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\begin{document}
\title{Torus Actions Determined by Simple Polytopes}
\author{Victor M. Buchstaber}
\address{Department of Mathematics and Mechanics, Moscow
State University,\newline 119899 Moscow, RUSSIA}
\email{buchstab@mech.math.msu.su}
\author{Taras E. Panov}
\email{tpanov@mech.math.msu.su}
\thanks{Partially supported by
the Russian Foundation for Fundamental Research, grant no. 99-01-00090,
and INTAS, grant no. 96-0770.}
\subjclass{Primary 57R19, 57S25; Secondary 14M25, 52B05}
\begin{abstract}
An $n$-dimensional polytope $P^n$ is called {\it simple} if exactly $n$
codimension-one faces meet at each vertex. The lattice of faces of a simple
polytope $P^n$ with $m$ codimension-one faces defines an arrangement of
coordinate subspaces in $\C^m$. The group $\R^{m-n}$ acts on the complement
of this arrangement by dilations. The corresponding quotient is a smooth
manifold $\ZP$ invested with a canonical action of the compact torus $T^m$
with the orbit space $P^n$. For each smooth projective {\it toric variety}
$M^{2n}$ defined by a simple polytope $P^n$ with the given lattice of faces
there exists a subgroup $T^{m-n}\subset T^m$ acting freely on $\ZP$ such that
$\ZP/T^{m-n}=M^{2n}$. We calculate the cohomology ring of $\ZP$ and show that
it is isomorphic to the cohomology of the {\it Stanley--Reisner ring} of
$P^n$ regarded as a module over the polynomial ring. In this way the
cohomology of $\ZP$ acquires a {\it bigraded} algebra structure, and the
additional grading allows to catch combinatorial invariants of the polytope.
At the same time this gives an example of explicit calculation of the
cohomology ring for the complement of a subspace arrangement defined by
simple polytope, which is of independent interest.
\end{abstract}
\maketitle
\section*{Introduction}
A convex $n$-dimensional polytope $P^n$ is called {\it simple} if exactly $n$
codimension-one faces meet at each vertex. Such polytopes are generic points
in the variety of all $n$-dimensional convex polytopes. One can associate to
each simple polytope $P^n$ a smooth $(m+n)$-dimensional manifold $\ZP$ with
canonical action of the torus $T^m$ on it; here $m$ is the number of
co\-di\-men\-si\-on-one faces of $P^n$.
A number of manifolds playing an important role in different aspects of
topology, algebraic and symplectic geometry are quotients $\ZP/T^k$ for the
action of some subgroup $T^k\subset T^m$. The most well-known class of such
manifolds are the (smooth, projective) {\it toric varieties} in algebraic
geometry. From the viewpoint of our approach, these toric varieties (or
toric manifolds) correspond to those simple polytopes $P^n$ for which there
exists a torus subgroup in $T^m$ of maximal possible rank $m-n$ that acts
freely on $\ZP$. Thus, all toric manifolds can be obtained as quotients of
$\ZP$ by a torus of above type.
The manifolds $\ZP$ were firstly introduced in~\cite{DJ} via certain
equivalence relation $\sim$ as $\ZP=T^m\times P^n/\sim$ (see
Definition~\ref{defzp}). We propose another approach to defining $\ZP$ based
on a construction from the algebraic geometry of toric varieties. This
construction was initially used in~\cite{Ba} (see also~\cite{Au},~\cite{Co})
for definition of toric manifolds. Namely, the combinatorial structure of
$P^n$ defines an algebraic set $U(P^n)\subset\C^m$ (the complement of a
certain arrangement of coordinate subspaces, see Definition~\ref{defU}) with
action of the group $(\C^{*})^m$ on it. Toric manifolds appear when one can
find a subgroup $D\subset(\C^{*})^m$ isomorphic to $(\C^{*})^{m-n}$ that acts
freely on $U(P^n)$. However, it turns out that it is {\it always} possible to
find a subgroup $R\subset(\C^{*})^m$ isomorphic to $\R^{m-n}$ that acts on
$U(P^n)$ freely. Then, for each subgroup $R$ of such kind the corresponding
quotient $U(P^n)/R$ is homeomorphic to $\ZP$.
One of our main goals here is to study relationships between the
combinatorial structure of simple polytopes and the topology of the above
manifolds. One aspect of this relation is the existence of a certain bigraded
complex calculating the cohomology of $\ZP$. This bigraded complex arises
from the interesting geometric structure on $\ZP$, which we call {\it
bigraded cell structure}. This structure is defined by the torus action and
the combinatorics of polytope. Thus, it seems to us that the above manifolds
defined by simple polytopes could be also used as a powerful combinatorial
tool.
Part of results of this article were announced in~\cite{BP1}.
The authors express special thanks to Nigel Ray, since the approach to
studying the manifolds $\ZP$ described here was partly formed during the work
on the article~\cite{BR}.
\section{Manifolds defined by simple polytopes}
We start with reviewing some basis combinatorial objects associated with
simple polytopes. The good references here are~\cite{Br} and~\cite{Zi}.
For any simple $P^n$, let $f_i$ denote the number of faces of codimension
$(i+1)$, $0\le i\le n-1$. The integer vector $(f_0,\ldots,f_{n-1})$ is called
the {\it $f$-vector} of $P^n$. It is convenient to set $f_{-1}=1$. We will
also consider the another integral vector $(h_0,\ldots,h_n)$ called {\it
$h$-vector} of $P^n$, where $h_i$ are retrieved from the formula
\begin{align}
\label{hvector}
h_0t^n+\ldots+h_{n-1}t+h_n&=(t-1)^n+f_0(t-1)^{n-1}+\ldots+f_{n-1},
\notag\\
\intertext{that is, }
\sum_{i=0}^nh_it^{n-i}&=\sum_{i=0}^nf_{i-1}(t-1)^{n-i}.
\end{align}
This implies that
\begin{equation}
\label{hf}
h_k=\sum_{i=0}^k(-1)^{k-i}\binom{n-i}{k-i}f_{i-1}.
\end{equation}
Now let ${\mathcal F}=(F^{n-1}_1,\ldots,F^{n-1}_m)$ be the set of all
codimension-one faces of $P^n$, so $m=f_0$. We fix a commutative ring $\k$,
which we refer to the as ground ring.
\begin{definition}
\label{fr}
The {\it face ring} (or the {\it Stanley--Reisner ring})
$\k(P^n)$ is defined to be the ring $\k[v_1,\ldots,v_m]/I$, where
$$
I=\left(v_{i_1}\ldots v_{i_s}\;:\quad i_1i_e>> (D^2)^m\\
@VVV @VVV\\
P^n @>i_P>> I^m,\\[2mm]
\end{CD}
$$
\end{theorem}
\begin{proof}
Let $\rho:{\mathcal Z}_P\to P^n$ be the orbit map. It easily follows from the
definition of ${\mathcal Z}_P$ that for each cube $I^n_v\subset P^n$ (see
Lemma~\ref{Pcub}) we have $\rho^{-1}(I^n_v)\cong(D^2)^n\times T^{m-n}$. Here
$(D^2)^n$ is the unit poly-disk in $\C^n$ with diagonal action of $T^n$.
Hence, $\ZP$ is represented as the union of blocks $B_v=\rho^{-1}(I^n_v)$,
each of which is isomorphic to $(D^2)^n\times T^{m-n}$. These blocks $B_v$
are glued together along their boundaries to get the smooth $T^m$-manifold
$\ZP$.
Now, let us prove the second part of the theorem concerning the equivariant
embedding. First, we fix a numeration of codimension-one faces of $P^n$:
$F^{n-1}_1,\ldots,F^{n-1}_m$. Take the block
$$
B_v\cong(D^2)^n\times T^{m-n}=
D^2\times\ldots\times D^2\times S^1\times\ldots\times S^1
$$
corresponding to a vertex $v\in P^n$. Each factor $D^2$ or $T^1$ in $B_v$
corresponds to a codimension-one face of $P^n$ and therefore acquires a
number (index) $i$, $1\le i\le m$. Note that $n$ factors $D^2$ acquire the
indices corresponding to those codimension-one faces containing $v$, while
other indices are assigned to $m-n$ factors $T^1$. Now we numerate the
factors $D^2\subset(D^2)^m$ of the poly-disk in any way and embed each block
$B_v\subset\ZP$ into $(D^2)^m$ according to the indexes of its factors. It
can be easily seen that the embedding of a face $I^n$ given by $m-n$
equations of type $y_j=1$ (as in Proposition~\ref{cubmap}) into the cube
$I^m$ is covered by the above constructed embedding of $B_v\cong(D^2)^n\times
T^{m-n}$ into $(D^2)^m$. Then it follows from Proposition~\ref{cubmap} that
the set of embeddings $(D^2)^n\times T^{m-n}\cong B_v\hookrightarrow (D^2)^m$
defines an equivariant embedding $i_e:\ZP\hookrightarrow (D^2)^m$. By the
construction, this embedding covers the embedding $i_P:P^n\hookrightarrow
I^m$ from Lemma~\ref{Pcub}.
\end{proof}
\begin{example}
\label{sphdec}
If $P^n=\Delta^1$ is an 1-dimensional simplex (a segment), then
$B_v=D^2\times S^1$ for each of the two vertices, and we obtain the
well-known decomposition $\mathcal Z_{\D^1}\cong S^3=D^2\times S^1\cup
S^1\times D^2$. If $P^n=\Delta^n$ is an $n$-dimensional simplex, we obtain
the similar decomposition of a $(2n+1)$-sphere into $n+1$ ``blocks"
$(D^2)^n\times S^1$.
\end{example}
\section{Connections with toric varieties and subspace arrangements}
The above constructed embedding $i_e:\ZP\hookrightarrow (D^2)^m\subset\C^m$
allows us to connect the manifold $\ZP$ with one construction from the theory
of toric varieties. Below we describe this construction, following~\cite{Ba}.
\begin{definition}
\label{defU}
Let $I=\{i_1,\ldots,i_p\}$ be an index set, and let $A_I\subset\C^m$
denote the coordinate subspace $z_{i_1}=\dots=z_{i_p}=0$. Define the
arrangement $\mathbf A(P^n)$ of subspaces of $\mathbb C^m$ as
$$
\mathbf A(P^n)=\bigcup_I A_I,
$$
where the union is taken over all $I=\{i_1,\ldots,i_p\}$ such that
$F_{i_1}\cap\cdots\cap F_{i_p}=\varnothing$ in $P^n$. Put
$$
U(P^n)=\mathbb C^m\setminus \mathbf A(P^n).
$$
\end{definition}
Note that the closed set $\mathbf A(P^n)$ has codimension at least 2 and is
invariant with respect to the diagonal action of $(\mathbb C^{*})^m$ on
$\mathbb C^m$. (Here $\C^{*}$ denote the multiplicative group of non-zero
complex numbers). Hence, $(\mathbb C^{*})^m$, as well as the torus
$T^m\subset(\mathbb C^{*})^m$, acts on $U(P^n)\subset\mathbb C^m$.
It follows from Proposition \ref{cubmap} that the image of $\ZP$ under the
embedding $i_e:\ZP\to\C^m$ (see Theorem~\ref{manif}) does not intersect
$\mathbf A(P^n)$, that is, $i_e(\ZP)\subset U(P^n)$.
We put
$$
\R^m_>=\{(\alpha_1,\ldots,\alpha_m)\in\R^n:\alpha_i>0\}.
$$
This is a group with respect to multiplication, which acts by dilations on
$\R^m$ and $\C^m$ (an element $(\alpha_1,\ldots,\alpha_m)\in\R^m_>$ takes
$(y_1,\ldots,y_m)\in\R^m$ to $(\alpha_1y_1,\ldots,\alpha_my_m)$). There is
the isomorphism $\exp:\R^m\to\R^m_>$ between additive and multiplicative
groups, which takes $(t_1,\ldots,t_m)\in\R^m$ to
$(e^{t_1},\ldots,e^{t_m})\in\R^m_>$.
Remember that the polytope $P^n$ is a set of points $x\in\R^n$ satisfying $m$
linear inequalities:
\begin{equation}
\label{ptope}
P^n=\{x\in\R^n:\langle l_i,x\rangle\ge-a_i,\; i=1,\ldots,m\},
\end{equation}
where $l_i\in(\R^n)^*$ are normal (co)vectors of facets. The set of
$(\mu_1,\ldots,\mu_m)\in\R^m$ such that $\mu_1l_1+\ldots+\mu_ml_m=0$ is
an $(m-n)$-dimensional subspace in $\R^m$. We choose a basis
$\{w_i=(w_{1i},\ldots,w_{mi})^\top\}$, $1\le i\le m-n$, in this subspace and
form the $m\times(m-n)$-matrix
\begin{equation}
\label{wmatrix}
W=\begin{pmatrix}
w_{11}&\ldots&w_{1,m-n}\\
\ldots&\ldots&\ldots\\
w_{m1}&\ldots&w_{m,m-n}
\end{pmatrix}
\end{equation}\\
of maximal rank $m-n$. This matrix satisfies the following property.
\begin{proposition}
\label{fprop}
Suppose that $n$ facets $F^{n-1}_{i_1},\ldots,F^{n-1}_{i_n}$ of $P^n$ meet
at the same vertex $v$: $F^{n-1}_{i_1}\cap\cdots\cap F^{n-1}_{i_n}=v$.
Then the minor $(m-n)\times(m-n)$-matrix $W_{i_1\ldots i_n}$ obtained from
$W$ by deleting $n$ rows $i_1,\ldots,i_n$ is non-degenerate: $\det
W_{i_1\ldots i_n}\ne0$.
\end{proposition}
\begin{proof}
Suppose $\det W_{i_1,\ldots,i_n}=0$, then one can find a zero non-trivial
linear combination of vectors $l_{i_1},\ldots,l_{i_n}$. But this is
impossible: since $P^n$ is simple, the set of normal vectors of facets
meeting at the same vertex constitute a basis of~$\R^n$.
\end{proof}
The matrix $W$ defines the subgroup
$$
R_W=\{(e^{w_{11}\tau_1+\cdots+w_{1,m-n}\tau_{m-n}},\ldots,
e^{w_{m1}\tau_1+\cdots+w_{m,m-n}\tau_{m-n}})\in\R^m_>\}\subset\R^m_>,
$$
where $(\tau_1,\ldots,\tau_{m-n})$ runs over $\R^{m-n}$. This subgroup is
isomorphic to $\R^{m-n}_>$. Since $U(P^n)\subset\C^m$ (see
Definition~\ref{defU}) is invariant with respect to the action of
$\R^m_>\subset(\mathbb C^{*})^m$ on $\C^m$, the subgroup $R_W\subset\R_>^m$
also acts on $U(P^n)$.
\begin{theorem}
\label{zu}
The subgroup $R_W\subset\R^m_>$ acts freely on $U(P^n)\subset\C^m$. The
composite map $\ZP\to U(P^n)\to U(P^n)/R_W$ of the embedding $i_e$ and the
orbit map is a homeomorphism.
\end{theorem}
\begin{proof}
A point from $\C^m$ may have the non-trivial isotropy subgroup with respect
to the action of $\R^m_>$ on $\C^m$ only if at least one of its coordinates
vanishes. As it follows from Definition~\ref{defU}, if a point $x\in U(P^n)$
has some zero coordinates, then all of them correspond to facets of $P^n$
having at least one common vertex $v\in P^n$. Let
$v=F^{n-1}_{i_1}\cap\cdots\cap F^{n-1}_{i_n}$. Then the isotropy subgroup of
the point $x$ with respect to the action of $R_W$ is non-trivial only if some
linear combination of vectors $w_1,\ldots,w_{m-n}$ lies in the coordinate
subspace spanned by $e_{i_1},\ldots,e_{i_n}$. But this means that $\det
W_{i_1\ldots i_n}=0$, which contradicts Proposition~\ref{fprop}. Thus, $R_W$
acts freely on $U(P^n)$.
Now, let us prove the second part of the theorem. Here we use both embeddings
$i_e:\ZP\to(D^2)^m\subset\C^m$ from Theorem~\ref{manif} and $i_P:P^n\to
I^m\subset\R^m$ from Lemma~\ref{Pcub}. It is sufficient to prove that each
orbit of the action of $R_W$ on $U(P^n)\subset\C^m$ intersects the image
$i_e(\ZP)$ at a single point. Since the embedding $i_e$ is equivariant,
instead of this we may prove that each orbit of the action of $R_W$ on the
real part $U_{\R}(P^n)=U(P^n)\cap\R^m_+$ intersects the image $i_P(P^n)$ in a
single point. Let $y\in i_P(P^n)\subset\R^m$. Then $y=(y_1,\ldots,y_m)$ lies
in some $n$-face $I^n_v$ of the unit cube $I^m\subset\R^m$ as described by
Proposition~\ref{cubmap}. We need to show that the $(m-n)$-dimensional
subspace spanned by the vectors $(w_{11}y_1,\ldots,w_{m1}y_m)^\top,\ldots,
(w_{1,m-n}y_1,\ldots,w_{m,m-n}y_m)^\top$ is in general position with the
$n$-face $I^n_v$. But this follows directly from Propositions~\ref{cubmap}
and~\ref{fprop}.
\end{proof}
The above theorem gives a new proof of the fact that $\ZP$ is a smooth
manifold, which allows a $T^m$-equivariant embedding into $\C^m\cong\R^{2m}$
with trivial normal bundle.
\begin{example}
\label{sphere}
Let $P^n=\D^n$ (an $n$-simplex). Then $m=n+1$,
$U(P^n)=\C^{n+1}\setminus\{0\}$, $R^{m-n}_>$ is $\R_>$, and $\alpha\in\R_>$
takes $z\in\C^{n+1}$ to $\alpha z$. Thus, we have $\ZP=S^{2n+1}$ (this
could be also deduced from Definition~\ref{defzp}; see also
Example~\ref{sphdec}).
\end{example}
Now, suppose that all vertices of $P^n$ belong to the integer lattice
$\Z^n\subset\R^n$. Such an integral simple polytope $P^n$ defines a projective
toric variety $M_P$~(see~\cite{Fu}). Normal (co)vectors $l_i$ of facets of
$P^n$ (see~(\ref{ptope})) can be taken integral and primitive. The toric
variety $M_P$ defined by $P^n$ is smooth if for each vertex
$v=F_{i_1}\cap\ldots\cap F_{i_n}$ the vectors $l_{i_1},\ldots,l_{i_n}$
constitute an integral basis of $\Z^n$. As before, we may construct the
matrix $W$ (see~(\ref{wmatrix})) and then define the subgroup
$$
C_W=\{(e^{w_{11}\tau_1+\cdots+w_{1,m-n}\tau_{m-n}},\ldots,
e^{w_{m1}\tau_1+\cdots+w_{m,m-n}\tau_{m-n}})\}\subset (\C^{*})^m,
$$
where $(\tau_1,\ldots,\tau_{m-n})$ runs over $\C^{m-n}$. This subgroup is
isomorphic to $(\C^{*})^{m-n}$. It can be shown (see~\cite{Ba}) that $C_W$
acts freely on $U(P^n)$ and the toric manifold $M_P$ is identified with the
orbit space (or the geometric quotient) $U(P^n)/C_W$. Thus, we have the
commutative diagram
$$
\begin{CD}
U(P^n) @>R_W\cong\R^{m-n}_{>}>> \ZP\\
@VC_W\cong(\C^{*})^{m-n}VV @VVT^{m-n}V\\
M^{2n} @= M^{2n}.\\[2mm]
\end{CD}
$$
Since $\ZP$ can be viewed as the orbit space of $U(P^n)$ with respect to the
action of $R_W\cong\R^{m-n}_>$, the manifold $\ZP$ and the complement
$U(P^n)$ of an arrangement of planes are of same homotopy type. In the next
section we calculate the cohomology ring of $\ZP$ (or $U(P^n)$).
\section{Cohomology ring of $\ZP$}
The following lemma follows readily from the construction of $\ZP$.
\begin{lemma}
\label{prod}
If $P^n$ is the product of two simple polytopes: $P^n=P^{n_1}_1\times
P^{n_2}_2$, then ${\mathcal Z}_P={\mathcal Z}_{P_1}\times{\mathcal
Z}_{P_2}$. If $P_1^{n_1}\subset P^n$ is a face, then ${\mathcal Z}_{P_1}$
is a submanifold of ${\mathcal Z}_P$.\ep
\end{lemma}
The space $BT^m=(\C P^{\infty})^m$ has a canonical cellular decomposition
(that is, each $\C P^\infty$ has one cell in each even dimension). For each
index set $I=\{i_1,\ldots,i_k\}$ we introduce the cellular subcomplex
$BT^k_I=BT^k_{i_1,\ldots,i_k}\subset BT^m$ homeomorphic to~$BT^k$.
\begin{definition}
\label{tbtk}
Define the cellular subcomplex $\widetilde{B_TP}\subset BT^m$ to be the
union of $BT^k_I$ over all $I=\{i_1,\ldots,i_k\}$ such that
$F_{i_1}\cap\cdots\cap F_{i_p}\ne\varnothing$ in $P^n$.
\end{definition}
\begin{theorem}
\label{cell}
The cellular embedding $i:\widetilde{B_TP}\hookrightarrow BT^m$ (see
{\rm Definition~\ref{tbtk}}) and the fibration $p:B_TP\to BT^m$
(see~{\rm (\ref{btp})}) are homotopically equivalent. In particular,
$\widetilde{B_TP}$ and $B_TP$ are of same homotopy type.
\end{theorem}
\begin{proof}
The proof can be found in \cite{BP1}.
\end{proof}
\begin{corollary}
The cohomology ring of $B_TP$ is isomorphic to the face ring $\k(P^n)$ (see
{\rm Definition~\ref{fr}}). The projection
$p:B_TP\hookrightarrow BT^m$ induces the quotient epimorphism
$p^{*}:\k[v_1,\ldots,v_m]\to\k(P^n)=\k[v_1,\ldots,v_m]/I$ in the
cohomology.\ep
\end{corollary}
A simple polytope $P^n$ with $m$ codimension-one faces is called {\it
$q$-neighbourly}~\cite{Br} if the $(q-1)$-skeleton of the simplicial complex
$K^{n-1}_P$ dual to the boundary $\partial P^n$ coincides with the
$(q-1)$-skeleton of an $(m-1)$-simplex. Equivalently, $P^n$ is $q$-neighbourly
if any $q$ codimension-one faces of $P^n$ have non-empty intersection. Note
that any simple polytope is 1-neighbourly. The next theorem about the
homotopy groups of $\ZP$ and $B_TP$ follows easily from cellular structure of
$B_TP$ and exact homotopy sequence of the bundle $p:B_TP\to BT^m$ with
fibre $\ZP$.
\begin{theorem}
\label{homot}
For any simple polytope $P^n$ with $m$ codimension-one faces we have:
\begin{enumerate}
\item $\pi_1({\mathcal Z}_P)=\pi_1(B_TP)=0$.
\item $\pi_2({\mathcal Z}_P)=0,\;\pi_2(B_TP)=\Z^m$.
\item $\pi_q({\mathcal Z}_P)=\pi_q(B_TP)$ for $q\ge 3$.
\item If $P^n$ is $q$-neighbourly, then $\pi_i({\mathcal Z}_P)=0$ for
$i<2q+1$, and $\pi_{2q+1}(\ZP)$ is a free Abelian group whose generators
correspond to monomials $v_{i_1}\cdots v_{i_{q+1}}\in I$ (see {\rm
Definition~\ref{fr}}).\ep
\end{enumerate}
\end{theorem}
From (\ref{btp}) we obtain the commutative square
$$
\begin{CD}
\ZP @>>> B_TP\\
@VVV @VVpV\\
{*} @>>> BT^m.
\end{CD}
$$
The Eilenberg--Moore spectral sequence~\cite{Sm} of this square has the
$E_2$-term
$$
E_2\cong\Tor_{\k[v_1,\ldots,v_m]}\bigl(\k(P^n),\k\bigr),
$$
where $\k(P^n)$ is regarded as a $\k[v_1,\ldots,v_m]$-module by means of
quotient projection $\k[v_1,\ldots,v_m]\to\k[v_1,\ldots,v_m]/I=\k(P^n)$.
This spectral sequence converges to the cohomology of $\ZP$. It turns out
that the spectral sequence collapses at the $E_2$ term, and moreover, the
following statement holds:
\begin{theorem}
\label{mult}
Provided that $\k$ is a field, we have an isomorphism of algebras:
$$
H^{*}(\ZP)\cong\Tor_{\k[v_1,\ldots,v_m]}\bigl(\k(P^n),\k\bigr).
$$
The additive structure of the cohomology is thus given by the
isomorphisms
$$
H^r(\ZP)\cong\bigoplus_{2j-i=r}
\Tor^{-i,2j}_{\k[v_1,\ldots,v_m]}\bigl(\k(P^n),\k\bigr),\quad i,j\ge0.
$$
\end{theorem}
\begin{proof}
The proof of the theorem uses some results of~\cite{Sm} on the
Eilenberg--Moore spectral sequences and Theorem~\ref{cell}. This proof can be
found in~\cite{BP1}.
\end{proof}
Suppose that there is at least one smooth toric variety $M^{2n}$ whose orbit
space with respect to the $T^n$-action has combinatorial type of the simple
polytope $P^n$. Then one can find a subgroup $T^{m-n}\cong H\subset T^m$ that
acts freely on the manifold $\ZP$ such that $M^{2n}=\ZP/H$. In general case
such a subgroup may fail to exist; however, sometimes one can find a subgroup
of dimension less than $m-n$ that acts freely on $\ZP$. So, let $H\cong T^r$
be such a subgroup. Then the inclusion $s:H\hookrightarrow T^m$ is determined
by an integral $(m\times r)$-matrix $(s_{ij})$ such that the $\Z$-module
spanned by its columns $s_j=(s_{1j},\ldots,s_{mj})^\top$, $j=1,\ldots,r$, is
a direct summand in $\Z^m$. Choose any basis $t_i=(t_{i1},\ldots,t_{im})$,
$i=1,\ldots,m-r$, in the kernel of the dual map
$s^{*}:(\Z^m)^{*}\to(\Z^r)^{*}$. Then we have the following result describing
the cohomology ring of the manifold ${\mathcal Y}=\ZP/H$.
\begin{theorem}
\label{quot}
The following isomorphism of algebras holds:
$$
H^{*}({\mathcal Y})\cong\Tor_{\k[t_1,\ldots,t_{m-r}]}
\bigl(\k(P^n),\k\bigr),
$$
where the $\k[t_1,\ldots,t_{m-r}]$-module structure on
$\k(P^n)=\k[v_1,\ldots,v_m]/I$ is defined by the map
\begin{align*}
\k[t_1,\ldots,t_{m-r}]&\to\k[v_1,\ldots,v_m]\\
t_i&\to t_{i1}v_1+\ldots+t_{im}v_m.
\end{align*}
\end{theorem}
\begin{proof}
This theorem, as well as the previous one, can be proved by considering a
certain Eilenberg--Moore spectral sequence. See~\cite[Theorem~4.13]{BP1}.
\end{proof}
In the case of toric varieties (that is, $r=m-n$ in Theorem~\ref{quot}) we
obtain
$$
H^{*}(M^{2n})\cong\Tor_{\k[t_1,\ldots,t_n]}\bigl(\k(P^n),\k\bigr).
$$
It can be shown that in this case $\k(P^n)$ is a {\it free}
$\k[t_1,\ldots,t_n]$-module (which implies that $t_1,\ldots,t_n$ is a {\it
regular} sequence, and $\k(P^n)$ is a so-called {\it Cohen--Macaulay ring}).
Thus, we have
$$
H^{*}(M^{2n})\cong\k(P^n)/J=\k[v_1,\ldots,v_m]/I{+}J,
$$
where $J$ is the ideal generated by $t_{i1}v_1+\ldots+t_{im}v_m$,
$i=1,\ldots,n$. This result (the description of the cohomology ring of a
smooth toric variety) is well known in algebraic geometry as the
Danilov--Jurkiewicz theorem~\cite{Da}.
\medskip
In order to describe the cohomology ring of $\ZP$ more explicitly, we apply
some constructions from homological algebra.
Let $\G=\k[y_1,\ldots,y_n]$, $\deg y_i=2$, be a graded polynomial algebra,
and let $A$ be any graded $\G$-module. Let $\Lambda[u_1,\ldots,u_n]$ denote
the exterior algebra on generators $u_1,\ldots,u_n$ over $\k$, and consider
the complex
$$
\mathcal E=\G\otimes\Lambda[u_1,\ldots,u_n].
$$
This is a bigraded differential algebra; its gradings and differential are
defined by
\begin{align*}
\bideg(y_i\otimes1)&=(0,2),&d(y_i\otimes1)&=0;\\
\bideg(1\otimes u_i)&=(-1,2),&d(1\otimes u_i)&=y_i\otimes1
\end{align*}
and requiring that $d$ be a derivation of algebras. The differential adds
$(1,0)$ to bidegree, hence, the components $\mathcal E^{-i,*}$ form a cochain
complex. It is well known that this complex is a $\G$-free resolution of
$\k$ (regarded as a $\G$-module) called the {\it Koszul resolution}
(see~\cite{Ma}). Thus, for any $\G$-module $A$ we have
$$
\Tor_{\Gamma}(A,\k)=H\bigl[A\otimes_{\Gamma}\Gamma\otimes
\Lambda[u_1,\ldots,u_n],d\bigr]=
H\bigl[A\otimes\Lambda[u_1,\ldots,u_n],d\bigr],
$$
where $d$ is defined as $d(1\otimes u_i)=y_i\otimes1$.
Applying this construction to the case $\G=\k[v_1,\ldots,v_m]$, $A=\k(P^n)$
and using Theorem~\ref{mult}, we get the following statement.
\begin{theorem}
\label{cos}
The following isomorphism of graded algebras holds:
\begin{gather*}
H^{*}(\ZP)\cong H\bigl[\k(P^n)\otimes\Lambda[u_1,\ldots,u_m],d\bigr],\\
\bideg v_i=(0,2),\quad\bideg u_i=(-1,2),\\
d(1\otimes u_i)=v_i\otimes 1,\quad d(v_i\otimes 1)=0,
\end{gather*}
where $\k(P^n)=\k[v_1,\ldots,v_m]/I$ is the face ring.\ep
\end{theorem}
\begin{corollary}
The Leray--Serre spectral sequence of the $T^m$-bundle
$$
\ZP\times ET^m\to B_TP=\ZP\times_{T^m}ET^m
$$
collapses at the $E_3$ term.\ep
\end{corollary}
Theorems \ref{mult} and \ref{cos} show that instead of usual grading, the
cohomology of $\ZP$ has bigraded algebra structure with bigraded components
$$
H^{-i,2j}(\ZP)\cong
\Tor^{-i,2j}_{\k[v_1,\ldots,v_m]}\bigl(\k(P^n),\k\bigr),\quad i,j\ge0,
$$
satisfying $H^r(\ZP)=\bigoplus_{2j-i=r}H^{-i,2j}(\ZP)$.
Since $\ZP$ is a manifold, there is the Poincar\'e duality in $H^{*}(\ZP)$.
This Poincar\'e duality has the following combinatorial interpretation.
\begin{lemma}
\label{pd}\
\begin{enumerate}
\item The Poincar\'e duality in $H^{*}\bigl(\ZP)$ regards the bigraded
structure defined by theorems~\ref{mult} and~\ref{cos}. More
precisely, if $\alpha\in H^{-i,2j}(\ZP)$ is a cohomology class, then its
Poincar\'e dual $D\alpha$ belongs to $H^{-(m-n)+i,2(m-j)}$.
\item Let $v=F^{n-1}_{i_1}\cap\cdots\cap F^{n-1}_{i_n}$ be a vertex of the
polytope $P^n$, and let $j_1<\ldots