Complex Cobordism of quasitoric orbifolds
orbcplxcob.tex
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\begin{document}

\title{Complex Cobordism of quasitoric orbifolds}

\author[S. Sarkar]{Soumen Sarkar}
\address{\it{Department of Mathematics and Statistics, University of Regina,
Regina S4S 6A7, Canada}}

\email{soumensarkar20@gmail.com}

\subjclass[2010]{55N22, 57R90}

\keywords{quasitoric orbifolds, stable and almost complex structure, complex cobordism}

\abstract We construct manifolds and orbifolds with quasitoric boundary. We show that these
manifolds and orbifolds with boundary has a stable complex structure. These induce explicit
(orbifold) complex cobordism relations among quasitoric manifolds and orbifolds. In particular,
we show that a quasitoric orbifold is complex cobordant to some copies of fake weighted projective
spaces. The famous problem of Hirzebruch is that which complex cobordism classes in $\Omega^U$ contain connected
nonsingular algebraic varieties? We give some sufficient conditions to show when a
complex cobordism class may contains an almost complex quasitoric manifold. Andrew Wilfong give some
necessary condition of this problem up to dimension 8.
\endabstract
\maketitle

\section{Introduction}\label{intro}
Cobordism was explicitly introduced by Lev Pontryagin \cite{Pon1} in geometric work on manifolds.
%In \cite{Tho}, Ren\'{e} Thom showed that cobordism groups could be computed by results of homotopy theory.
 In \cite{Tho} Thom showed that the cobordism classes
of $G$-manifolds, for a Lie group $G$, are in one to one correspondence with the
elements of the homotopy group of the Thom space of the group $G \subseteq O_m$.
After few years, Atiyah \cite{At1} showed that complex cobordism is a generalized
cohomology theories using the Thom spectrum. Following the geometric definition of
complex cobordism for manifolds using stably complex manifolds we may define the
complex cobordism of stably complex orbifolds. In this article we show that a
quasitoric orbifold is complex cobordant to some copies of fake weighted
projective spaces, Theorem \ref{qbd}. To show these, we construct stably complex
orbifolds with quasitoric boundary in Theorem \ref{comcob}. These produce explicit
cobordism relations among quasitoric manifolds and orbifolds.

Quasitoric orbifolds are orbifold generalization of quasitoric manifolds which were
introduced by Davis and Januskiewicz in \cite{DJ}. An orbifold with quasitoric boundary
is an orbifold with boundary where the boundary is a disjoint union of some quasitoric
orbifolds.
% The strategy of our proof is to first construct some compact orientable manifolds with quasitoric boundary.
%Then identifying suitable boundary components using certain equivariant homeomorphisms
%we obtain oriented manifolds with the boundary $\CP^{2k+1}$ for each $k \geq 0$, see theorem \ref{gr}.
The famous problem of Hirzebruch is that which complex cobordism classes in $\Omega^U$ contain connected
nonsingular algebraic varieties? In Corollary \ref{hp}, we give some sufficient conditions to show when a
complex cobordism class may contains an almost complex quasitoric manifold. In \cite{W}, Andrew Wilfong give some
necessary condition of this problem up to dimension 8.

%Interestingly, we show that such a orbifold with quasitoric boundary could be viewed as the quotient space
%of a torus orbifold corresponding to a certain circle action on it followed by a natural question, see Section \ref{ccob}.

\section{orbifolds}\label{orb}
Orbifolds were introduced by Satake \cite{Sat}, who called them $V$-manifolds.
Orbifolds are singular spaces that are locally look like as a quotient of an
open subset of Euclidean space by an action of a finite group. The readers are requested to
consult Section 1.1 in \cite{ALR} for the definitions and basic facts concerning effective orbifolds.
The reader may also consult \cite{MM} for an excellent exposition of the foundations of the theory of
(reduced) differentiable orbifolds.

Similarly as the definition of manifold with boundary, we can talk about orbifold with
boundary see Definition 1.3 in \cite{Dru1}. Definition 1.7 in this article discuss
about orientation on orbifolds.

Many concepts in orbifold theory are defined in the context of groupoid, see \cite{ALR} to enjoy
this approach. For example, Section 2.3 in \cite{ALR} discuss orbifold vector bundles in the language
of groupoid. Most relevant examples of orbibundle of an orbifold is its tangent bundle. Explicit
description of tangent bundle of an effective orbifold is given in Section 1.3 of \cite{ALR}.


\begin{defn}
Let $Y$ be a smooth orbifold with tangent bundle $(TY, p_Y, Y)$ where $p_Y: TY \to Y$ is the projection map.
\begin{enumerate}
\item An almost complex structure on $Y$ is an endomorphism $J : TY \to TY$ such that $J^2 = -Id$.

\item A stable almost complex structure on $Y$ is an endomorphism
\begin{equation}
J : TY \oplus (Y \times \RR^k) \to TY \oplus (Y \times \RR^k)
\end{equation}
such that $J^2 = -Id$ for some positive integer $k$. Let $\xi$ be a complex bundle isomorphic, as a real bundle, to
$TY \oplus (Y \times \RR^k)$. We denote the pair $(Y, \xi)$ as a stable complex structure on $Y$.

%\item We call $\mathcal{Y}$ a complex orbifold if all the defining maps are holomorphic. In this case $\mathcal{Y}$
%has a canonical stable complex structure, namely the pair $(\mathcal{Y}, T\mathcal{Y})$.
\end{enumerate}
\end{defn}


\section{Quasitoric orbifolds}\label{qo}
In this Section we review the definitions of quasitoric orbifolds following \cite{PS}.
An $n$-dimensional {\em simple polytope} in $\RR^n$ is a convex polytope where exactly $n$ bounding hyperplanes meet at 
each vertex. The codimension one faces of a convex polytope are called $facets$.
Let $\mathcal{F}(P)$ be the set of facets of an $n$-dimensional simple polytope $P$.
Let $\TT^n = (\ZZ^n \tensor_{\ZZ} \RR)/\ZZ^n$ and $\TT_M = (M \tensor_{\ZZ} \RR)/M$ for a free $\ZZ$-module $M$. 

\begin{defn}
A $2n$-dimensional quasitoric orbifold $ Y$ is a smooth orbifold with a $\TT^n$ action,
%where $N$ is a fixed free $\ZZ$-module of rank $n$,
such that the orbit space is (diffeomorphic to) a simple $n$-dimensional polytope $P$.
Denote the projection map from $Y$ to $P$ by $\pi: Y \to P$. Furthermore every point $x \in Y$ has
\begin{itemize}
\item[A1)] a $\TT^n$-invariant neighborhood $V$,
\item[A2)] an associated free  $\ZZ$-module $M$ of rank $n$ with an
  isomorphism $\theta: \TT_M \to U(1)^n$ and an injective module homomorphism $\iota: M \to \ZZ^n$
  which induces a surjective covering homomorphism $\iota_M : \TT_M \to \TT^n $,
\item[A3)] an orbifold chart $(W, G, \eta)$ over $V$ where $W$ is $\theta$-equivariantly diffeomorphic to an
    open set in $\CC^n$,  $G = {\ker} \iota_M $ and $\eta: W \to V$ is an equivariant map i.e.
   $\eta(t \cdot y)= \iota_M (t) \cdot \eta(y)$ inducing a homeomorphism between $W/G$ and $V$.
\end{itemize}
\end{defn}

Note that this definition is a generalization of the axiomatic definition, see Section 1 in \cite{DJ},
of a quasitoric manifold using the notion of locally standard action.

\begin{defn}\label{dich}
A function $\xi : \mathcal{F}(P) \to \ZZ^n$ is called a di-characteristic function if the 
vectors $\xi_{j_1}, \ldots, \xi_{j_l}$ are linearly independent over $\ZZ$ whenever
the intersection of the facets $F_{j_1}, \ldots, F_{j_l}$ is nonempty.

The vectors $ \xi(F_{j})$ are called di-characteristic vectors and the pair $(P, \xi)$ is
called a characteristic model of $P$.
\end{defn}

In Subsection 2.1 of \cite{PS}, the authors construct a quasitoric orbifold from the characteristic
model $(P, \xi)$. Also given a quasitoric orbifold we can associate a characteristic model to it up
to choice of signs of di-characteristic vectors.

\begin{defn} 
Let $\delta : \TT^n \to \TT^n$ be an automorphism. Two quasitoric orbifolds $X_1$ and $X_2$ over the same
polytope $P$ are called $\delta$-equivariantly homeomorphic if there is a homeomorphism $ f : X_1 \to X_2$
such that $f(t \cdot x) = \delta(t)\cdot f(x)$ for all $(t, x ) \in \TT^n \times X_1$.
\end{defn}
The automorphism $\delta$ induces an automorphism $\delta_{\ast}$ of $\ZZ^n$.
For this automorphism $\delta$, a $\delta$-$translation$ of characteristic model $(P, \xi)$
is defined by the characteristic model $(P, \delta_{\ast} \circ \xi)$. Following classification result is the 
Proposition 5.14 in \cite{BP} for quasitoric manifolds and Lemma 2.2 in \cite{PS} for quasitoric orbifolds.
\begin{prop}\label{probi}
For any automorphism $\delta$ of $\TT^n$, there is a bijection between $\delta$-equivariant homeomorphism
classes of quasitoric orbifolds over $P$ and $\delta$-translations of characteristic model $(P, \xi)$.
\end{prop}
Suppose $\delta$ is the identity automorphism of $\TT^n$. From Proposition \ref{probi} we have two quasitoric
orbifolds are equivariantly homeomorphic if and only if their di-characteristic functions are the same.

There are important closed $\TT^n$-invariant suborbifolds of a quasitoric orbifold $X$ over the polytope $P$.
These suborbifolds correspond to the faces of $P$. If $F$ is a face of $P$ of codimension $k$, then
define $X(F) := \pi^{-1}(F)$. With subspace topology, $X(F)$ is a quasitoric orbifold of dimension $2n-2k$.
The space $X(F)$ is called a {\it characteristic suborbifold} of $X$, if $F$ is a facet of $P$. Note that
a choice of orientation on $\TT^n $ and $ P$ gives an orientation on the orbifold $X$.
\begin{defn}
 An {\it omniorientation} of a quasitoric orbifold $X$ is a choice of orientation
for $X$ as well as an orientation for each characteristic suborbifold of $X$.
\end{defn}
Clearly the isotropy group of a characteristic suborbifold is a circle subgroup of $\TT^n$.
So there is a natural $S^1$-action on the normal bundle of that characteristic suborbifold.
Thus the normal bundle has a complex structure and consequently an orientation. Whenever the
sign of the characteristic vector of a facet is reverse, we get the opposite orientation on
the normal bundle. An orientation on the normal bundle together with an orientation on $X$
induces an orientation on the characteristic suborbifold. So a di-characteristic function
determines a natural omniorientation. We call this omniorientation as the {\it characteristic omniorientation}.

Fake weighted projective is a holomorphic generalization of weighted projective space, see \cite{Ka}.
These spaces are defined by a complete fan associated to a simplex. Fake weighted projective spaces
are quasitoric orbifolds over a simplex. As a topological counter part, we may give following definition.
\begin{defn}
 A quasitoric orbifold over a simplex is called fake weighted projective space.
\end{defn}

\section{Construction of orbifolds with quasitoric boundary}\label{def}
In this section we construct some oriented orbifolds with boundary whose boundaries are quasitoric
orbifolds. This construction is a generalization of the construction of manifolds with boundary of Section
$4$ in \cite{Sar2}.
\begin{defn}
An $(n+1)$-dimensional simple polytope $Q$ in $\RR^{n+1}$ is said to be polytope with exceptional facets
$\{Q_1, \ldots, Q_k\}$ if $Q_i \cap Q_j$ is empty for $i \neq j$ and $V(Q) = \cup_{i=1}^k V(Q_i)$.
\end{defn}
%Clearly, the vertex cut of an edge-simple polytope $P$ with $k$ vertices is a face-simple manifold with exceptional
%facets $\{P_{H_i}: i=1, \ldots, k\}$. Let $Y$ be an $(n+1)$-dimensional face-simple manifold with exceptional facets.
Let $ \mathcal{F}(Q) = \{F_1, \ldots, F_m\} \cup \{Q_1, \ldots, Q_k\}$ be the facets of $Q$ where $\{Q_1,
\ldots, Q_k\}$ are the exceptional facets.
\begin{defn}\label{isofun}
A function $ \lambda \colon \{F_1, \ldots, F_m\} \to \ZZ^{n}$ is called an isotropy function of $Q$ except the facets
$\{Q_1, \ldots, Q_k\}$ if
the set of vectors $ \{\lambda(F_{i_1}), \ldots,$ $ \lambda(F_{i_{k}})\}$ is a linearly independent set in $\ZZ^{n}$
whenever the intersection of the facets $ \{F_{i_1}, \ldots,$ $ F_{i_k}\}$ is nonempty. The vector $\lambda_i :=
\lambda(F_i)$ is called an isotropy vector assign to the facet $F_i$ for $i=1, \ldots, m$.
\end{defn}
We define isotropy functions of some polytopes with exceptional facets in Example \ref{egchar2}.
%\subsection{Manifolds with Quasitoric Boundary}\label{defqb}
\begin{remark}
 Since $\ZZ^n$ is not a union of finitely many proper submodule over $\ZZ$, we can define isotropy function
of any polytope with exceptional facets.
\end{remark}

Let $ F $ be a face of $ Q $ of codimension $l$ with $ 0< l \leq (n+1)$. If $F$ is a face of $Q_i$ for some
$i \in \{1, \ldots, k\}$, then $ F $ is the intersection of a unique collection of $ l $ facets $F_{i_1}, F_{i_2},
\ldots, F_{i_{l-1}}, Q_i$ of $ Q $. Otherwise, $ F $ is the intersection of a unique collection of $ l $
facets $F_{i_1}, F_{i_2}, \ldots, F_{i_{l}}$ of $Q$. Let $M(F)$ denote the submodule of $\ZZ^{n}$ generated
by the isotropy vectors $\{\lambda_{i_j} : j = 1, \ldots, l-1\}$ if $F$ is a face of $Q_i$ and $\{\lambda_{i_j}
: j=1, \ldots, l\}$ otherwise. So
\begin{equation}
\TT_F := (M(F) \tensor_{\ZZ} \RR)/M(F)
\end{equation}
is a compact torus of dimension $l-1$ and $l$ if $F$ is a face of $Q_i$ for some $i \in \{1, \ldots, k\}$
and otherwise respectively. We will adopt the convention that $\TT_{M(Q)} = 1= \TT_{M(Q_i)}$ for $i=1,
\ldots, k$.  The inclusion $M(F) \to \ZZ^n$ induces a natural homomorphism $$f_F : \TT_{F} \to \TT^{n}$$
for any face $F$ of $Q$. We denote the image of $f_F$ by $Im(f_F)$. We define an equivalence relation
$\sim_b$ on the product $ \TT^{n} \times Q$ as follows.
\begin{equation}\label{equilam}
(t, x) \sim_b (u, y) ~ \mbox{if and only if} ~  x = y ~ \mbox{ and} ~ tu^{-1} \in Im(f_F)
\end{equation}
where $ F $ is the unique face of $ Q $ containing $ y $ in its relative interior. We denote the quotient
space $ (\TT^{n} \times Q)/ \sim_b $ and the equivalence class of $(t, x)$ by $ W(Q, \lambda)$ and $[t, x]^{\sim_b}$
respectively. The space $W(Q, \lambda)$ is a $\TT^{n}$ space where the action is induced by the group
operation in $\TT^{n}$. Let $\pi: W(Q, \lambda) \to Q$ be the projection map defined by $\pi([t,x]^{\sim_b}) = x$.
We consider the standard orientation of $T^{n}$ and the orientation of $Q$ induced from the ambient space $\RR^{n+1}$.

\begin{lemma}\label{orbbd}
The space $ W(Q, \lambda) $ is a $(2n+1)$-dimensional oriented orbifold with boundary. The boundary is a
disjoint union of $2n$-dimensional quasitoric orbifolds.
\end{lemma}
\begin{proof}
Let $C(Q_j)$ be the union of all faces $F$ of $Q$ such that $F \cap Q_j$ is empty.
Let $$U(Q_j) = Q - C(Q_j)$$ for $j=1, \ldots, k$. Since $Q$ is a simple polytope,
$U(Q_j)$ is diffeomorphic to $Q_j \times [0, 1)$ and $Q = \cup_{j=1}^k U(Q_{j})$.
Let $$ f_j : U(Q_j) \to Q_j \times [0,1)$$ be a diffeomorphism. Note that the facets of $Q_j$ are
$\{Q_j \cap F_{j_1}, \ldots, Q_{j} \cap F_{j_l}\}$ for some facets $F_{j_1}, \ldots, F_{j_l} \in \{F_1, \ldots, F_m\}$.
The restriction of the isotropy function $\lambda$ on the facets of $Q_j$ is given by
$$\xi^j(Q_j \cap F_{j_i})= \lambda_{j_i} ~\mbox{for} ~ i = 1, \ldots, l.$$
By definition of $\lambda$, we can see that $\xi^j$ is a di-characteristic function on $Q_j$.
So from Subsection 2.1 of \cite{PS} we get that the quotient $(T^{n} \times Q_j)/\sim_b$ is a $2n$-dimensional
quasitoric orbifold $X(Q_j, \xi^j)$ for $j=1, \ldots, k$.
From the equivalence relation $\sim_b$ in (\ref{equilam}), it is clear that $f_j$ induces a
homeomorphism $$h_j: (T^{n} \times U(Q_j))/\sim_b \to ((T^{n} \times Q_j)/ \sim_b)\times [0,1)
= X(Q_j, \xi^j) \times [0, 1).$$

So we get $W(Q, \lambda)= \bigcup_{j=1}^k (T^{n} \times U(Q_j)/\sim_b$. Hence $W(Q, \lambda)$
is a manifold with boundary where the boundary is the disjoint union of quasitoric orbifolds
$\{X(Q_j, \xi^j) : j = 1, \ldots, k\}$. Clearly orientations of $T^{n}$ and $Q$ induce an
orientation of $W(Q, \lambda)$.
\end{proof}

If $\lambda$ satisfy the following condition: the set of vectors $ \{\lambda(F_{i_1}), \ldots,$ $ \lambda(F_{i_{k}})\}$
is a part of a basis of $\ZZ^{n}$ over $\ZZ$ whenever the intersection of the facets $ \{F_{i_1}, \ldots,$ $ F_{i_k}\}$
is nonempty, then all the quasitoric orbifolds in the proof of Lemma \ref{orbbd} are quasitoric manifolds. So, in this
case we have.
\begin{corollary}
With the assumption in the above paragraph, the space $ W(Y, \lambda) $ is a $(2n+1)$-dimensional oriented manifold
with boundary. The boundary is a disjoint union of quasitoric manifolds.
\end{corollary}

\begin{example}\label{egchar2}
Some isotropy functions of the polytopes with exceptional facets $Y_1$ and $Y_2$ are given in
the Figure \ref{egc2} $(a)$ and $(b)$ respectively. In the first figure all $Q_1, Q_2, Q_3, Q_4$ are exceptional
facets which are triangles. The restriction of the isotropy function on $Q_{i}$ gives that the space
$(\TT^2 \times Q_i)/\sim_b$ is a fake weighted projective space $M_i^4$ for $i\in \{1,2,3,4\}$.
So $W(Y_1, \lambda)$ is an oriented orbifold with boundary where the boundary is the disjoint union
of distinct $4$-dimensional generalized weighted projective spaces.

In the second figure, $Q_1,Q_{2}, Q_{3}, Q_{4}$ and $Q_5$ are exceptional facets where $Q_1, \ldots, Q_4$
are triangles and $Q_{5}$ is a rectangle. The restriction of the isotropy function on $Q_{j}$ gives that the space
$(\TT^2 \times Q_j)/\sim_b$ is a fake weighted projective space $M_i$ for $i\in \{1,2,3,4\}$ and
$(\TT^2 \times Q_5)/\sim_b$ is a quasitoric orbifold $X(Q_5, \lambda^5)$. Hence the space
$\sqcup_{i=1}^4 M_i \sqcup X(Q_5, \lambda^5)$ is oriented $\TT^2$-equivariant boundary of $W(Y_2, \lambda)$.

\begin{figure}[ht]
        \centerline{
           \scalebox{0.60}{
            \input{egc2.pstex_t}
            }
          }
       \caption {Isotropy functions of some polytopes with exceptional facets.}
        \label{egc2}
      \end{figure}
\end{example}

\section{Stable complex structure and complex cobordism}\label{ccob}
In this section first, we show the existence of stable complex structure on orbifolds with quasitoric boundary $W(Q, \lambda)$.
Then, we construct the complex orbifold cobordism class of a quasitoric orbifold
explicitly. Recall definitions of stable and almost complex structure of orbifolds from Section \ref{orb}.
Similarly as the manifolds case, we may define complex cobordism of orbifolds.
\begin{defn}
 Let $Y$ be a topological space and $\mathcal{X}_1$, $\mathcal{X}_2$ be two $n$-dimensional stable complex orbifolds.
Let $ h_i : \mathcal{X}_i \to Y$ be continuous map for $i=1, 2$. Then $h_1$ and $h_2$ are bordant if there exists a
stable complex orbifold $\mathcal{Z}$ of dimension $n+1$ with $\partial{\mathcal{Z}} = \mathcal{X}_1 \sqcup \mathcal{X}_2$
and a continuous map $H: \mathcal{Z} \to Y$ such that $H|_{\partial{\mathcal{Z}}}= h_1 \sqcup h_2$. 
\end{defn}
This induces an equivalence relation on the collection $$\{(\mathcal{X}, h) : \mathcal{X}~ \mbox{is a stable orbifold and}~ 
h : \mathcal{X} \to Y ~ \mbox{is a continuous map}\}.$$ We denote the equivalence class of $(\mathcal{X}, h)$ by $[\mathcal{X}, h]$.
Let $OB_n(Y)= \{[\mathcal{X}, h] : \dim{X} =n\}$. This has a group structure under the operation of disjoint union. $OB_n(Y)$
is called the $n$-th complex orbifold bordism group of $Y$. Let $OB_{\ast}(Y) = \cup OB_n(Y)$. Then product endow the structure
of a graded ring on $OB_{\ast}(Y)$, called the complex orbifold bordism ring of $Y$.
\begin{defn}
The complex orbifold bordism groups  and ring of a point are called complex orbifold cobordism groups and ring respectively.
In this case we denote the cobordism class $[\mathcal{X}, h]$ by $[\mathcal{X}]$. 
\end{defn}

Stable complex structure of quasitoric manifolds and orbifolds were studied in \cite{DJ} and \cite{PS}
respectively. Now we show that the orbifold with boundary $W(Q, \lambda)$ which
is constructed in Section \ref{def} has stable complex structure.
\begin{lemma}\label{qto}
The orbifold with boundary $W(Q, \lambda)$ is an orbit space of a circle action on a quasitoric orbifold.
\end{lemma}
\begin{proof}
Let $Q$ be a simple polytope with exceptional facets $\{Q_1, \ldots, Q_{k}\}$.
Let $$ \mathcal{F}({Q}) := \{ F_{i} \colon i= 1, 2, \ldots, m\} \cup \{Q_j \colon j = 1, 2, \ldots, k\}$$ be
the facets of $Q$. Let $\lambda$ be an isotropy function except the facets $\{Q_1, \ldots, Q_{k}\}$ of $Q$.
We define a function $ \eta \colon \mathcal{F}(Q) \to \ZZ^{n}$ as follows.
\begin{equation}\label{charp}
\eta( F ) = \left\{ \begin{array}{ll} ( 0, \ldots, 0, 1 ) \in \ZZ^{n+1} & \mbox{if} ~ F = Q_j~ \mbox{and} ~ j~ \in \{1, \ldots, k\} \\
{(\lambda_i, 0)} \in \ZZ^{n} \times\{0\} \subset \ZZ^{n+1} & \mbox{if}~ F = F_i ~ \mbox{and} ~ i \in \{ 1, 2, \ldots, m\}
\end{array} \right.
\end{equation}
So the function $ \eta $ is a di-characteristic function (see Definition \ref{dich}) of the $(n+1)$-dimensional simple
polytope $Q$. Hence from the di-characteristic pair $(Q, \eta)$ we can construct quasitoric orbifold $ X(Q, \eta)$.
There is a natural $ \TT^{n+1} $ action on $X(Q, \eta)$ induce by the group operation on $\TT^{n+1}$. Let
$$\pi_Q : X(Q, \eta) \to Q$$ be the orbit of this action. Let $ \TT_Q $
be the circle subgroup of $\TT^{n+1}$ determined by the submodule $\{0\} \times \ldots \times \{0\} \times \ZZ $ of
$\ZZ^{n+1}$. From the definition of $\eta$ it is clear that $W(Q, \lambda ) $ is the orbit space of the circle
$ \TT_Q $ action on $ X( Q, \eta ) $.
\end{proof}
\begin{remark}
The quotient map $ \phi_Q \colon X( Q, \eta ) \to W( Q, \lambda) $ is not a fiber bundle map.
\end{remark}

Now we prove our main theorem.
\begin{theorem}\label{comcob}
Let $ Q $ be an $(n+1)$-polytope with exceptional facets $Q_1, \ldots,$ $ Q_k$ in $\RR^{n+1}$.
Let $\lambda$ be an isotropy function of $Q$ except the facets $\{Q_1, \ldots, Q_k\}$. Then there is a stable
complex structure on $W(Q, \lambda)$. Moreover, $[X(Q_1, \xi^1)] + \cdots + [X(Q_k, \xi^k)] = 0$ where $\xi^i$
is the restriction of the isotropy function $\lambda$ on the facets of $Q_i$ for $i=1, \ldots, k$.
\end{theorem}
\begin{proof}
From the pair $(Q, \lambda)$, we can construct a characteristic pair $(Q, \eta)$, see Equation \ref{charp}.
So by Lemma \ref{qto}, we can construct a quasitoric orbifold $X(Q, \eta)$.
We fix an orientation of $\TT^{n+1}$ and the polytope $Q$. So choice of signs of di-characteristic
vectors determines the omniorientation of $X(Q, \eta)$. Let $m$ be the number of facets of $Q$ and the 
omniorientation of $X(Q, \eta)$ is the characteristic omniorientation. Following Section 6 of \cite{PS},
we get that this omniorientation determines a stably complex
structure on $X(Q, \eta)$ by means of the following isomorphism of orbifold real $2m$-bundles
\begin{equation}
T(X(Q, \eta)) \oplus \RR^{2(m-n-1)} \cong \rho_1 \oplus \cdots \oplus \rho_m
\end{equation}
where the orbifold complex line bundles $\rho_i$'s can be interpreted in the following way.
The orientation of the orbifold normal bundle $\mu_i$ over the characteristic suborbifold $X_i$
defines a rational Thom class in the cohomology group $H^2({\bf T}(\mu_i), \QQ)$, represented by
a complex line bundle over the Thom complex ${\bf T} (\mu_i)$. We pull this back along the
Pontryagin-Thom collapse $X(Q, \eta) \to {\bf T}(\mu_i)$, and denote the resulting bundle $\rho_i$. 

Cut off a neighborhood of each facets $ Q_j, j= 1, 2, \ldots, k $ of $Q$
by an affine hyperplane $ H_i, i= 1, 2, \ldots, k $ in $\RR^{n+1}$ such that $H_i \cap H_j \cap Q$
are empty sets for $i \neq j$. Then the remaining subset, denoted by $Q_P$, of $Q$ is an $(n+1)$-dimensional
simple polytope which is diffeomorphic to $Q$. Suppose $ Q_{H_i} = Q \cap H_i =
H_i \cap Q_P$ for $ i=1, 2, \ldots k $. Then $ Q_{H_i}$ is a facet of $Q_P$
called the facet corresponding to the facet $Q_i$ for each $i= 1, \ldots, k $.
Note that $Q_{H_i}$ is diffeomorphic to $Q_i$ for $i=1, \ldots, k$.
Let $\bar{W}$ be the pullback of the following diagram:
\begin{equation}
\begin{CD}
\bar{W} @>>> X(Q, \eta)\\
@V\pi_v VV  @V\pi_Q VV \\
Q_P @>\iota >> Q.
\end{CD}
\end{equation}
Clearly, $W(Q_P, \lambda)= (\TT^{n+1} \times Q_P)/\sim_b$ is diffeomorphic to $W(Q, \lambda)$ and
$\bar{W} = W(Q_P, \lambda) \times T_Q$ where $T_Q$ is the circle subgroup of $\TT^{n+1}$ determined by
the vector $(0, \ldots, 0, 1)$ in $\ZZ^{n+1}$. We have the following commutative diagram of complex orbibundles.
\begin{equation}
\begin{CD}
E_3 @>>> E_2 @>>> E_1 @>>> T(X(Q, \eta)) \oplus \RR^{2(m-n-1)}\\
@VVV @VVV @VVV  @VVV \\
X(Q_i, \xi^i) @>>>W(Q_P, \lambda) @>>> \bar{W} @>>> X(Q, \eta)
\end{CD}
\end{equation}
where $E_1$ and $E_2$ are the pullback bundles. Since $\bar{W} = W(Q, \lambda) \times T_Q$, the bundle $E_2$ is isomorphic
to $T(W(Q_P, \lambda)) \oplus \RR^{2(m-n)+1}$. Hence for a choice of sign of isotropy vectors of $Q$ the orbifold
with boundary $W(Q, \lambda)$ has a stable complex structure.

Observe that the bundle $E_3$ is isomorphic to
\begin{equation}
T(X(Q_i, \xi^i)) \oplus \RR^{2(m-n)} \cong \rho_{i_1} \oplus \cdots \oplus \rho_{i_l} \oplus \RR^{2 (m-l)}
\end{equation}
where $\rho_{i_j}$ is the complex line bundle corresponding to the $i_j$-th characteristic suborbifold of $X(Q_i, \xi^i)$.
Hence $[X(Q_1, \xi^1)] + \cdots + [X(Q_k, \xi^k)] = 0$ in the orbifold complex cobordism group $OB_n(pt)$.
\end{proof}

In \cite{BR}, Buchstaber and Ray showed that any complex cobordism class contains a quasitoric manifold.
Non-singular projective toric varieties are almost complex quasitoric manifolds. Now we give a computation
when a complex cobordism class may contains an almost complex quasitoric manifold.

Let $M$ be a $2n$-dimensional almost complex quasitoric manifold over a simple polytope $P$. Let
$\xi : \mathcal{F}(P) \to \ZZ^n$ be the corresponding characteristic function of $M$. Let $Q$ be
an $(n+1)$-dimensional polytope with exceptional facets $\{P, \bigtriangleup^n_1, \ldots, \bigtriangleup^n_k\}$
where $\{\bigtriangleup_1^n, \ldots, \bigtriangleup_k^n\}$ are $n$-dimensional simplices in $\RR^{n+1}$.
Let facets of $Q$ are
$$\{F_1, \ldots, F_m\} \cup \{P, \bigtriangleup^n_1, \ldots, \bigtriangleup^n_k\}. $$ 
Let $\lambda: \{F_1, \ldots, F_m\} \to \ZZ^n$ be an isotropy function of $Q$ except the facets $P, \bigtriangleup_1^n,
\ldots, \bigtriangleup_k^n$ such that the following holds.
\begin{enumerate}
 \item $\lambda(F_i)=\xi(F_i \cap P)$ if $F_i \cap P$ is nonempty.
\item If $e$ is an edge of $Q$ not contained in $\cup_{i=1}^k \bigtriangleup^n_i \cup P$ then
$\{\lambda(F_{i_1}), \ldots, \lambda(F_{i_n})\}$ is a basis of $\ZZ^n$ where $e = F_{i_1} \cap \cdots \cap F_{i_n}$.
\end{enumerate}
Let $\xi^i : \mathcal{F}(\bigtriangleup^n_i) \to \ZZ^n$ be the map defined by 
$$\xi^i(F) = \lambda(F_j) ~ \mbox{if} ~ F= F_j \cap \bigtriangleup^n_i.$$
So $\xi^i$ is a characteristic function for a quasitoric manifold $X(\bigtriangleup^n_i, \xi^i)$.
Note that $X(\bigtriangleup_i^n, \xi^i)$ is $\delta$-equivariantly diffeomorphic to $\CP^n$ for some
$\delta \in \Aut(\TT^n)$.


\begin{corollary}\label{hp}
The complex cobordism class of $\sqcup_{i=1}^k X(\bigtriangleup^n_i, \xi^i)$
contains the almost complex quasitoric manifold $M$.
\end{corollary}
\begin{proof}
 This follows from Theorem \ref{comcob}.
\end{proof}
\begin{remark}
Since smooth projective spaces are almost complex quasitoric manifolds, the above corollary gives a partial answer
of the Hirzebruch problem which is mentioned in introduction.
\end{remark}


\begin{theorem}\label{qbd}
Let $X$ be a $2n$-dimensional quasitoric orbifold over the simple polytope $P$. Then $[X] = [M_1]
+ \cdots + [M_l]$ for some fake weighted projective spaces $M_1, \ldots, M_l$.
\end{theorem}
\begin{proof}
Let $\mathcal{F}(P) = \{F_1, \ldots, F_m\}$ be the facets and $\{v_1, \ldots, v_k\}$ be the vertices
of $P \subseteq \RR^n$.  Let $Q = P \times [0, 1]$. So $Q$ is an $(n+1)$-dimensional simple polytope in
$\RR^{n+1}$. Cut off a neighborhood of each vertex $v_j \times 0$ of $Q$ by an affine hyperplane $H_j$
for $j=1, \ldots, k$ in $\RR^{n+1}$ such that
$$H_i \cap H_j \cap Q ~ \mbox{is empty for} ~ i \neq j, ~ \mbox{and} ~ H_j \cap P ~\mbox{is empty for any} ~j \in \{1, \ldots, k\}.$$
Then the remaining subset, denoted by $Q_P$, of $Q$ is an $(n+1)$-dimensional simple polytope. Observe
that the facets $Q \cap H_i = H_i \cap Q_P$ of $Q_P$ is an $n$-dimensional simplex $\bigtriangleup^n_i$
in $\RR^{n+1}$ for $i=1, \ldots, k$. So $Q_P$ is polytope with exceptional facets $\{P \times 1, \bigtriangleup^n_1,
\ldots, \bigtriangleup^n_k \}$. Let
$$\bar{F}_0 = Q_P \cap P \times 0, ~\mbox{and} ~ \bar{F}_i = Q_P \cap (F_i \times [0, 1]) ~\mbox{for} ~ i=1, \ldots, m.$$
So $\mathcal{F}(Q_P)=\{\bar{F}_i : i=0, \ldots, m\} \cup \{P \times 1, \bigtriangleup^n_1, \ldots, \bigtriangleup^n_k\}$.

Let $\xi : \mathcal{F}(P) \to \ZZ^n$ be the di-characteristic function of $X$. Let $E(P)$ be the edges of $P$ and $e \in E(P)$.
Then $e = F_{i_1} \cap \cdots \cap F_{i_{n-1}}$ for a unique collection of facets $F_{i_1}, \ldots, F_{i_{n-1}}$.
Let $\ZZ(e)$ be the submodule of $\ZZ^n$ generated by $\{\xi(F_{i_1}), \ldots, \xi(F_{i_{n-1}})\}$. So $\ZZ(e)$ is
free $\ZZ$-module of rank $n-1$ for any $e \in E(P)$. Let
$$\lambda_0 \in \ZZ^n - \bigcup_{e \in E(P)} \ZZ(e)$$ be a primitive vector in $\ZZ^n$. Then $\{\lambda_0, \xi(F_{i_1}),
\ldots, \xi(F_{i_{n-1}})\}$ is a linearly independent set in $\ZZ^n$ for the edge $e \in E(P)$. 
So the function $\lambda$ defined below is an isotropy function of $Q_P$ except the facets
$\{P \times 1, \bigtriangleup^n_1, \ldots, \bigtriangleup^n_k\}$. 
\begin{equation}
\lambda(F) = \left\{ \begin{array}{ll} \xi(F_i) & \mbox{if}~  F= \bar{F}_i \\
\lambda_0 & \mbox{if}~ F = \bar{F}_0.
 \end{array} \right.
\end{equation}
Let $\xi^j : \mathcal{F}(\bigtriangleup^n_j) \to \ZZ^n$ be the restriction of $\lambda$ on the facets
of $\bigtriangleup^n_j$ for $j=1, \ldots, k$. Let $M_j$ be the quasitoric orbifold constructed from the 
characteristic model $(Q_P, \xi^j)$ for $j=1, \ldots, k$. So by Lemma \ref{orbbd}, the boundary of
$(2n+1)$-dimensional oriented orbifold $W(Q_P, \lambda)$ is $X \sqcup M_1 \sqcup \cdots \sqcup M_k$.
 Hence by Theorem \ref{comcob} we get the orbifold complex cobordism relation $[X]= [M_1] + \cdots + [M_k]$.
\end{proof}

\begin{obser}
Let $A = \{[M] : M ~ \mbox{is a quasitoric orbifold over}~ \bigtriangleup^n\}$. We show that $A$ is not a
linearly independent set in orbifold complex cobordism group. Let $Q$ be an $(n+1)$-dimensional simple
polytope in $\RR^{n+1}$ with vertices $\{v_1, \ldots, v_k\}$ and facts $\{F_1, \ldots, F_m\}$. We delete
a neighborhood of each vertex $v_i$ by cutting with an hyperplane $H_i$ in $\RR^{n+1}$ for $i= 1, \ldots, k$ such that $H_i \cap H_j \cap Q$
is empty for $i \neq j$. Then the remaining subset, denoted by $Q_V$, of $Q$ is an $(n+1)$-dimensional
polytope with exceptional facets $\{Q \cap H_i = Q_V \cap H_i : i=1, \ldots, k\}$. Since $Q$ is an
$(n+1)$-dimensional simple polytope, $Q \cap H_i$ is an $n$-simplex $\bigtriangleup^n_i$ for $i=1, \ldots, k$.
Let $\bar{F}_i = F_i \cap Q_V$ for $i=1, \ldots, k$. 
So $$\mathcal{F}(Q_V)=\{\bar{F}_1, \ldots, \bar{F}_m\} \cup \{\bigtriangleup_1^n, \ldots, \bigtriangleup_k^n\}$$ are the
facets of $Q_V$. Since $\ZZ^n$ is not a union of finitely many proper submodule over $\ZZ$, we can define
isotropy function $$\lambda: \{\bar{F}_1, \ldots, \bar{F}_m\} \to \ZZ^n$$ of the polytope $Q_V$ except the facets
$\{\bigtriangleup_1^n, \ldots, \bigtriangleup_k^n\}$.
Since $Q$ is a simple $(n+1)$-polytope, each vertex $v_i$ is the intersection of unique collection
of facets $\{F_{i_1}, \ldots, F_{i_{n+1}}\}$ for $i=1, \ldots, k$. So the facets of $\bigtriangleup^n_i$ are
$\mathcal{F}(\bigtriangleup_i^n) = \{F_{i_j} \cap \bigtriangleup_i^n : j = 1, \ldots, n+1\}$. We define a map
$\xi^i : \mathcal{F}(\bigtriangleup_i^n) \to \ZZ^n$ by
\begin{equation}
\xi^i(F_{i_j} \cap \bigtriangleup_i^n) = \lambda(\bar{F}_{i_j}).
\end{equation}
Then $\xi^i$ is a di-characteristic function of $\bigtriangleup_i^n$. Let $X(\bigtriangleup_i^n, \xi^i)$ be the 
quasitoric orbifold for the characteristic pair $(\bigtriangleup_i^n, \xi^i)$ for $i=1, \ldots, k$. 
So by Lemma $W(Q_V, \lambda)$ is an oriented orbifold with boundary where the boundary $\partial{\bigtriangleup^{n+1}_V}$
is a disjoint union of $\{X(\bigtriangleup_i^n, \xi^i) : i=1, \ldots, k\}$. So by theorem \ref{comcob}, we have
$[X(\bigtriangleup_i^n, \xi^1)] + \cdots + [X(\bigtriangleup_i^n, \xi^k)] =0$ in $OB_n(pt)$.
Hence the set of vectors in $A$ are not linearly independent. Now we may ask the following.
\end{obser}
\begin{ques}
What are the other complex cobordism relations among the quasitoric orbifolds over a simplex?
\end{ques}


{\bf Acknowledgement.} The author would like to thank Professor Mainak Poddar and 
Professor Goutam Mukherjee for helpful suggestions and stimulating discussions.
He thanks University of Regina and Pacific Institute for the Mathematical Sciences
for support. 

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