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\begin{document}

\title[Moment-angle complexes and simplicial manifolds]
{Moment-angle complexes and combinatorics of simplicial manifolds}
\author{Victor M. Buchstaber and Taras E. Panov}
\thanks{Partially supported by
the Russian Foundation for Fundamental Research grant no. 99-01-00090}
\address{Department of Mathematics and Mechanics, Moscow
State University, 119899 Moscow, RUSSIA}
\email{buchstab@mech.math.msu.su \quad tpanov@mech.math.msu.su}

\maketitle

Let $\rho:(D^2)^m\to I^m$ be the orbit map for the diagonal action of torus
$T^m$ on the unit poly-disk $(D^2)^m\subset\C^m$. Each face of the cube
$I^m=[0,1]^m$ (viewed as a cubical complex) has the form
$$
  F_{I\subset J}=\{(y_1,\ldots,y_m)\in I^m\: : \: y_i=0\text{ if }i\in
  I,\; y_j=1\text{ if }j\notin J\},
$$
where $I\subset J$ are two subsets of the index set $[m]=\{1,\ldots,m\}$. For
each face $F_{I\subset J}$ put $B_{I\subset J}:=\rho^{-1}(F_{I\subset J})$.
If $\#I=i$, $\#J=j$, then $B_{I\subset J}\cong(D^2)^{j-i}\times T^{m-j}$.
\begin{definition}
\label{ma}
  Let $C$ be a cubical subcomplex of $I^m$. The {\it moment-angle complex}
  $\ma(C)$ is the $T^m$-invariant decomposition of the subset
  $\rho^{-1}(|C|)\subset(D^2)^m$ into blocks $B_{I\subset J}$ corresponding
  to the faces $F_{I\subset J}$ of complex $C$.
\end{definition}
Many combinatorial problems concerning cubical complexes may be treated by
studying the equivariant topology of moment-angle complexes. In the present
paper we realize this approach in the case of cubical complexes determined by
simplicial complexes.  Let $K^{n-1}$ be an $(n-1)$-dimensional simplicial
complex with $m$ vertices, and $|K|$ the corresponding polyhedron. If
$I=\{i_1,\ldots,i_k\}\subset[m]$ is a simplex of $K$, then we would write
$I\in K$. Define the following two cubical subcomplexes of $I^m$:
$$
  \cub(K)=\{ F_{I\subset J}\::\:J\in K, I\ne\varnothing \},
  \quad \cc(K)=\{ F_{I\subset J}\::\:J\in K\}.
$$
\begin{lemma}
  As a topological space, the complex $\cub(K)$ is homeomorphic to $|K|$,
  while $\cc(K)$ is homeomorphic to the cone $|\cone(K)|$.
\end{lemma}
The cubical complex $\cc(K)$ was introduced in~\cite{BP1} and then studied
in~\cite{BP2}. The cubical complex $\cub(K)$ appeared in~\cite{SS}.

\medskip

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  \put(2,7){\small $T$}
  \put(26,7){\small 1}
  \put(9,13){\small $D$}
  \put(-4,8){a)}
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\qquad\qquad\qquad
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  \put(19,11){\small $I$}
  \put(9,13){\small $D$}
  \put(-4,8){b)}
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Denote the moment-angle complexes corresponding to $\cub(K)$ and $\cc(K)$ by
$\wk$ and $\zk$ respectively. Consider the cellular decomposition of the
poly-disk $(D^2)^m$ that is obtained by subdividing each factor $D^2$ into
0-dimensional cell $1$, 1-dimensional cell $T$, and 2-dimensional cell $D$,
see Fig.~a). Each cell of $(D^2)^m$ is a product of cells $D_i$, $T_i$,
$1_i$, $i=1,\ldots,m$, i.e., can be written as $D_IT_J1_{[m]\setminus I\cup
J}$, where $I,J$ are disjoint subsets of $[m]$.  Set
$D_IT_J:=D_IT_J1_{[m]\setminus I\cup J}$. Now it can be easily seen that
$\zk$ is cellular subcomplex of $(D^2)^m$ consisting of all cells $D_IT_J$
such that $I\in K$.

\begin{lemma}
  The embedding $T^m=\rho^{-1}(1,\ldots,1)\hookrightarrow\zk$ is a cellular
  map homotopic to the map to a point.
\end{lemma}

As it was shown in~\cite{BP2}, for any field $\k$ there is the following
isomorphism of algebras:
\begin{equation}
\label{toralg}
  H^{*}(\zk)\cong\Tor_{\k[v_1,\ldots,v_m]}\bigl(\k(K),\k\bigr)
  =H^{*}\bigl[\k(K)\otimes\Lambda[u_1,\ldots,u_m],d\bigr],
\end{equation}
where $\k(K)$ is the Stanley--Raisner ring of complex $K$, and the
differential $d$ is defined by $d(v_i)=0$, $d(u_i)=v_i$, $i=1,\ldots,m$. The
$\Tor$-algebra from~(\ref{toralg}) is naturally a {\it bigraded} algebra with
$\bideg(v_i)=(0,2)$, $\bideg(u_i)=(-1,2)$. The calculation of the ring
$H^{*}(\zk)$ allowed to describe the multiplicative structure in the
cohomology of the complement of a coordinate subspace arrangement in
$\C^m$~\cite{BP2}.

In~\cite{BP1} there was introduced the subcomplex $\mathcal C^{*}(K)\subset
\k(K)\otimes\Lambda[u_1,\ldots,u_m]$ spanned by monomials $u_J$ and
$v_Iu_J$ such that $I\cap J=\varnothing$, $I\in K$. It was also shown there
that the cohomology of $\mathcal C^{*}(K)$ is also isomorphic to that of
$\zk$. Denote by $\mathcal C_{*}(\zk)$ and $\mathcal C^{*}(\zk)$ the chain
and the cochain complex for type~a) cellular decomposition of $\zk$
respectively.

\begin{theorem}
\label{hom}
  Let $(D_IT_J)^{*}\in\mathcal C^{*}(\zk)$ denote the cellular cochain dual
  to the cell $D_IT_J\in\zk$. The correspondence $v_Iu_J\mapsto(D_IT_J)^{*}$
  defines a canonical isomorphism of complexes $\mathcal C^{*}(K)$ and
  $\mathcal C^{*}(\zk)$, each of which calculates $H^{*}(\zk)$.
\end{theorem}

The pair $(\zk,T^m)$ acquires a bigraded cellular structure by setting
$\bideg(D_i)=(0,2)$, $\bideg(T_i)=(-1,2)$, $\bideg(1_i)=(0,0)$. Put
$b_{-q,2p}(\zk,T^m)=\dim H_{-q,2p}[\mathcal C_{*}(\zk,T^m)]$. Consider the
new cellular decomposition of the poly-disc $(D^2)^m$ that is obtained by
subdividing each factor $D^2$ into 5 cells $D$, $T$, $I$, 1, 0, see~Fig.~b).
This allows to introduce a bigraded cellular structure on $\wk$ and define
the numbers $b_{q,2p}(\wk)=\dim H_{q,2p}[\mathcal C_{*}(\wk)]$. Put
$$
  \chi(\zk,T^m;t)=\sum_{p,q}(-1)^qb_{-q,2p}(\zk,T^m)t^{2p},\quad
  \chi(\wk;t)=\sum_{p,q}(-1)^qb_{q,2p}(\wk)t^{2p}.
$$

Let $f_i$ be the number of $i$-simplices of $K$, and
$(h_0,\ldots,h_n)$ the $h$-{\it vector} of $K$ determined from the equation
$h_0t^n+\ldots+h_{n-1}t+h_n=(t-1)^n+f_0(t-1)^{n-1}+\ldots+f_{n-1}$.
\begin{theorem}
\label{chi}
Put $h(t)=h_0+h_1t+\cdots+h_nt^n$. Then
$$
\begin{aligned}
  \chi(\zk,T^m;t)&=(1-t^2)^{m-n}h(t^2)-(1-t^2)^m,\\
  \chi(\wk;t)&=(1-t^2)^{m-n}h(t^2)+(-1)^{n-1}h_n(1-t^2)^m.
\end{aligned}
$$
\end{theorem}
\begin{lemma}
  If $|K|\cong S^{n-1}$ (i.e., $K$ is a simplicial sphere), then $\zk$
  is a closed manifold.
\end{lemma}
Suppose now that $K^{n-1}$ is a simplicial manifold. Then the complex $\zk$
generally fails to be a manifold. However, removing from $\zk$ a small
neighbourhood $U_\varepsilon(T^m)$ of the orbit $\rho^{-1}(1,\ldots,1)\cong
T^m$ we obtain manifold $W_K=\zk\setminus U_\varepsilon(T^m)$ with
boundary $\partial W_K=|K|\times T^m$.

\begin{theorem}
  The manifold (with boundary) $W_K$ is equivariantly homotopy equivalent to
  the complex $\wk$. There is a canonical homeomorphism of pairs
  $(W_K,\partial W_K)\to(\zk,T^m)$.
\end{theorem}
The relative Poincar\'e duality isomorphisms for $W_K$ imply
$$
  \chi(\wk;t)=(-1)^{m-n}t^{2m}\chi(\zk,T^m;\textstyle\frac1t).
$$
Taking into account Theorem~\ref{chi} we obtain
\begin{corollary}
\label{DSman}
  Let $K^{n-1}$ be a simplicial manifold. Then
  $$
    h_{n-i}-h_i=(-1)^i(h_n-1){\textstyle\binom ni}
    =(-1)^i\bigl(\chi(K^{n-1})-\chi(S^{n-1})\bigr)
    {\textstyle\binom ni},\quad i=0,1,\ldots,n,
  $$
  where $\chi(\cdot)$ denotes the Euler number.
\end{corollary}
Rewriting the equation~(\ref{DSman}) in terms of the $f$-vector we come to
more complicated equations, which were deduced in~\cite{CY}, \cite{Kl}. For
$|K|=S^{n-1}$ Corollary~\ref{DSman} gives the classical Dehn--Sommerville
equations. In the particular case of PL-manifolds the topological invariance
of numbers $h_{n-i}-h_i$ (which follows directly from Corollary~\ref{DSman})
was firstly observed by Pachner in~\cite[(7.11)]{Pa}.

The extended version of this article is {\tt
http://xxx.lanl.gov/abs/math.AT/0005199}. The authors wish to express
special thanks to Oleg Musin for stimulating discussions and helpful
comments, in particular, for drawing our attention to the results
of~\cite{CY} and~\cite{Pa}.

\begin{thebibliography}{BP2}

\bibitem{BP1}
V.\,M.~Buchstaber and T.\,E.~Panov.
Proceedings of the Steklov Institute of
Mathematics~{\bf 225} (1999), 87--120.

\bibitem{BP2}
V.\,M.~Buchstaber and T.\,E.~Panov.
(Russian),
Zap. Nauchn. Semin. POMI {\bf 266} (2000), 29--50.

\bibitem{SS}
M.\,A.~Shtan'ko and M.\,I.~Shtogrin.
Russian Math. Surveys {\bf 47} (1992), no.~1, 267--268.

\bibitem{CY}
B.~Chen, M.~Yan.
Proceedings of the Steklov Institute of Mathematics~{\bf 221} (1998),
305--319.

\bibitem{Kl}
V.~Klee.
Canadian J. Math. {\bf 16} (1964), 517--531.

\bibitem{Pa}
U.~Pachner.
European J. Combinatorics {\bf 12} (1991), 129--145.

\end{thebibliography}

\end{document}