{ j_1,\cdots, j_s\} \subset [k]$ with $j_1 < \cdots < j_s$, 
    let
     $$t_{\mathrm{J}} := t_{j_1}\cdots t_{j_s} \in 
         \Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K]. $$           
   We introduce a multi-grading $\mathrm{mdeg}^{\upalpha}$ 
   on $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]
                  \otimes \mathbf{k}[\K]$ by
   \begin{equation}\label{Equ:Multi-Grading-1}
       \mathrm{mdeg}^{\upalpha}(t_{\J}) = (-1, \J) \in (\Z, [k]),\ \ \,
       \mathrm{mdeg}^{\upalpha}(\mathbf{v}_{\sigma}) =
       (0 ,\mathrm{I}_{\upalpha}(\sigma)) \in (\Z,[k]).
     \end{equation} 
   Here we define the sum of two element $(n, \J)$, $(n',\J') \in (\Z, [k])$ to be
      \begin{equation} \label{Equ:Multi-Grading-Sum}
          (n,\J) + (n',\J') := (n+n', \J\cup \J').
       \end{equation}   
   So the multi-grading of the product of two monomials in 
   $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]
                  \otimes \mathbf{k}[\K]$ is equal to the sum 
    of their multi-gradings.  \n  
    
                  
      Next, we identify  
   $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K]$
   with the cellular cochain complexes of some spaces.
   These spaces are obtained by replacing each vertex 
   $v$ of $\K$ in~\eqref{Equ:Decomp-Real-form-star} and~\eqref{Equ:Decomp-Complex-form-star}
    by an infinitely dimensional sphere $S^{\infty}$, i.e.
     \begin{equation} \label{Equ:Decomp-Real-form-star-S-inf}
       \mathcal{X}(\K, \upalpha, S^0) := \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} 
       \big(S^0_{(i)} * \underset{v\in V(\sigma)\cap \alpha_i}{\Asterisk} S^{\infty}_v \big)\times
        \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} S^0_{(i)} \Big),
     \end{equation}    
  \begin{equation} \label{Equ:Decomp-Complex-form-star-S-inf}
      \mathcal{X}(\K, \upalpha, S^1) := \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} 
       \big(S^1_{(i)} * \underset{v\in V(\sigma)\cap \alpha_i}{\Asterisk} S^{\infty}_v \big)\times
        \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} S^1_{(i)} \Big),
   \end{equation}  
   where $S^{\infty}_v$ denotes the copy of $S^{\infty}$ corresponding to the vertex $v$, and
   $S^0_{(i)}$ and $S^1_{(i)}$ 
   denote the copy of $S^0$ and $S^1$ corresponding to the index $i\in [k]$, respectively.\n
   
   Note that $S^{\infty}$ is a contractible space which has a cell decomposition
   with exactly one cell in each dimension. The boundary of every $2j$-dimensional cell $f^{2j}$ 
   is the closure of the $(2j-1)$-dimensional cell $f^{2j-1}$, while the boundary of 
   each $f^{2j-1}$ is zero.  For any vertex
   $v$ of $\K$, the cells in $S^{\infty}_v$ are denoted by $\{ f^n_{v} \}_{n \geq 0}$.
\n 
         
   By identifying each vertex $v\in V(\K)$ as the $0$-cell 
   $e^0_v\in S^{\infty}_v$, we can think of $X(\K,\lambda_{\upalpha})$
   and $X(\K,\Lambda_{\upalpha})$ as subspaces of $\mathcal{X}(\K, \upalpha, S^0)$ 
    and $\mathcal{X}(, \upalpha, S^1)$, respectively. The deformation retraction of
   each $S^{\infty}_v$ to $e^0_v = v$ obviously induces deformation retractions of
    $\mathcal{X}(\K, \upalpha, S^0)$ to $X(\K,\lambda_{\upalpha})$ and
  $\mathcal{X}(\K, \upalpha, S^1)$ to $X(\K,\Lambda_{\upalpha})$, respectively. So we have
  isomorphisms of cohomology rings 
  \begin{equation} \label{Equ:Deform-Cohomology-Iso-Real}
      H^*(\mathcal{X}(\K, \upalpha, S^0);\mathbf{k}) \cong 
      H^*(X(\K,\lambda_{\upalpha});\mathbf{k}).  
  \end{equation}
  \begin{equation} \label{Equ:Deform-Cohomology-Iso-Complex}
     H^*(\mathcal{X}(\K, \upalpha, S^1);\mathbf{k}) \cong
          H^*(X(\K,\Lambda_{\upalpha});\mathbf{k}).
   \end{equation}      
 To give explicit descriptions of the cell structures of $\mathcal{X}(\K, \upalpha, S^0)$
 and $\mathcal{X}(\K, \upalpha, S^1)$, we introduce a new notion below.
 
 \begin{defi}[Weighted Simplex]
    An $r$-simplex $\Delta^r$ along with
    a function $\mathbf{w} : V(\Delta^r) \rightarrow \Z_{\geq 0}$ is called a 
    \emph{weighted $r$-simplex}, denoted by $(\Delta^r,\mathbf{w})$.
    We call $\mathbf{w}$ a \emph{weight} of $\sigma$. For any face
    $\sigma$ of $\Delta^r$, let $\mathbf{w}_{\sigma}$ denote the restriction of $\mathbf{w}$ to
    $V(\sigma)$, and we call $(\sigma,\mathbf{w}_{\sigma})$ a \emph{weighted face} of 
    $(\Delta^r,\mathbf{w})$. Any weight of the empty face $\hat{0}$ and the ghost
    face $-\hat{1}$ of $\widehat{\Delta}^r$ is null. So for brevity, we use $\hat{0}$ and
    $-\hat{1}$ to denote the weighted simplices $(\hat{0},\mathbf{w}_{\hat{0}})$ and 
    $(-\hat{1},\mathbf{w}_{-\hat{1}})$. 
   \end{defi}
 
 For any $d \geq 1$, let $\{ e^0, e^d \}$ 
   denote the cell decomposition of the $d$-sphere $S^d$ where $e^0$ is a $0$-cell and
    $e^d$ is a $d$-cell.
  When $S^0=\{ a^0,a^1\}$, we define $e^0=a^0$ and $\bar{e}^0 = a^1-a^0$ and formally think of
  $\{ e^0, \bar{e}^0\}$ as a cell decomposition of $S^0$.  \n
  
  For any $d\geq 0$ and a weighted simplex $(\Delta^r,\mathbf{w})$, we define an open ball
  \[ S^d * (\Delta^r,\mathbf{w}) = S^d * \underset{v\in V(\Delta^r)}{\Asterisk}
    f^{\mathbf{w}(v)}_v  \subset S^d * \underset{v\in V(\Delta^r)}{\Asterisk} S^{\infty}_v. \]
   
     We adopt the following conventions in the rest.
 \begin{itemize}
    \item when $d=0$,  define $S^0 * \hat{0} = \bar{e}^0$, $S^0 * -\hat{1} = e^0$.\n
    
     \item when $d\geq 1$,  define $S^0 * \hat{0} = 
                  e^d$, $S^0 * -\hat{1} = e^0$.
  \end{itemize} 
 \n
 
 Then a cell decomposition of 
  $S^d * \underset{v\in V(\Delta^r)}{\Asterisk} S^{\infty}_v$ is given by:
   \[ \{ S^d * (\sigma,\mathbf{w}_{\sigma}) \,|\, 
  \sigma\in\widehat{\Delta}^r,\ \mathbf{w}_{\sigma} \ \text{is an arbitrary 
    weight of}\ \sigma
      \}   \]
 Here we write the argument for
        an arbitrary $d\geq 0$ because we will generalize our discussion
      to a wider range of spaces later (see Theorem~\ref{Thm:Main-2-gen} and
      Theorem~\ref{Thm:Main-3-gen}).\n
            
 Now we can construct cell decompositions of $\mathcal{X}(\K, \upalpha, S^0)$ and
  $\mathcal{X}(\K, \upalpha, S^1)$ as follows.
 For any weighted simplex $(\sigma,\mathbf{w}_{\sigma})$, $\sigma\in \K$ 
    and $\J\subset [k]$ with $\J\cap \I_{\upalpha}(\sigma)=\varnothing$,
    \[ \mathbf{B}(\sigma,\mathbf{w}_{\sigma},\J) :=
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^0_{(i)} * 
       (\sigma_{(i)},\mathbf{w}_{\sigma_{(i)}})
       \times \underset{i\in \J}{\prod}\ S^0_{(i)} * \hat{0}_{(i)} \times  
         \prod_{i\in [k]\backslash
        (\mathrm{I}_{\upalpha}(\sigma)\cup \J)}  S^0_{(i)} * -\hat{1}_{(i)},\]
    \[ \widehat{\mathbf{B}}(\sigma,\mathbf{w}_{\sigma},\J) :=
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^1_{(i)}
         * (\sigma_{(i)},\mathbf{w}_{\sigma_{(i)}})\times \underset{i\in \J}{\prod}\  
         S^1_{(i)} * \hat{0}_{(i)} \times   \prod_{i\in [k]\backslash
        (\mathrm{I}_{\upalpha}(\sigma)\cup \J)} S^1_{(i)} * -\hat{1}_{(i)},\]
     where $\sigma_{(i)} = \sigma\cap \Delta^{\alpha_i}$ and $\hat{0}_{(i)}$, $-\hat{1}_{(i)}\in
     \widehat{\Delta}^{\alpha_i}$.      
     In particular, when $\sigma =\hat{\mathbf{0}}\in \K$, 
     $\mathbf{w}_{\hat{\mathbf{0}}}$ is null and $\I_{\upalpha}(\hat{\mathbf{0}})
     =\varnothing$. So we have:
     \[ \mathbf{B}(\hat{\mathbf{0}},\mathbf{w}_{\hat{0}},\J)=
       \underset{i\in \J}{\prod}\ S^0* \hat{0}_{(i)} \times  
        \prod_{i\in [k]\backslash \J}  S^0 * -\hat{1}_{(i)},\ \forall \J\subset [k], \]
    \[ \widehat{\mathbf{B}}(\hat{\mathbf{0}},\mathbf{w}_{\hat{0}},\J)=
       \underset{i\in \J}{\prod}\  S^1 * \hat{0}_{(i)} \times   
        \prod_{i\in [k]\backslash \J} S^1 * -\hat{1}_{(i)}, \ \forall \J\subset [k].\]
     Let $B(\sigma,\mathbf{w}_{\sigma},\J)$
     and $\widehat{B}(\sigma,\mathbf{w}_{\sigma},\J)$ denote the relative interiors of 
     $\mathbf{B}(\sigma,\mathbf{w}_{\sigma},\J)$ and
      $\widehat{\mathbf{B}}(\sigma,\mathbf{w}_{\sigma},\J) $, respectively.
       It is easy to see that the set of all such
       cells $\{ B(\sigma,\mathbf{w}_{\sigma},\J) \}$
     and $\{ \widehat{B}(\sigma,\mathbf{w}_{\sigma},\J) \}$ give us cell decompositions
     of $\mathcal{X}(\K, \upalpha, S^0)$ and
  $\mathcal{X}(\K, \upalpha, S^1)$, respectively.
  
  \begin{rem}
   In fact $B(\sigma,\mathbf{w}_{\sigma},\J)$ is a union of
    $2^{|\J|}$ open balls. But it does not hurt to treat 
     $\{ B(\sigma,\mathbf{w}_{\sigma},\J) \}$ as 
     a cell decomposition of $\mathcal{X}(\K, \upalpha, S^0)$ for the calculation of 
    the cohomology groups of $\mathcal{X}(\K, \upalpha, S^0)$ because,
    $\{ B(\sigma,\mathbf{w}_{\sigma},\J) \}$ form a basis of the
    cellular chain complex of 
    an anthentic cell decomposition of $\mathcal{X}(\K, \upalpha, S^0)$.
    A similar construction is 
     used in our discussion of $X(\K,\lambda_{\upalpha})$ in section 2.
  \end{rem}
 
        
     Let $C^*(\mathcal{X}(\K, \upalpha, S^0);\mathbf{k})$ and
  $C^*(\mathcal{X}(\K, \upalpha, S^1);\mathbf{k})$ denote the cellular cochain complexes
  corresponding to the above cell decompositions. 
  Let $\mathbf{x}^{(\sigma,\mathbf{w}_{\sigma},\J)}$ and 
  $\mathbf{y}^{(\sigma,\mathbf{w}_{\sigma},\J)}$ denote the basis of
  $C^*(\mathcal{X}(\K, \upalpha, S^0);\mathbf{k})$ and
  $C^*(\mathcal{X}(\K, \upalpha, S^1);\mathbf{k})$ that are dual to 
  $B(\sigma,\mathbf{w}_{\sigma},\J)$
     and $\widehat{B}(\sigma,\mathbf{w}_{\sigma},\J)$, respectively.
\n
  
 For any simplex $\sigma$ in $\K$, 
 let $\mathbf{w}^0_{\sigma}$ denote the weight on $\sigma$ that assigns $0$ to
  all vertices of $\sigma$. Then under the inclusions of 
  $X(\K,\lambda_{\upalpha})$ into  $\mathcal{X}(\K, \upalpha, S^0)$ and
  $X(\K,\Lambda_{\upalpha})$ into $\mathcal{X}(\K, \upalpha, S^1)$,  the cell
  $\mathbf{x}^{\Phi^{\J}_{\sigma}}$ is identified with 
  $\mathbf{x}^{(\sigma,\mathbf{w}^0_{\sigma},\J)}$ and,
  $\mathbf{y}^{\Phi^{\J}_{\sigma}}$ is identified with 
  $\mathbf{y}^{(\sigma,\mathbf{w}^0_{\sigma},\J)}$.\n
  
 On the other hand, by the definition of $\mathbf{k}[\K]$ in~\eqref{Equ:Face-Ring-Def-1}, 
  every element
  of $\mathbf{k}[\K]$ can be uniquely written as a linear combination of monomials of the form 
  $v^{n_1}_{i_1}\cdots v^{n_s}_{i_s}$ where $\{ v_{i_1},\cdots, v_{i_s}\}$ span a
  simplex in $\K$ and $n_1,\cdots, n_s > 0$. The coefficients ring $\mathbf{k}$ is generated by
  the empty monomial which corresponds to the empty simplex $\hat{\mathbf{0}}\in \K$.
   So we obtain an additive basis of 
   $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K]$ over $\mathbf{k}$:
   $$\{ t_{\mathrm{J}} v^{n_1}_{i_1}\cdots v^{n_s}_{i_s}\,|\, 
            \mathrm{J}\subset [k],\,
            \{ v_{i_1},\cdots, v_{i_s}\} \ \text{spans a simplex in}\ \K,
            \ n_1,\cdots n_s\in \Z_+ \}.$$
  Suppose $\{ v_{i_1},\cdots, v_{i_s} \}=V(\sigma)$ for some simplex $\sigma\in \K$.
  Then corresponding to 
  $t_{\mathrm{J}} v^{n_1}_{i_1}\cdots v^{n_s}_{i_s}$,  we have the 
  cochains $\mathbf{x}^{(\sigma,\mathbf{w}_{\sigma},\J)}$ and 
  $\mathbf{y}^{(\sigma,\mathbf{w}_{\sigma},\J)}$ 
 where the weight $\mathbf{w}_{\sigma}$ is:
    $$ \mathbf{w}_{\sigma} (v_{i_j}) = n_j -1, \  1\leq j \leq s\ \, (\text{if}\ 
      \sigma \neq \hat{\mathbf{0}}\in\K).$$
   In particular, we have the correspondences:
  \[ \quad t_i\ \longleftrightarrow\ 
    \mathbf{x}^{(\hat{\mathbf{0}},\mathbf{w}_{\hat{\mathbf{0}}},\{ i\})} \ \text{and}\ 
                         \mathbf{y}^{(\hat{\mathbf{0}},\mathbf{w}_{\hat{\mathbf{0}}}, \{i\})},
           \ \forall\, i\in[k] 
                          \]
  \[   \mathbf{v}_{\sigma}  \ \longleftrightarrow \
  \mathbf{x}^{(\sigma, \mathbf{w}^0_{\sigma},\varnothing)}\ \text{and}\ 
  \mathbf{y}^{(\sigma, \mathbf{w}^0_{\sigma},\varnothing)},\ \forall\, \sigma\in \K.
   \]
   \n
 
 Similarly to the $\mathbf{x}^{\Phi^{\J}_{\sigma}}$ and $\mathbf{y}^{\Phi^{\J}_{\sigma}}$ 
 in section 2, 
 we
 can write  $\mathbf{x}^{(\sigma,\mathbf{w}_{\sigma},\J)}$ and 
  $\mathbf{y}^{(\sigma,\mathbf{w}_{\sigma},\J)}$ as cross products of cellular cochains
  in $C^*(\mathcal{X}(\K, \upalpha, S^0);\mathbf{k})$ and
  $C^*(\mathcal{X}(\K, \upalpha, S^1);\mathbf{k})$, and write out their differentials.
  In particular, we have
    $$d(\mathbf{x}^{(\hat{\mathbf{0}},\mathbf{w}_{\hat{\mathbf{0}}},\{ i\})}) =
     \sum_{v\in\alpha_i} \mathbf{x}^{(v, \mathbf{w}^0_{v},\varnothing)} , \ \
     d(\mathbf{y}^{(\hat{\mathbf{0}},\mathbf{w}_{\hat{\mathbf{0}}},\{ i\})}) =
     \sum_{v\in\alpha_i} \mathbf{y}^{(v, \mathbf{w}^0_{v},\varnothing)} $$
     $$ d (\mathbf{x}^{(\sigma, \mathbf{w}^0_{\sigma},\varnothing)}) =       
            \underset{\dim(\omega)=\dim(\sigma)+1} 
            {\sum_{\sigma\subset \omega,\, \mathrm{I}_{\upalpha}(\omega)
                  = \mathrm{I}_{\upalpha}(\sigma)}} 
            \pm \mathbf{x}^{(\omega, \mathbf{w}^0_{\omega},\varnothing)},\ \ 
            d (\mathbf{y}^{(\sigma, \mathbf{w}^0_{\sigma},\varnothing)}) =       
            \underset{\dim(\omega)=\dim(\sigma)+1} 
            {\sum_{\sigma\subset \omega,\, \mathrm{I}_{\upalpha}(\omega)
                  = \mathrm{I}_{\upalpha}(\sigma)}} 
            \pm \mathbf{y}^{(\omega, \mathbf{w}^0_{\omega},\varnothing)}. $$ 
 
 \n
 It is easy to check that when $\mathbf{k}=\Z_2$,
 the above correspondences define cochain complex isomorphisms from
 $\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K]$ to
  $C^*(\mathcal{X}(\K, \upalpha, S^0);\Z_2)$ and
  $C^*(\mathcal{X}(\K, \upalpha, S^1);\Z_2)$.
  The reason why we have to assume the $\Z_2$-coefficients here is: for any $\J\subset [k]$,
   $$ d_{\upalpha}(t_{\J})= \sum_{i\in \J} \kappa(i,\mathrm{J})\cdot t_{\J\backslash \{i\}}
   \sum_{v_j\in \alpha_i} \mathbf{v}_{\{j\}}. $$ 
   But the sign $\kappa(i,\mathrm{J})$ is not compatible with the sign
    $\varepsilon(\sigma,\omega)$ appearing in the boundary of a simplex in $\K$ unless $\upalpha$
    is the trivial partition 
      (also see Remark~\ref{Rem:Sign-Problem}).\n

     Combining the above isomorphisms of cochain complexes with the 
    cohomology isomorphisms in~\eqref{Equ:Deform-Cohomology-Iso-Real} 
    and~\eqref{Equ:Deform-Cohomology-Iso-Complex}, we obtain the following
    additive isomorphisms. 
    \[  H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k]\otimes  \Z_2[\K], d_{\upalpha}) \cong
             H^*(\mathcal{X}(\K, \upalpha, S^0);\Z_2) \cong
       H^*(X(\K,\lambda_{\upalpha});\Z_2)  \]
       \[  H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k]\otimes  \Z_2[\K], d_{\upalpha}) \cong
             H^*(\mathcal{X}(\K, \upalpha, S^1);\Z_2) \cong
       H^*(X(\K,\Lambda_{\upalpha});\Z_2)  \]
    These prove part of the claims in Theorem~\ref{Thm:Main-3}.    \\
    
        

  \section{Cohomology ring structures of $X(\K,\lambda_{\upalpha})$ and 
     $X(\K,\Lambda_{\upalpha})$}
         
    It is shown in~\cite{BP02} that for any coefficients $\mathbf{k}$,
    the cohomology ring $H^*(\mathcal{Z}_{\K};\mathbf{k})$
     of a moment-angle complex $\mathcal{Z}_{\K}$
    is isomorphic to the Tor algebra of the face ring $\mathbf{k}[\K]$ of $\K$.
    In this section, we obtain parallel results for 
     $X(\K,\Lambda_{\upalpha})$ under $\Z_2$-coefficients. 
     Similarly to the discussion of $\mathcal{Z}_{\K}$ in~\cite{BP02} and~\cite{Panov08},
      we first introduce some differential 
     graded algebras as auxiliary means for our discussion.\n     
  
         
     \subsection{A differential graded algebra $R^*_{\mathbf{k}}(\K,\upalpha)$}\ \n
         
  Suppose
    $\upalpha=\{\alpha_1,\cdots,\alpha_k\}$ is a partition of the vertex set
    $[m]=\{ v_1,\cdots, v_m\}$ of a simplicial complex $\K$.  
          Using the definition~\eqref{Equ:Face-Ring-Def-2}
       of $\mathbf{k}[\K]$,  we define
     a differential (graded) commutative algebra $R^*_{\mathbf{k}}(\K,\upalpha)$ 
     over $\mathbf{k}$ as follows.
     \begin{equation}\label{Equ:R-def}
       R^*_{\mathbf{k}}(\K,\upalpha) := \Lambda_{\mathbf{k}}[t_1,\cdots, t_k]
                  \otimes \mathbf{k}[\K]\slash \mathcal{J}_{\upalpha}
     \end{equation}   
      where $\mathcal{J}_{\upalpha}$ is 
      the ideal generated by the following subset of 
      $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]
                  \otimes \mathbf{k}[\K]$ 
      \[  \{ t_i \mathbf{v}_{\sigma}\,;\,  V(\sigma)\subset \alpha_i,\, 1\leq i \leq k\} \cup\{ \mathbf{v}_{\sigma} \mathbf{v}_{\tau} \,;\, \I_{\upalpha}(\sigma)\cap \I_{\upalpha}(\tau) \neq \varnothing
           \} \]     
          We denote the quotient map in~\eqref{Equ:R-def} by
      \begin{equation}\label{Equ:rho-Phi}
        \rho_{\upalpha}: \Lambda_{\mathbf{k}}[t_1,\cdots, t_k]\otimes
        \mathbf{k}[\K] \longrightarrow R^*_{\mathbf{k}}(\K,\upalpha).
      \end{equation}  
     Let $\bar{t}_i$ and $\bar{\mathbf{v}}_{\sigma}$
     denote the image of $t_i$ and $\mathbf{v}_{\sigma}$ in 
        $R^*_{\mathbf{k}}(\K,\upalpha)$ under $\rho_{\upalpha}$, respectively.
       $R^*_{\mathbf{k}}(\K,\upalpha)$ inherits the multi-gradings from 
       $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]\otimes
        \mathbf{k}[\K]$ which gives:
    \begin{equation}\label{Equ:Multi-Grading-2}
       \mathrm{mdeg}^{\upalpha}(\bar{t}_i) = 
       \mathrm{mdeg}^{\upalpha}(t_i) = (-1, \{i\}) \in (\Z, [k]),
    \end{equation}    
    \begin{equation}\label{Equ:Multi-Grading-3}
       \mathrm{mdeg}^{\upalpha}(\bar{\mathbf{v}}_{\sigma})
       =\mathrm{mdeg}^{\upalpha}(\mathbf{v}_{\sigma})=
       (0 ,\mathrm{I}_{\upalpha}(\sigma)) \in (\Z,[k]).
     \end{equation}    
 Moreover,  $R^*_{\mathbf{k}}(\K,\upalpha)$ inherits a differential $\bar{d}_{\upalpha}$ 
 from $d_{\upalpha}$ on
       $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]\otimes \mathbf{k}[\K]$.
      \begin{equation}\label{Equ:R-boundary-2}
        \bar{d}_{\upalpha}(\bar{t}_i) = \sum_{v_j\in \alpha_i} \bar{\mathbf{v}}_{\{j\}}, 
             \ 1\leq i \leq n,\ \ \text{and}
      \ \ \bar{d}_{\upalpha}(\bar{\mathbf{v}}_{\sigma})= 
      \underset{\dim(\omega) = \dim(\sigma)+1}
       {\sum_{\sigma\subset \omega,\, 
          \mathrm{I}_{\upalpha}(\omega) =\mathrm{I}_{\upalpha}(\sigma)}} 
        \varepsilon(\sigma,\omega)\cdot  \bar{\mathbf{v}}_{\omega}.
     \end{equation}  
    It is clear that
    $\rho_{\upalpha}$ is a chain map between 
    $\Lambda_{\mathbf{k}}[t_1,\cdots, t_k]\otimes
        \mathbf{k}[\K]$ and $R^*_{\mathbf{k}}(\K,\upalpha)$. 
       \n
 
 
     
       
    For any subset $\mathrm{J}=\{ j_1,\cdots, j_s\} \subset [k]$ with $j_1 < \cdots < j_s$, 
    let
     $$ \bar{t}_{\mathrm{J}} := \bar{t}_{j_1}\cdots \bar{t}_{j_s} \in 
          R^*_{\mathbf{k}}(\K,\upalpha). $$  
          By~\cite[Lemma 5.4]{MasPanov06}, any element of $\mathbf{k}[\K]$
  can be uniquely written as a linear combination of monomials 
  $\mathbf{v}^{l_1}_{\sigma_1}\mathbf{v}^{l_2}_{\sigma_s}\cdots \mathbf{v}^{l_s}_{\sigma_s}$
  corresponding to chains of totally ordered simplices $\sigma_1\subset \sigma_2
  \subset\cdots\subset \sigma_s$ of $\K\backslash \hat{\mathbf{0}}$. 
  Then the definition of $\mathcal{J}_{\upalpha}$
   implies that $R^*_{\mathbf{k}}(\K,\upalpha)$ is generated, as an abelian group, 
     by the set of monomials
   $$\{ \bar{t}_{\mathrm{J}} \bar{\mathbf{v}}_{\sigma}\,|\, 
     \mathrm{J}\cap \mathrm{I}_{\upalpha}(\sigma)=\varnothing,\,
        \mathrm{J}\subset [k],\,\sigma\in \K \},\ \ 
         \mathrm{mdeg}^{\upalpha}(\bar{t}_{\mathrm{J}} \bar{\mathbf{v}}_{\sigma} ) = 
         (-|\mathrm{J}|,  \mathrm{I}_{\upalpha}(\sigma)\cup \J).$$ 
    In particular, $R^*_{\mathbf{k}}(\K,\upalpha)$
    is a free abelian group of finite rank. \n
    
    For any simplex $\sigma\in \K$ and
      $\mathrm{J}\subset [k]$ with
    $\mathrm{J}\cap \mathrm{I}_{\upalpha}(\sigma)=\varnothing$, we obtain 
    from~\eqref{Equ:R-boundary-2} 
       \begin{align} \label{Equ:d-Phi-Formula}
         \bar{d}_{\upalpha}(\bar{t}_{\mathrm{J}}\bar{\mathbf{v}}_{\sigma}) 
          & =  \sum_{i\in \mathrm{J}} 
           \kappa(i,\mathrm{J})\,\bar{t}_{\mathrm{J}\backslash\{i\}} 
          \Big( \sum_{v_j\in \alpha_i} \bar{\mathbf{v}}_{\{j\}} 
           \bar{\mathbf{v}}_{\sigma} \Big)\ + \
              (-1)^{|\mathrm{J}|}\,
    \bar{t}_{\mathrm{J}} \Big( 
         \underset{\dim(\omega) = \dim(\sigma)+1}
       {\sum_{\sigma\subset\omega,\, \mathrm{I}_{\upalpha}(\omega) 
            =\mathrm{I}_{\upalpha}(\sigma)}} 
        \varepsilon(\sigma,\omega)\cdot  \bar{\mathbf{v}}_{\omega} \Big) \notag\\
        &=  \sum_{i\in \mathrm{J}} 
            \kappa(i,\mathrm{J})\,\bar{t}_{\mathrm{J}\backslash\{i\}} 
          \Big( \underset{\dim(\omega) = \dim(\sigma)+1}
          {\sum_{\sigma\subset\omega,\, \mathrm{I}_{\upalpha}(\omega) =
          \mathrm{I}_{\upalpha}(\sigma) 
          \cup\{i\}}} \bar{\mathbf{v}}_{\omega} \Big) \notag\\
          & \quad + (-1)^{|\J|}\,
        \bar{t}_{\mathrm{J}} \Big( 
         \underset{\dim(\omega) = \dim(\sigma)+1}
       {\sum_{\sigma\subset\omega,\, \mathrm{I}_{\upalpha}(\omega) 
           =\mathrm{I}_{\upalpha}(\sigma)}} 
        \varepsilon(\sigma,\omega)\cdot  \bar{\mathbf{v}}_{\omega} \Big).
       \end{align}   
          Note that $\bar{d}_{\upalpha}$ preserve the $[k]$-grading of 
         $R^*_{\mathbf{k}}(\K,\upalpha)$. 
         So for any $\mathrm{L}\subset [k]$, let
    $$R^{*,\mathrm{L}}_{\mathbf{k}}(\K,\upalpha) := \text{the $\mathbf{k}$-submodule of 
    $R^*_{\mathbf{k}}(\K,\upalpha)$
   generated by}$$
   $$\quad\qquad\ \,  \{ \bar{t}_{\mathrm{J}} \bar{\mathbf{v}}_{\sigma}\,|\, \mathrm{J}\cap 
   \mathrm{I}_{\upalpha}(\sigma)=\varnothing,\, \mathrm{J}\cup 
   \mathrm{I}_{\upalpha}(\sigma)= \mathrm{L}\}.$$
   
  Then $R^{*,\mathrm{L}}_{\mathbf{k}}(\K,\upalpha)$ is a cochain subcomplex 
   of $R^*_{\mathbf{k}}(\K,\upalpha)$ and we have 
    \begin{equation}\label{Equ:Decomp-R}
      R^*_{\mathbf{k}}(\K,\upalpha) = \bigoplus_{\mathrm{L}\subset [k]} 
      R^{*,\mathrm{L}}_{\mathbf{k}}(\K,\upalpha),\ \
    H^*(R^*_{\mathbf{k}}(\K,\upalpha)) =   \bigoplus_{\mathrm{L}\subset [k]} 
     H^*(R^{*,\mathrm{L}}_{\mathbf{k}}(\K,\upalpha)).
    \end{equation}    
  \n
 
   \begin{lem} \label{Lem:Cohomology-Isomor}
  Let $\upalpha=\{\alpha_1,\cdots, \alpha_k \}$ be a partition of the vertex set $V(\K)$ 
  of a simplicial complex $\K$.
  There are $\Z_2$-module isomorphisms 
   \[ H^*(R^*_{\Z_2}(\K,\upalpha))
    \cong H^*(X(\K,\lambda_{\upalpha});\Z_2),\ 
   H^*(R^*_{\Z_2}(\K,\upalpha))\cong 
   H^*(X(\K,\Lambda_{\upalpha});\Z_2). \]
 \end{lem}
 \begin{proof}
   By the decompositions in~\eqref{Equ:Decomp-R}, it is sufficient
  for us to show that for any subset $\mathrm{L}\subset [k]$
 there are cochain complex isomorphisms from 
 $R^{*,\mathrm{L}}_{\Z_2}(\K,\upalpha)$ to $ C^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\Z_2)$ and $C^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\Z_2)$.
 We define additive isomorphisms 
 \begin{align*}
    \psi^{\mathrm{L}}_{\upalpha} : 
       R_{\Z_2}^{*,\mathrm{L}}(\K,\upalpha) & \longrightarrow C^{*,\mathrm{L}}
         (X(\K,\lambda_{\upalpha});\Z_2),  \ \  \widehat{\psi}^{\mathrm{L}}_{\upalpha} : 
       R_{\Z_2}^{*,\mathrm{L}}(\K,\upalpha)\longrightarrow C^{*,\mathrm{L}}
        (X(\K, \Lambda_{\upalpha});\Z_2) \\
        \bar{t}_{\J} \bar{\mathbf{v}}_{\sigma} \ &\longmapsto \ \mathbf{x}^{\Phi^{\J}_{\sigma}} 
        \qquad\qquad\qquad\qquad\qquad\quad\ \bar{t}_{\J} \bar{\mathbf{v}}_{\sigma}\  \longmapsto \ \mathbf{y}^{\Phi^{\J}_{\sigma}} 
       \end{align*} 
   Note that $$ \dim(\mathbf{x}^{\Phi^{\J}_{\sigma}})=\mathrm{rank}(\sigma),\ \ \,
 \dim(\mathbf{y}^{\Phi^{\J}_{\sigma}})=\mathrm{rank}(\sigma)+|\I_{\upalpha}(\sigma)|+|\J|.$$ 
   So if we want to require $\psi^{\mathrm{L}}_{\upalpha}$
   and $\widehat{\psi}^{\mathrm{L}}_{\upalpha}$ to be dimension-preserving, we should define
   the dimensions of elements in $R^*_{\Z_2}(\K,\upalpha)$ correspondingly.
   \begin{align*}
     & \bullet \text{For $\psi^{\mathrm{L}}_{\upalpha}$, let
   $\dim(\bar{t}_{j}) =0 =\dim(S^0)$ 
        and $\dim(\bar{\mathbf{v}}_{\sigma})= \mathrm{rank}(\sigma)$}.
          \\   
    & \bullet \text{For $\widehat{\psi}^{\mathrm{L}}_{\upalpha}$, let
       $\dim(\bar{t}_j) =1=\dim(S^1)$ 
        and $\dim(\bar{\mathbf{v}}_{\sigma})= \mathrm{rank}(\sigma)+|\I_{\upalpha}(\sigma)|$}.
   \end{align*}
      In addition, we define the multi-gradings $\mathrm{mdeg}^{\upalpha}$ of 
      $\mathbf{x}^{\Phi^{\J}_{\sigma}}$ and $\mathbf{y}^{\Phi^{\J}_{\sigma}}$
       by:
       \[ \mathrm{mdeg}^{\upalpha}(\mathbf{x}^{\Phi^{\J}_{\sigma}}) = 
       \mathrm{mdeg}^{\upalpha}(\mathbf{y}^{\Phi^{\J}_{\sigma}})=
       (-|\mathrm{J}|,  \mathrm{I}_{\upalpha}(\sigma)\cup \J) \in (\Z,[k]). \ \ 
           \]
      Then $\psi^{\mathrm{L}}_{\upalpha}$ and $\widehat{\psi}^{\mathrm{L}}_{\upalpha}$
       also preserve the 
        multi-gradings of the elements in
         $R^{*,\mathrm{L}}_{\Z_2}(\K,\upalpha)$, 
         $C^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\Z_2)$
         and  $C^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\Z_2)$.\n

       
    Now we show that 
    $\psi^{\mathrm{L}}_{\upalpha} (\bar{d}_{\upalpha}(\bar{t}_{\J} \bar{\mathbf{v}}_{\sigma})) = 
     \bar{d}_{\upalpha}( \psi^{\mathrm{L}}_{\upalpha}(\bar{t}_{\J} \bar{\mathbf{v}}_{\sigma}))$
   for any $\bar{t}_{\J} \bar{\mathbf{v}}_{\sigma}\in R_{\Z_2}^{*,\mathrm{L}}(\K,\upalpha)$. 
   Because of the
   $\Z_2$-coefficents, we can ignore all the signs 
   in~\eqref{Equ:d-Phi-Formula}.   
   \begin{align*}
      \psi^{\mathrm{L}}_{\upalpha} (\bar{d}_{\upalpha}(\bar{t}_{\J} \bar{\mathbf{v}}_{\sigma})) 
      &= 
        \psi^{\mathrm{L}}_{\upalpha} \bigg( \sum_{i\in \mathrm{J}} 
         \bar{t}_{\mathrm{J}\backslash\{i\}} \Big(   \underset{\dim(\omega) = \dim(\sigma)+1}
          {\sum_{\sigma\subset\omega,\, \mathrm{I}_{\upalpha}(\omega) =
          \mathrm{I}_{\upalpha}(\sigma) 
          \cup\{i\}}} \bar{\mathbf{v}}_{\omega} \Big)\ + \
        \bar{t}_{\mathrm{J}} \Big( 
         \underset{\dim(\omega) = \dim(\sigma)+1}
       {\sum_{\sigma\subset\omega,\, \mathrm{I}_{\upalpha}(\omega) 
           =\mathrm{I}_{\upalpha}(\sigma)}} 
        \bar{\mathbf{v}}_{\omega} \Big) \bigg)\\
     &=   \psi^{\mathrm{L}}_{\upalpha} \bigg( \underset{\dim(\omega) 
         =\dim(\sigma)+1} {\sum_{\sigma\subset \omega,\, \mathrm{I}_{\upalpha}(\omega)
                  \subset\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}}} 
                  \bar{t}_{\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)} \bar{\mathbf{v}}_{\omega} \bigg) \\
      &  = \underset{\dim(\omega)=\dim(\sigma)+1} 
           {\sum_{\sigma\subset \omega,\, \mathrm{I}_{\upalpha}(\omega)
                  \subset\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}}}      
                \mathbf{x}^{\Phi_{\omega}^{\mathrm{I}_{\upalpha}(\sigma)\cup \mathrm{J}\backslash 
                      \mathrm{I}_{\upalpha}(\omega)}} 
      \overset{\eqref{Equ:Diff-Real-2}}{=} \bar{d}_{\upalpha}( \mathbf{x}^{\Phi^{\J}_{\sigma}})
        =  \bar{d}_{\upalpha}( \psi^{\mathrm{L}}_{\upalpha}(\bar{t}_{\J} 
        \bar{\mathbf{v}}_{\sigma})).
   \end{align*}
    A parallel argument shows that
     $\widehat{\psi}^{\mathrm{L}}_{\upalpha} (\bar{d}_{\upalpha}(\bar{t}_{\J} \bar{\mathbf{v}}_{\sigma})) = 
     \bar{d}_{\upalpha}( \widehat{\psi}^{\mathrm{L}}_{\upalpha}(\bar{t}_{\J} \bar{\mathbf{v}}_{\sigma}))$.
     So $\psi^{\mathrm{L}}_{\upalpha}$ and $\widehat{\psi}^{\mathrm{L}}_{\upalpha}$ are cochain complex isomorphisms. Now let
     \begin{equation}\label{Equ:psi-real}
        \psi_{\upalpha}= \bigoplus_{\mathrm{L}\subset [k]} \psi^{\mathrm{L}}_{\upalpha} :
      R^*_{\Z_2}(\K,\upalpha) \longrightarrow
       C^*(X(\K,\lambda_{\upalpha});\Z_2),
      \end{equation}
      \begin{equation} \label{Equ:psi-complex}
         \widehat{\psi}_{\upalpha}= 
      \bigoplus_{\mathrm{L}\subset [k]} \widehat{\psi}^{\mathrm{L}}_{\upalpha} :
      R^*_{\Z_2}(\K,\upalpha) \longrightarrow
       C^*(X(\K,\Lambda_{\upalpha});\Z_2).
      \end{equation}
      
    By our construction,
     $\psi_{\upalpha}$ and $\widehat{\psi}_{\upalpha}$ are cochain complex isomorphisms
     which preserve the multi-gradings 
    of the elements in
         $R^*_{\Z_2}(\K,\upalpha)$, 
         $C^*(X(\K,\lambda_{\upalpha});\Z_2)$
         and  $C^*(X(\K,\Lambda_{\upalpha});\Z_2)$. The theorem is proved.
   \end{proof}
 
 \begin{rem} \label{Rem:Sign-Problem}
    When 
  $\upalpha$ is a nontrivial partition of $V(\K)$ and $\mathbf{k}\neq \Z_2$,
 it seems that there is no proper way to define
         a chain isomorphism between 
 $R^{*,\mathrm{L}}_{\mathbf{k}}(\K,\upalpha)$ and 
 $ C^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\mathbf{k})$ or 
 $C^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\mathbf{k})$. This is 
  because when $\upalpha$ is nontrivial, the signs
  $\kappa(i,\J)$ and $\varepsilon(\sigma,\omega)$ that appear in the expressions of 
  $\bar{d}_{\upalpha}(\bar{t}_{\mathrm{J}}\bar{\mathbf{v}}_{\sigma})$,
    $d(\mathbf{x}^{\Phi^{\J}_{\sigma}})$ and $d (\mathbf{y}^{\Phi^{\J}_{\sigma}})$
    are not compatible.
 \end{rem} 
   \n


     
   \subsection{Cellular cup products of cochains in
     $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$}
   \ \n
   
     To understand the cohomology ring structures of 
     $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$, we need to
     construct explicit cup products of cellular cochains on these spaces.
     The usual way to do this is to choose cellular approximations of the diagonal maps
     of these spaces. Indeed, this is what people do in~\cite{BP02} for moment-angle complexes 
     and in~\cite{BBCG10-2} for generalized moment-angle complexes.
     In the following we first
      construct cellular cup products for some simple spaces, and then 
      write the cellular
     cup products on $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$.\nn
     
     $\bullet$ For $S^0 =\{ a^0, a^1\}$, let $(a^0)^*$ and $(a^1)^*$ be the 
     basis of $C^*(S^0;\mathbf{k})$. Then the cellular cup product is given by:
        \begin{equation} \label{Equ:Cup-Product-S-0}
           (a^0)^*\cup(a^0)^* = (a^0)^*, \ \ (a^1)^*\cup(a^1)^*= (a^1)^*, \ \ 
            (a^0)^*\cup (a^1)^* = 0. 
        \end{equation}  
      Observe that there are no elements in $C^*(S^0;\mathbf{k})$ with
      order two. \n
      
      $\bullet$ For any $d\geq 1 $, let $S^d =e^0\cup e^d$ be a cell decomposition 
      where $e^d$ is a $d$-cell.
      Let $(e^0)^* , (e^d)^* \in  C^*(S^d;\mathbf{k})$ be the dual basis of $e^0, e^d$.
      The cellular cup product on $C^*(S^d;\mathbf{k})$ is given by
       \begin{equation} \label{Equ:Cup-Product-Sd}
             (e^0)^*\cup (e^d)^*= (e^d)^*, \ \ (e^d)^*\cup (e^d)^*= 0.
        \end{equation}       
          \n
          
          
        $\bullet$ For any $r\geq 0$, let $V(\Delta^r)=\{ v_0,\cdots, v_r\}$.
       We decompose $\Delta^r$ as its open faces
         $$ \Delta^r=\bigcup_{\sigma\in\Delta^r} \sigma^{\circ}.$$ 
       Consider $\{ \sigma^{\circ} \,|\, \sigma\in\Delta^r \}$ as a basis of the
       cellular chain complex of $\Delta^r$ and let $\sigma^* \in C^*(\Delta^r;\mathbf{k})$
          denote the dual basis of $\sigma^{\circ}$. 
       Then by the Alexander-Whitney diagonal for $\Delta^r$ (see~\cite{Rotman}), we obtain
          a cellular cup product on $\Delta^r$ by
      $$ \sigma^* \cup \tau^* = \begin{cases}
                            (\sigma\vee \tau)^*,   &   \text{ $\sigma\cap \tau =\varnothing$ } \\
                         \ 0,     &   \text{ else. } \\
                       \end{cases} $$
          \n
      
      $\bullet$ The above cellular cup products on $S^d$ and $\Delta^r$ uniquely determine
      a cellular cup product on $S^d \times \Delta^r\times \Delta^1$ (with the product cell 
      structure) by the rule:
      \[ (f_1\times g_1)\cup (f_2\times g_2) = (-1)^{|g_1||f_2|} (f_1\cup f_2)\times (g_1\cup g_2). \] 
      Note that $S^d*\Delta^r$ is a cellular
       quotient space of $S^d \times \Delta^r\times \Delta^1$
      by collapsing $S^d \times \Delta^r\times \{ 0\}$ to $S^d$ and 
      $S^d \times \Delta^r\times \{ 1\}$ to $\Delta^r$. Then $S^d*\Delta^r$ inherits the
      cellular cup product from $S^d \times \Delta^r\times \Delta^1$, which can be 
      described explicitly as follows.\n
      
      Let $p: S^d \times \Delta^r\times \Delta^1 \rightarrow S^d*\Delta^r$ be the quotient map.           
      By our argument in Section 3,   
      $\{S^d * \sigma^{\circ} \, |\, \sigma \in \widehat{\Delta}^r \}$ is a basis 
      for the cellular chain complex of $S^d*\Delta^r$. 
  \begin{itemize}
   \item[--] When $d=0$, $S^0 * \hat{0} = a^1-a^0$ and $S^0* -\hat{1} = a^0$, where
    $S^0=\{ a^0,a^1\}$;\n
   
   \item[--] When $d\geq 1$, $S^d * \hat{0} = e^d$ and $S^d * -\hat{1}=e^0$.
   \end{itemize}
    Note that $p^{-1}(S^d * \sigma^{\circ}) = S^d\times \sigma^{\circ} \times [0,1]$
     for any nonempty face $\sigma$ of $\Delta^r$.\n
 
   Let $\{ z^{\sigma}\,|\, \sigma\in \widehat{\Delta}^r\} $
  be the basis of $C^*(S^d*\Delta^r;\mathbf{k})$ where $z^{\sigma}$ is the dual of 
  $S^d * \sigma^{\circ}$.  
  By the notations in Section 3,
  $z^{\sigma} = x^{\sigma}$ or $y^{\sigma}$ when $d=0$ or $1$.
  The differential of $z^{\sigma}$ is parallel 
  to~\eqref{Equ:Diff-Complex-0} and~\eqref{Equ:Diff-Complex-0-0}. 
 \n
  
  By~\eqref{Equ:Cup-Product-Sd}, the cellular cup product on $S^d*\Delta^r$ inherited from 
    $S^d \times \Delta^r\times \Delta^1$ reads:
 \begin{itemize}
   \item[--] when $d=0$, 
          $z^{\hat{0}}=(a^1)^*$ and $z^{-\hat{1}}=(a^0)^*+(a^1)^*$ 
           (see Remark~\ref{Rem:S0-Cochains}). So 
           $$z^{\hat{0}}\cup z^{\hat{0}} = z^{\hat{0}}, \ \ \,  
             z^{-\hat{1}}\cup z^{\hat{0}} = z^{\hat{0}},\ \ \,
             z^{-\hat{1}}\cup z^{-\hat{1}} =
       z^{-\hat{1}}\ \, \text{(by~\eqref{Equ:Cup-Product-S-0})}.$$
        For any nonempty
       faces $\sigma,\tau$ of $\Delta^r$, we have $z^{\sigma}\cup z^{\tau} =
       \begin{cases}
   z^{\sigma\vee\tau},   &   \text{ $\sigma\cap \tau=\varnothing$; } \\
   \ 0,     &   \text{ else. } 
 \end{cases} $  \nn
         
  \item[--] when $d\geq 1$, $z^{\sigma}\cup z^{\tau} = 0$ for any $\sigma,\tau \in 
        \widehat{\Delta}^r \backslash -\hat{1}$,
        and $z^{-\hat{1}}\cup z^{\sigma}=z^{\sigma}$ for any $\sigma\in\widehat{\Delta}^r$.\n
  \end{itemize} 
\n
 
               
 $\bullet$ Given any partition $\upalpha=\{\alpha_1,\cdots, \alpha_k \}$ of $V(\K)=[m]$,
       we obtain a cellular cup product on the Cartesian product 
       $\prod_{i\in [k]} S^d * \Delta^{\alpha_i}$ by:
       $$ ( z^{\sigma_1}_{1}\times \cdots \times z^{\sigma_k}_k )\cup
       ( z^{\tau_1}_{1}\times \cdots \times z^{\tau_k}_k ) = 
         ( z^{\sigma_1}_{1}\cup z^{\tau_1}_{1}) \times \cdots \times (z^{\sigma_k}_k 
         \cup z^{\tau_k}_k ). $$ 
      Then $C^*\big( \prod_{i\in [k]} S^d*\Delta^{\alpha_i};\mathbf{k}\big)$ is isomorphic to
       $\bigotimes_{i\in[k]} C^*(S^d*\Delta^{\alpha_i};\mathbf{k})$ as graded algebras
       with respect to the cup products.
   Moreover, by the inclusions (see~\eqref{Equ:Universal-Space})
       $$ X(\K,\lambda_{\upalpha}) \hookrightarrow \prod_{i\in [k]} S^0 * \Delta^{\alpha_i},\ \
       X(\K,\Lambda_{\upalpha}) \hookrightarrow 
     \prod_{i\in [k]} S^1 * \Delta^{\alpha_i},$$ 
     $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$ inherit 
     the cellular cup products from $\prod_{i\in [k]} S^0* \Delta^{\alpha_i}$ and
     $\prod_{i\in [k]} S^1 * \Delta^{\alpha_i}$, respectively.
       \nn
         
   Now we are ready to prove the following lemma which is the key to the proof 
   of Theorem~\ref{Thm:Main-3}.
      
     \begin{lem} \label{Lem:Cohomology-Isomor-ring}
      Let $\upalpha=\{\alpha_1,\cdots, \alpha_k \}$ be a partition of the vertex set $V(\K)$ 
      of a simplicial complex $\K$.
     There is a ring isomorphism 
   \[ H^*(R^*_{\Z_2}(\K,\upalpha);\Z_2)\cong 
   H^*(X(\K,\Lambda_{\upalpha});\Z_2). \]
  \end{lem}
     \begin{proof}
      It is sufficient to show that
       $\widehat{\psi}_{\upalpha}: R^*_{\Z_2}(\K,\upalpha) \longrightarrow
       C^*(X(\K,\Lambda_{\upalpha});\Z_2)$ defined in~\eqref{Equ:psi-complex} is a 
    ring isomorphism with respect to the product structure of 
    $ R_{\Z_2}^*(\K,\upalpha)$ and the cup products of cellular cochains
     in $X(\K,\Lambda_{\upalpha})$. In the following
      we proceed our proof along the same line as~\cite[Lemma 4.6]{Panov08}.\n
    
  For any coefficients $\mathbf{k}$, 
    notice that $R_{\mathbf{k}}^*(\Delta^{[m]},\upalpha)$ is isomorphic to the tensor product 
    $\underset{i\in [k]}{\bigotimes} R_{\mathbf{k}}^*(\Delta^{\alpha_i})$ as differential 
    algebras where 
     \[ R_{\mathbf{k}}^*(\Delta^{\alpha_i}) :=
          \Lambda_{\mathbf{k}}[t_i]
                  \otimes \mathbf{k}[\Delta^{\alpha_i}]\slash 
                  ( t_i \mathbf{v}_{\sigma} = \mathbf{v}_{\sigma}\mathbf{v}_{\tau}=0,\ 
                   \forall \sigma,\tau \in \Delta^{\alpha_i}).  \]
   The quotient images of $t_i$ and $\mathbf{v}_{\sigma}$ 
   are denoted by $\bar{t}_i$ and $\bar{\mathbf{v}}_{\sigma}$, respectively.
  The differential in $R_{\mathbf{k}}^*(\Delta^{\alpha_i})$ is defined 
  by~\eqref{Equ:R-boundary-2}. 
  It is easy to see that $R_{\mathbf{k}}^*(\Delta^{\alpha_i})$ is
 spanned as a $\mathbf{k}$-module by 
 $$\{ 1, \bar{t}_i, \bar{\mathbf{v}}_{\sigma};\, \sigma \ \text{is any
 nonempty face of}\ \Delta^{\alpha_i}\}.$$
    
 For any $d\geq 1$, we define a $\mathbf{k}$-linear map 
 $\psi^{S^d}_{\alpha_i}: 
 R_{\mathbf{k}}^*(\Delta^{\alpha_i}) \rightarrow C^*(S^d*\Delta^{\alpha_i};\mathbf{k})$ by:
 
  \begin{itemize}         
   \item  $\psi^{S^d}_{\alpha_i} ( 1 ) = z^{-\hat{1}}$ and
          $\psi^{S^d}_{\alpha_i} ( \bar{t}_i ) = z^{\hat{0}}$; \n
    
    \item  $ \psi^{S^d}_{\alpha_i} ( \bar{\mathbf{v}}_{\sigma}) = \ z^{\sigma} $    
             for any nonempty face $\sigma$ of $\Delta^{\alpha_i}$. 
  \end{itemize}  
 
  It is easy to check that
  $\psi^{S^d}_{\alpha_i}$ is a ring isomorphism with respect to the cellular cup product in 
  $C^*(S^d*\Delta^{\alpha_i};\mathbf{k})$ described above.
 Moreover, $\psi^{S^d}_{\alpha_i}$ is a chain map
  with respect to the differentials in $R_{\mathbf{k}}^*(\Delta^{\alpha_i})$ and
  $C^*(S^d*\Delta^{\alpha_i};\mathbf{k})$.\n
   
   Taking the tensor product of $\psi^{S^d}_{\alpha_i}$, $1\leq i \leq k$, we obtain a ring
    isomorphism
    $$\Psi^{S^d}_{\upalpha} : R_{\mathbf{k}}^*(\Delta^{[m]},\upalpha) 
    \overset{\underset{i\in[k]}{\bigotimes} \psi^{S^d}_{\alpha_i}}{\xrightarrow{\hspace*{1.2cm}}} 
     \bigotimes_{i\in[k]} C^*(S^d*\Delta^{\alpha_i};\mathbf{k}) 
     \overset{\cong\,}{\longrightarrow} C^*\Big( \prod_{i\in [k]}
                       S^d*\Delta^{\alpha_i};\mathbf{k}\Big).
       $$
 In addition, the inclusion $\K\hookrightarrow \Delta^{[m]}$ induces 
   a ring epimorphism 
   $$\eta_{\upalpha}: R_{\mathbf{k}}^*(\Delta^{[m]},\upalpha) \rightarrow 
    R_{\mathbf{k}}^*(\K,\upalpha).$$
    
     Let $\xi_{\upalpha}:  C^*\big(\prod_{i\in [k]}
    S^1*\Delta^{\alpha_i};\mathbf{k}\big) \rightarrow C^*(X(\K,\Lambda_{\upalpha});\mathbf{k})$ 
    be the ring homomorphism induced by the inclusion $X(\K,\Lambda_{\upalpha}) \hookrightarrow 
     \prod_{i\in [k]} S^1 * \Delta^{\alpha_i}$.
     Then from the definitions of 
    $\widehat{\psi}_{\upalpha}$ and $\Phi^{S^1}_{\upalpha}$,
   we have the following commutative diagram for $\mathbf{k}=\Z_2$.
      \[
        \xymatrix{
           R_{\Z_2}^*(\Delta^{[m]},\upalpha) \ar[r]^{\Psi^{S^1}_{\upalpha}\qquad }  \ \
           \ar[d]_{\eta_{\upalpha}} &  \ C^*\Big( \prod_{i\in [k]} 
                                    S^1*\Delta^{\alpha_i};\Z_2\Big)   
                                     \ar[d]_{\xi_{\upalpha}} \\
         R^*_{\Z_2}(\K,\upalpha) \ar[r]^{\widehat{\psi}_{\upalpha}\quad\ } \ \ & \ \ C^*(X(\K,\Lambda_{\upalpha});\Z_2)
        } 
        \]
      Next we show that $\widehat{\psi}_{\upalpha}$ is a ring isomorphism. 
        For any $\gamma_1,\gamma_2 \in R^*_{\Z_2}(\K,\upalpha)$,
        there exist $\gamma_1',\gamma_2'\in R_{\Z_2}^*(\Delta^{[m]},\upalpha)$ with
        $\eta_{\upalpha}(\gamma_1')=\gamma_1$ and $\eta_{\upalpha}(\gamma_2')=
        \gamma_2$ since $\eta_{\upalpha}$ is surjective. Then since $\eta_{\upalpha}$,
        $\Psi^{S^1}_{\upalpha}$ and $\xi_{\upalpha}$ are all ring homomorphisms,  we obtain
        \begin{align*}
           \widehat{\psi}_{\upalpha}(\gamma_1\gamma_2) & = \widehat{\psi}_{\upalpha}(\eta_{\upalpha}(\gamma_1')
             \eta_{\upalpha}(\gamma_2'))  = \widehat{\psi}_{\upalpha}
             (\eta_{\upalpha}(\gamma_1'\gamma_2')) \\
            & =\xi_{\upalpha}\big( \Psi^{S^1}_{\upalpha}(\gamma_1'\gamma_2') \big) 
              = \xi_{\upalpha} \big(\Psi^{S^1}_{\upalpha}(\gamma_1') 
                 \cdot \Psi^{S^1}_{\upalpha}(\gamma_2')\big) \\
             & = \xi_{\upalpha} \big(\Psi^{S^1}_{\upalpha}(\gamma_1') \big)
                 \cdot \xi_{\upalpha}\big(\Psi^{S^1}_{\upalpha}(\gamma_2')\big)\\
             &= \widehat{\psi}_{\upalpha}(\eta_{\upalpha}(\gamma_1'))\cdot 
             \widehat{\psi}_{\upalpha}(\eta_{\upalpha}(\gamma_2')) = \widehat{\psi}_{\upalpha}(\gamma_1)
              \widehat{\psi}_{\upalpha}(\gamma_2) 
         \end{align*}     
     So $\widehat{\psi}_{\upalpha}$ is a ring isomorphism. The lemma is proved.     
  \end{proof}
  
  \begin{rem}
   The map $\psi^{S^d}_{\alpha_i}$ 
 is well defined only when $d\geq 1$ because when $d=0$, the element
  $z^{\hat{0}} \in C^*(S^0*\Delta^{\alpha_i};\mathbf{k})$ is not of order two.
 This is 
 why we do not obtain an algebra isomorphism from
  $H^*(R^*_{\mathbf{k}}(\K,\upalpha);\Z_2)$ to
   $H^*(X(\K,\lambda_{\upalpha});\Z_2)$ (also 
   see the remark after~\cite[Theorem 8.17]{BP02} and~\cite[Remark 9]{LuZhi09}). 
   In addition, it is constructed in~\cite[Theorem 1.4]{Cai-13}
   a non-commutative differential graded algebra which
       describes the cellular cochain algebra of 
     a real moment-angle complex $\R\mathcal{Z}_{\K}$.
  \end{rem}
       
  \begin{rem}
     By Lemma~\ref{Lem:Cohomology-Isomor},
     $C^*(X(\K,\Lambda_{\upalpha});\Z_2)$ is a graded commutative
     ring with respect to the cellular cup products described above. This
     is a very special property of $X(\K,\Lambda_{\upalpha})$ since,  
     a general CW-complex may not
     admit a (graded) commutative cellular cup product of cochains.
  \end{rem}         
  
   
   
  Combining the isomorphisms
  in Lemma~\ref{Lem:Cohomology-Isomor-ring} and Theorem~\ref{Thm:Main-1}, we obtain
   an additive isomorphism 
  \[ f= \widehat{\psi}^{-1}_{\upalpha} \circ \widehat{\varphi}_{\upalpha}:\ 
        \bigoplus_{q\geq 0}\underset{\mathrm{L} \subset [k]}{\bigoplus} 
                   \widetilde{H}^{q-1}(\K_{\upalpha, \mathrm{L}};\Z_2)
                   \longrightarrow  \bigoplus_{q\geq 0} 
                   \underset{\mathrm{L} \subset [k]}{\bigoplus}  
                   H^{q+|\mathrm{L}|}(R^{*,\mathrm{L}}_{\Z_2}(\K,\upalpha)). \]
        Since $R^*_{\mathbf{k}}(\K,\upalpha)$ is an algebra,
         $f$ induces an algebra structure on the left. 
           Consider the products of simplicial cochains 
        of the full subcomplexes $\K_{\upalpha, \mathrm{L}}$ given by:
     \begin{align} \label{Equ:Product-Simplicial}
         C^{p-1}(\K_{\upalpha,\mathrm{L}};\Z_2) \otimes 
             C^{q-1}(\K_{\upalpha,\mathrm{L}'};\Z_2) 
            & \longrightarrow
              C^{p+q-1}(\K_{\upalpha,\mathrm{L}\cup\mathrm{L}'};\Z_2) \notag\\
         \sigma^* \otimes \tau^* \ \  & \longmapsto  
         \begin{cases}
                (\sigma\vee\tau)^* ,   &   \text{if $\mathrm{L}\cap\mathrm{L}' =\varnothing$;} \\
                  0,     &   \text{otherwise.} 
       \end{cases}
  \end{align}
 \n
  
  \begin{prop}
    The product in the direct sum $\underset{q\geq 0}{\bigoplus}\,
       \underset{\mathrm{L} \subset [k]}{\bigoplus} 
                   \widetilde{H}^{q-1}(\K_{\upalpha, \mathrm{L}};\Z_2)$ 
                   induced from the isomorphisms in 
    Lemma~\ref{Lem:Cohomology-Isomor} and Theorem~\ref{Thm:Main-1} coincides with the product
    defined in~\eqref{Equ:Product-Simplicial}.
  \end{prop}
  \begin{proof}
     The argument is completely parallel to~\cite[Proposition 3.2.10]{BP13}. 
     By the definitions of $\widehat{\varphi}_{\upalpha}$ and
     $\widehat{\psi}_{\upalpha}$, 
     for any simplex $\sigma\in \K_{\upalpha,\mathrm{L}}$, 
     $$f(\sigma^*) = \bar{t}_{\mathrm{L}\backslash \I_{\upalpha}(\sigma)} 
              \bar{\mathbf{v}}_{\sigma}.$$
    Then for any simplices $\sigma\in \K_{\upalpha,\mathrm{L}}$ and $\tau\in
    \K_{\upalpha,\mathrm{L}'}$, the induced product is given by: 
     $$\sigma^*\bullet \tau^* := f^{-1}(f(\sigma^*)f(\tau^*)) =
     f^{-1} \big( \bar{t}_{\mathrm{L}\backslash \I_{\upalpha}(\sigma)} 
              \bar{\mathbf{v}}_{\sigma} \bar{t}_{\mathrm{L}'\backslash \I_{\upalpha}(\tau)} 
              \bar{\mathbf{v}}_{\tau} \big). $$ 
    If $\mathrm{L} \cap \mathrm{L}'\neq \varnothing$, then  
    $\bar{t}_{\mathrm{L}\backslash \I_{\upalpha}(\sigma)} 
              \bar{\mathbf{v}}_{\sigma} \bar{t}_{\mathrm{L}'\backslash \I_{\upalpha}(\tau)} 
              \bar{\mathbf{v}}_{\tau}$ is zero in $R^*_{\Z_2}(\K,\upalpha)$. Otherwise, we have
              $$\bar{t}_{\mathrm{L}\backslash \I_{\upalpha}(\sigma)} 
              \bar{\mathbf{v}}_{\sigma} \bar{t}_{\mathrm{L}'\backslash \I_{\upalpha}(\tau)} 
              \bar{\mathbf{v}}_{\tau} = \bar{t}_{(\mathrm{L}\cup\mathrm{L}')\backslash 
               (\I_{\upalpha}(\sigma)\cup\I_{\upalpha}(\tau))} 
              \bar{\mathbf{v}}_{\sigma\vee\tau} =
              \bar{t}_{(\mathrm{L}\cup\mathrm{L}')\backslash 
               \I_{\upalpha}(\sigma\vee\tau)} 
              \bar{\mathbf{v}}_{\sigma\vee\tau}.$$   
     So when $\mathrm{L} \cap \mathrm{L}'= \varnothing$, the product $\sigma^*\bullet \tau^* =
     (\sigma\vee\tau)^*$. The claim is proved.    
   \end{proof}
   \n
   
  \begin{lem} \label{Equ:Isom-Kozul-Quotient}
     For any partition
    $\upalpha=\{\alpha_1,\cdots,\alpha_k\}$ of the vertex set
      of a simplicial complex $\K$,  the 
    homomorphism $\rho_{\upalpha}: \Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes 
    \Z_2[\K]\rightarrow R^*_{\Z_2}(\K,\upalpha)$ induces a ring
     isomorphism of cohomology.
  \end{lem}
   \begin{proof}
   Our argument is parallel to the proof of~\cite[Theorem 3.6]{LuPanov11}. 
   From our discussion in section 4, we obtain the following commutative diagram
    \begin{equation} \label{Equ:Diagram-Koszul-complex}
          \xymatrix{
           \Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K] 
                \ar[r] \ar[d]_{\rho_{\upalpha}} &
            C^*(\mathcal{X}(\K, \upalpha, S^1);\Z_2)
            \ar[d] \\
         R^*_{\Z_2}(\K,\upalpha) \ar[r]^{\widehat{\psi}_{\upalpha}\quad } &  C^*(X(\K,\Lambda_{\upalpha});\Z_2)
        } 
     \end{equation}
  where the upper horizontal arrow is an isomorphism
   of cochain complexes, and the
  right vertical arrow induces an isomorphism of cohomology rings. By 
  Lemma~\ref{Lem:Cohomology-Isomor},
 $\widehat{\psi}_{\upalpha}$ induce isomorphisms of cohomology groups. 
  So $\rho_{\upalpha}$
  induces an isomorphism of cohomology groups. Then since 
  $\rho_{\upalpha}$ is a ring homomorphism, it induces 
  a cohomology ring isomorphism.
\end{proof}
   
   \begin{rem}
    It is very likely that Lemma~\ref{Equ:Isom-Kozul-Quotient}
     holds for arbitrary coefficients $\mathbf{k}$ as well. 
    Specifically, it might be possible to show that $\rho_{\upalpha}$ is a cochain equivalence
     using the strategy in the proof of~\cite[Lemma 4.4]{Panov08}. 
    But this approach may involve very cumbersome formulae. 
    For the trivial partition $\upalpha^*$ of $V(\K)$, 
    the proof is given in~\cite[Lemma 4.4]{Panov08} and~\cite[Theorem 3.6]{LuPanov11}.
   \end{rem}
 
   
  
 \begin{proof}[\textbf{Proof of Theorem~\ref{Thm:Main-3}}]   \ \n
  By Lemma~\ref{Equ:Isom-Kozul-Quotient}, we obtain
   a multi-grading on $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes 
    \Z_2[\K],d_{\upalpha})$ as follows. For any subset $\mathrm{L}\subset [k]$, define
    $$ H^{*,\mathrm{L}}(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes 
    \Z_2[\K],d_{\upalpha}) = \rho_{\upalpha}^{-1}
    \big(H^*(R^{*,\mathrm{L}}_{\mathbf{k}}(\K,\upalpha)) \big). $$
   So by Lemma~\ref{Lem:Cohomology-Isomor},
   there is an isomorphism from
   $H^{*,\mathrm{L}}(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K], d_{\upalpha})$ 
     to $ H^{*,\mathrm{L}}(X(\K,\lambda_{\upalpha});\Z_2)$ and 
    $ H^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\Z_2)$ (as $\Z_2$-modules). \n
          
   Moreover, Lemma~\ref{Lem:Cohomology-Isomor-ring} and Lemma~\ref{Equ:Isom-Kozul-Quotient}
    imply that
    $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes 
    \Z_2[\K],d_{\upalpha})$
     is isomorphic (via $\widehat{\psi}_{\upalpha}\circ \rho_{\upalpha}$)
      to $H^*(X(\K,\Lambda_{\upalpha});\Z_2)$ as multigraded $\Z_2$-algebras. This finishes
      the proof of Theorem~\ref{Thm:Main-3}.
     \end{proof}  
     
  
   Suppose $\upalpha$ is a non-degenerate partition of $V(\K)$. In Section 1, we have shown
    that
     $\mathrm{Tor}_{\Z_2[u_1,\cdots, u_k]}(\Z_2[\K], \Z_2) \cong H^*(
     \Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K],d_{\upalpha})$ in this case.
    Let
  $ \mathrm{Tor}^{*,\mathrm{L}}_{\mathbf{k}[u_1,\cdots, u_k]}(\mathbf{k}[\K], \mathbf{k})$
  denote
   $H^{*,\mathrm{L}}(\Lambda_{\mathbf{k}}[t_1,\cdots, t_k] \otimes \mathbf{k}[\K],d_{\upalpha})$.   
   Then by  Lemma~\ref{Lem:Cohomology-Isomor} and Theorem~\ref{Thm:Main-1}, we obtain the following group isomorphisms.
   \[ \mathrm{Tor}^{*,\mathrm{L}}_{\Z_2[u_1,\cdots, u_k]}(\Z_2[\K], \Z_2) \cong
    H^*(R^{*,\mathrm{L}}_{\Z_2}(\K,\upalpha)) \cong 
    H^{*,\mathrm{L}}(X(\K,\Lambda_{\upalpha});\Z_2) \cong
       \widetilde{H}^*(\K_{\upalpha,\mathrm{L}};\Z_2). \]
  
   \n
  
  \subsection{Connection to Moment-angle Complexes of Simplicial Posets}\ \n
     A poset (partially ordered set) $\mS$ with the order relation
    $\leq $ is called \emph{simplicial} if it has an initial element
    $\hat{0}$ and for each $\sigma \in \mS$ the lower segment
    $$[\hat{0},\sigma]=\{ \tau\in \mS: \hat{0} \leq \tau \leq \sigma\}$$
    is the face poset of a simplex. We refer to $\sigma\in\mS$ as a simplex
    of dimension $l$ if $[\hat{0},\sigma]$ is the face poset of an $l$-simplex.
   \n
        
    For each $\sigma \in \mS$ we assign a geometric simplex $\Delta^{\sigma}$
    whose face poset is $[\hat{0},\sigma]$, and glue these geometric simplices
    together according to the order relation in $\mS$. We get a cell
    complex $\Delta^{\mS}$ in which the closure of each cell is identified with a simplex
    preserving the face structure, and all attaching maps are inclusions.    
      We call $\Delta^{\mS}$ the \emph{geometric realization} of $\mS$.
       For convenience, we still use $\Delta^{\sigma}$ to denote the image of each
    geometric simplex $\Delta^{\sigma}$ in $\Delta^{\mS}$. Then $\Delta^{\sigma}$
    is a \emph{maximal simplex} of $\Delta^{\mS}$ 
    if and only if $\sigma$ is a maximal element of $\mS$.
  \n
      
    The notion of moment-angle complex $\mathcal{Z}_{\mS}$ associated to a 
    simplicial poset $\mS$ is introduced and thoroughly studied 
    in~\cite{LuPanov11}. Note that the barycentric subdivision also makes sense
    for $\Delta^{\mS}$. Let $P_{\mS}$ denote the cone of the barycentric
    subdivision of $\Delta^{\mS}$. Let the vertex set of $\Delta^{\mS}$ be
     $V(\Delta^{\mS})=\{ v_1,\cdots, v_m\}$.
     Let $\Lambda: V(\Delta^{\mS}) \rightarrow \Z^m$ be a map so that 
     $\{ \Lambda(v_i), \, 1\leq i \leq m\}$ is a basis of $\Z^m$.  
    Then we can construct
      $\mathcal{Z}_{\mS}$ from $P_{\mS}$ and $\Lambda$ 
     via the same rule in~\eqref{Def:Complex-Quotient}. So we also denote
      $\mathcal{Z}_{\mS}$ by $X(\mS,\Lambda)$. \n
      
      \begin{prop} \label{Prop:MAC-Poset}
         For any finite simplicial poset $\mS$, there always exists a finite 
      simplicial complex
      $\K$ and a partition $\upalpha$ of $V(\K)$ so that $\mathcal{Z}_{\mS}$
      is homotopy equivalent to $X(\K,\Lambda_{\upalpha})$.
      \end{prop}
      \begin{proof}
       We can construct $\K$ and $\upalpha$ in the following way. 
       Let $p: \Delta^{\mS} \times \R \rightarrow \Delta^{\mS}$ be the projection
       where $\Delta^{\mS}$ is identified with $\Delta^{\mS}\times \{0\}$. 
        Then for each maximal simplex
       $\Delta^{\sigma}\subset \Delta^{\mS}$, we 
        can choose a simplex $\widetilde{\Delta}^{\sigma}\subset \Delta^{\mS} \times \{ k \} $
        for some $0\leq k \leq n$ 
       where $n$ is a large enough integer, so that
       \begin{itemize}
           \item $p$ maps $\widetilde{\Delta}^{\sigma}$ simplicially isomorphically onto 
       $\Delta^{\sigma}$.\n
             
       \item The vertices of $\widetilde{\Delta}^{\sigma}$ are in 
       $V(\Delta^{\mS})\times \{ 0,\cdots, n\}$.    \n
       
        \item $\widetilde{\Delta}^{\sigma}\cap \widetilde{\Delta}^{\tau} =\varnothing$ for any 
              maximal elements $\sigma$ and $\tau$ in $\mS$.  
        \end{itemize} 
      We call $\widetilde{\Delta}^{\sigma}$ a \emph{horizontal lifting} of $\Delta^{\sigma}$.
       We consider the interval $[0,n]\subset \R$
       as a $1$-dimensional simplicial complex whose vertices are $\{0,\cdots, n\}$, and
        consider $\Delta^{\sigma} \times [0,n]$ 
       as the the Cartesian product of $\Delta^{\sigma}$ and $[0,n]$ as 
       simplicial complexes (see~\cite[Construction 2.11]{BP02}).
       If $\sigma$ and $\tau$ are both maximal, 
       $\Delta^{\sigma}\cap \Delta^{\tau}$ is the geometric realization of 
       $\sigma\wedge\tau$.
       Now define
         $$\K= \Big(\underset{\text{maximal}}{\bigcup_{\sigma\in \mS}} 
             \widetilde{\Delta}^{\sigma} \Big)
           \bigcup \Big( \underset{\text{maximal}}{\bigcup_{\sigma,\tau\in \mS}}
            (\Delta^{\sigma} \cap \Delta^{\tau} )\times [0,n] \Big).
            $$     
     Then $\K$ is clearly 
     a simplicial complex whose vertex set is $V(\mS)\times \{ 0,\cdots, n\}$.
    We call $\K$ a \emph{stretch of $\mS$} (see Figure~\ref{p:Lifting} for example).
     Let $V(\Delta^{\mS})=\{ v_1,\cdots, v_m\}$.  
      We 
       define a partition $\upalpha=\{ \alpha_1,\cdots,\alpha_m\}$ of $V(\K)$
       by 
       $$\alpha_i=\{ v_i \times \{ j\} \,;\, 0\leq j \leq n \},\ 1\leq i \leq m.$$
       
        \begin{figure}
         \includegraphics[width=0.9\textwidth]{Lifting.eps}\\
          \caption{A stretch of a simplicial poset} 
          \label{p:Lifting}
      \end{figure}
       
      Observe that the deformation retraction of $(\Delta^{\sigma}\cap \Delta^{\tau}) 
      \times [0,n]$ to
      $(\Delta^{\sigma}\cap \Delta^{\tau}) \times\{0\}$ for 
      any maximal simplices $\sigma,\tau\in\mS$
      gives us a homotopy equivalence between
      $\K$ and $\Delta^{\mS}$. This induces a homotopy equivalence between
       $P_{\K}$ and $P_{\mS}$,
      which further induces a homotopy equivalence between
       $X(\K,\Lambda_{\upalpha})$ and $X(\mS,\Lambda)=\mathcal{Z}_{\mS}$ via
       the construction~\eqref{Def:Complex-Quotient}.
       \end{proof}
      
      The notion of the face ring $\mathbf{k}[\mS]$ of a 
      simplicial poset $\mS$ is introduced by Stanley in~\cite{St91}.
       It is shown in~\cite{LuPanov11} that the integral cohomology ring of $\mathcal{Z}_{\mS}$
       is isomorphic to 
       $\mathrm{Tor}_{\Z[v_1,\cdots, v_m]}(\Z[\mS],\Z)$ as multi-graded
       algebras.
       Then Proposition~\ref{Prop:MAC-Poset} 
       implies that there is a multi-graded algebra isomorphism 
       from $\mathrm{Tor}_{\Z_2[v_1,\cdots, v_m]}(\Z_2[\mS],\Z_2)$ to 
       $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_m] \otimes \Z_2[\K], d_{\upalpha})$
       for some simplicial complex $\K$ and a partition $\upalpha=\{ \alpha_1,\cdots,\alpha_m\}$
        of $V(\K)$. \\
   
  
   \section{Some generalizations}
 
    We can generalize our discussion on $X(\K,\lambda_{\upalpha})$
    and $X(\K,\Lambda_{\upalpha})$ to a wider range of spaces defined below.\n
    
  \begin{itemize}
  
  \item[(I)] We can replace the simplicial complex $\K$ 
  by a simplicial poset $\mS$ in the constructions~\eqref{Def:Real-Quotient} 
      and~\eqref{Def:Complex-Quotient}. Then from a geometric realization of $\mS$, 
      we can define the simple polyhedral complex $P_{\mS}$ and,
      construct spaces $X(\mS,\lambda_{\upalpha})$
    and $X(\mS,\Lambda_{\upalpha})$ from any partition $\upalpha$ of the vertex set of $\mS$.
    The parallel statements of Theorem~\ref{Thm:Main-1}, 
     Theorem~\ref{Thm:Stable-Decomp},  Theorem~\ref{Thm:Main-2}
    and Theorem~\ref{Thm:Main-3}
   (by replacing $\K$ by $\mS$) also hold for these more general spaces. Their proofs are
   similar to the case of simplicial complexes. In particular, we can consider the simplicial 
   poset $\mS$ from a categorical viewpoint and take ``colimit'' of all the constructions 
   in our proofs (see~\cite{LuPanov11} for the details of this kind of argument).\nn

     
   \item[(II)] For a
  partition $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ of $V(\K)$, 
  we can replace the $S^0$ in~\eqref{Equ:Decomp-Real-form} and 
  the $S^1$ in~\eqref{Equ:Decomp-Complex-form} 
  by a sequence of spheres $\mathbb{S}=( S^{d_1},\cdots, S^{d_k})$ and define
  \[ \quad\ \qquad  X(\K, \upalpha,\mathbb{S}) = \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} S^{d_i}*(\sigma\cap \Delta^{\alpha_i}) \times
        \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} 
        S^{d_i} \Big) \subset \prod_{i\in [k]} 
        S^{d_i}*\Delta^{\alpha_i}.
  \]
   We obtain the following results for $X(\K, \upalpha,\mathbb{S})$ which are 
   parallel to Theorem~\ref{Thm:Main-1},  Theorem~\ref{Thm:Stable-Decomp},
   Theorem~\ref{Thm:Main-2} and Theorem~\ref{Thm:Main-3}.
    \begin{thm} \label{Thm:Main-1-gen}
       For any coefficients $\mathbf{k}$, there is a $\mathbf{k}$-module isomorphism: 
      $$\qquad \ \ \quad H^q(X(\K, \upalpha,\mathbb{S});\mathbf{k}) \cong \underset{\mathrm{L}
           \subset [k]}{\bigoplus} 
                   \widetilde{H}^{q-1-\sum_{i\in\mathrm{L}}d_i}
                   (\K_{\upalpha, \mathrm{L}};\mathbf{k}),\ \forall q\geq 0.$$
      \end{thm} 
      
       \begin{thm}\label{Thm:Stable-Decomp-gen}
            There is a homotopy equivalence
         $$ 
           \mathbf{\Sigma}( X(\K, \upalpha,\mathbb{S}) )
            \simeq \bigvee_{\mathrm{L}\subset [k]} 
           \mathbf{\Sigma}^{(\sum_{i\in \mathrm{L}} d_i) +2}( \K_{\upalpha,\mathrm{L}}).$$
       \end{thm}

      
    \begin{thm}\label{Thm:Main-2-gen}
    Let $\upalpha$ be a non-degenerate partition of $V(\K)$. \n
    \begin{itemize}
     \item[(i)] For any family $\mathbb{S}=(S^{d_1},\cdots, S^{d_k})$,
     $H^*(X(\K, \upalpha,\mathbb{S});\Z_2)$ is isomorphic to
      $\mathrm{Tor}_{\Z_2[u_1,\cdots, u_k]}(\Z_2[\K],\Z_2)$
      as 
        multigraded $\Z_2$-modules.\n
        
     \item[(ii)] For $\mathbb{S}=(S^{d_1},\cdots, S^{d_k})$ with
      $d_i \geq 1$, $i=1,\cdots, k$, $H^*(X(\K, \upalpha,\mathbb{S});\Z_2)$
     is isomorphic to
     $\mathrm{Tor}_{\Z_2[u_1,\cdots, u_k]}(\Z_2[\K],\Z_2)$ as 
        multigraded $\Z_2$-algebras.  \n
    \end{itemize}
  \end{thm}
    
      \begin{thm}\label{Thm:Main-3-gen}
    Let $\upalpha$ be an arbitrary partition of $V(\K)$. \n
    \begin{itemize}
     \item[(i)] For any family $\mathbb{S}=\{ S^{d_1},\cdots, S^{d_k} \}$,
     $H^*(X(\K, \upalpha,\mathbb{S});\Z_2)$ is isomorphic to
      $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K], d_{\upalpha})$
      as multigraded $\Z_2$-modules.\n
        
     \item[(ii)] For $\mathbb{S}=(S^{d_1},\cdots, S^{d_k})$ with
      $d_i \geq 1$, $i=1,\cdots, k$, $H^*(X(\K, \upalpha,\mathbb{S});\Z_2)$
     is isomorphic to $H^*(\Lambda_{\Z_2}[t_1,\cdots, t_k] \otimes \Z_2[\K], d_{\upalpha})$ as 
        multigraded $\Z_2$-algebras.  
    \end{itemize}
  \end{thm}
   Moreover, we can make the above isomorphisms in Theorem~\ref{Thm:Main-2-gen}
   and Theorem~\ref{Thm:Main-3-gen}
        dimension-preserving by defining 
  $$ \qquad\qquad \dim(u_i)= d_i, 1\leq i \leq k, \ \text{and}\ 
  \dim(\mathbf{v}_{\sigma}) = 
  \mathrm{rank}(\sigma) +\sum_{i\in \I_{\upalpha}(\sigma)} d_i, \ \forall\sigma\in\K.$$
   
  The proofs of the above four theorems are left to the
  reader. Especially, we need to modify  
  our definition of $\kappa(i,\mathrm{L})$ and $\kappa(\sigma,\mathrm{L})$ 
   (see~\eqref{Equ:Kappa}) to adapt to the proof of Theorem~\ref{Thm:Main-1-gen}.
   Indeed,  for $X(\K, \upalpha,\mathbb{S})$ we should redefine $\kappa(i,\mathrm{L})$  as follows
   and change $\kappa(\sigma,\mathrm{L})$ accordingly.
     $$\qquad\ \ \ \kappa(i,\mathrm{L})= (-1)^{r_{\mathbb{S}}(i,\mathrm{L})}, \ \text{where}\
     r_{\mathbb{S}}(i,\mathrm{L})= \sum_{j\in \mathrm{L},\, j< i} d_j, \ 
     \forall i\in \mathrm{L} \subset [k]. 
        $$  
            
 \begin{rem}
 When $\upalpha^*$ is the trivial partition of
 $V(\K)$, $X(\K, \upalpha^*,\mathbb{S})$ is nothing but the 
 polyhedral-product $\K^{(\mathbb{D},\mathbb{S})}$ where
 $$\qquad \ \ (\mathbb{D},\mathbb{S})= \{ (D^{d_1+1},S^{d_1}),\cdots, (D^{d_k+1},S^{d_k}) \}.$$ 
 In these cases, Theorem~\ref{Thm:Main-1-gen} coincides 
   with~\cite[Theorem 4.2]{LuZhi09}.
   Besides, Theorem~\ref{Thm:Main-2-gen} (ii) verifies a claim 
   at the end of~\cite[Remark 9]{LuZhi09} for $\K^{(\mathbb{D},\mathbb{S})}$.
  \end{rem}
 \n
 
 \item[(III)] For a
  partition $\upalpha=\{\alpha_1,\cdots, \alpha_k\}$ of the vertex set of a finite simplicial complex $\K$ and 
  a sequence of spaces $\mathbb{A}=( A_1,\cdots, A_k)$,  define
  \[ \quad \qquad  X(\K, \upalpha,\mathbb{A}) = \bigcup_{\sigma\in \K} \Big(
       \prod_{i\in \mathrm{I}_{\upalpha}(\sigma)} A_i*(\sigma\cap \Delta^{\alpha_i}) \times
        \prod_{i\in [k]\backslash \mathrm{I}_{\upalpha}(\sigma)} 
        A_i \Big) \subset \prod_{i\in [k]} 
        A_i*\Delta^{\alpha_i}.
  \]
  Similarly to $X(\K,\lambda_{\upalpha})$ and $X(\K,\Lambda_{\upalpha})$, 
  we have the following stable decomposition for 
  $ X(\K, \upalpha,\mathbb{A})$:
    \begin{equation}\label{Equ:Stable-Decomp-General}
      \qquad \mathbf{\Sigma}(  X(\K, \upalpha,\mathbb{A}) ) \simeq 
       \mathbf{\Sigma} \Big( \bigvee_{\mathrm{L}\subset [k]} 
           \bigvee_{\sigma\in \K_{\upalpha,\mathrm{L}}}
            |\Delta((\K_{\upalpha,\mathrm{L}})_{>\sigma})| *
      \mathbf{W}^{\mathbb{A}}_{\upalpha,\mathrm{L}}(\sigma) \Big) 
    \end{equation}  
      \[ \text{where}\ \ \mathbf{W}^{\mathbb{A}}_{\upalpha,\mathrm{L}}(\sigma) := 
   \bigwedge_{i\in \mathrm{I}_{\upalpha}(\sigma)\cap \mathrm{L}} 
    A_i*(\sigma\cap \Delta^{\alpha_i}) 
    \wedge \bigwedge_{i\in \mathrm{L} \backslash 
    (\mathrm{I}_{\upalpha}(\sigma)\cap \mathrm{L})} A_i . \] 
   Since $\mathbf{W}^{\mathbb{A}}_{\upalpha,\mathrm{L}}(\sigma)$ is
   contractible whenever $\sigma\neq 
  \hat{\mathbf{0}}\in \K_{\upalpha,\mathrm{L}}$, we 
  can simplify the right side of~\eqref{Equ:Stable-Decomp-General} and obtain
  the following theorem which is parallel to Theorem~\ref{Thm:Stable-Decomp}.
  \begin{thm} 
  There is a homotopy equivalence 
      \begin{equation}
         \mathbf{\Sigma}(  X(\K, \upalpha,\mathbb{A}) ) \simeq \mathbf{\Sigma} \Big( \bigvee_{\mathrm{L}\subset [k]} 
            |\K_{\upalpha,\mathrm{L}}| * \bigwedge_{i\in \mathrm{L}} A_i
     \Big).  
     \end{equation}
  \end{thm}   
  We can compute the cohomology groups
  of $X(\K, \upalpha,\mathbb{A})$ via the above stable decomposition.
   In particular for the trivial partition $\upalpha^*$ of $V(\K)$ where $V(\K)=[m]$,
    $X(\K, \upalpha^*,\mathbb{A})$ is nothing but the polyhedral product 
   $(\mathrm{Cone}(\mathbb{A}),\mathbb{A})^{\K}$ where 
  $$ \qquad (\mathrm{Cone}(\mathbb{A}),\mathbb{A}) = \{ (\mathrm{Cone}(A_1),A_1),
   \cdots, (\mathrm{Cone}(A_m),A_m) \}.$$
   The reader is referred to~\cite{BBCG10} and~\cite{Wang-Zheng-13} for
   the calculations of cohomology groups of polyhedral products in general.\\

 \end{itemize}
{"config": "arxiv", "file": "1406.7392/Quotients-MAC-V2.tex"}