# GKM graph locally modeled by $T^{n}\times S^{1}$-action on
$T^{*}\mathbb{C}^{n}$ and its graph equivariant cohomology

Shintarô KUROKI  Okayama University of Science, 1-1 Ridai-cho Kita-ku Okayama-
shi Okayama 700-0005, JAPAN<EMAIL_ADDRESS>and  Vikraman UMA
Department of Mathematics, Indian Institute of Technology, Madras, Chennai
600036, INDIA<EMAIL_ADDRESS>
###### Abstract.

We introduce a class of labeled graphs (with legs) which contains two classes
of GKM graphs of $4n$-dimensional manifolds with $T^{n}\times S^{1}$-actions,
i.e., GKM graphs of the toric hyperK${\rm\ddot{a}}$hler manifolds and of the
cotangent bundles of toric manifolds. Under some conditions, the graph
equivariant cohomology ring of such a labeled graph is computed. We also give
a module basis of the graph equivariant cohomology by using a shelling
structure of such a labeled graph, and study their multiplicative structure.

###### Key words and phrases:

Equivariant cohomology, GKM-graph, toric hyperK${\rm\ddot{a}}$hler manifold,
cotangent bundle of a toric manifold, toric topology

###### 1991 Mathematics Subject Classification:

Primary 55N91; Secondary 05C65, 57S12

The 1st author was supported by JSPS KAKENHI Grant Number 17K14196 and
21K03262.

##  1\. Introduction

A GKM graph is a labeled graph defined by the special but wide class of
manifolds with torus actions, called GKM manifolds. From the torus action on a
GKM manifold, a GKM graph is defined by its zero and one dimensional orbits
together with the labels on edges defined by the tangential representations
around fixed points. Goresky-Kottwicz-MacPherson in [GKM] show that if a GKM
manifold satisfies a certain condition, called equivariant formality, then its
equivariant cohomology is isomorphic to an algebra defined from its GKM graph.
We call this algebra a graph equivariant cohomology in this paper. Motivated
by the work of Goresky-Kottwicz-MacPherson, Guillemin-Zara in [GZ] introduce
the abstract GKM graph without considering any GKM manifolds, and they
translate some geometric properties of GKM manifolds into combinatorial ones
of GKM graphs. After the works of Guillemin-Zara, a GKM graph can be regarded
as a combinatorial approximation of space with torus action, and it has been
studied by some mathematicians, e.g. see [MMP, GHZ, GSZ, FIM, FY, Ku19, DKS].
In this paper, we introduce a certain class of GKM graphs with legs and
attempt to unify two slightly different classes of manifolds from the GKM
theoretical point of view, i.e., toric hyperKähler manifolds and cotangent
bundles of toric manifolds, where a leg is a half-line whose boundary
corresponds to the initial vertex. We briefly recall toric hyperKähler
manifolds and introduce the motivation of the present paper. We shall then
state our main results and outline the organization of this paper.

A toric hyperKähler variety is defined by the hyperK${\rm\ddot{a}}$hler
quotient of a torus action on the cotangent bundle $T^{*}\mathbb{C}^{m}$. This
space is introduced by Goto and Bielwasky-Dancer in [G, BD] as the
hyperK${\rm\ddot{a}}$hler analogue of the symplectic toric manifolds. The non-
singular toric hyperK${\rm\ddot{a}}$hler varieties are $4n$-dimensional non-
compact manifolds with $T^{n}$-action. They are completely determined by some
class of hyperplane arrangements in $\mathbb{R}^{n}$ (see [BD]) like
symplectic toric manifolds are completely determined by Delzant polytopes in
$\mathbb{R}^{n}$ (see [D]). The equivariant topology and geometry of toric
hyperK${\rm\ddot{a}}$hler manifolds are studied by some mathematicians, e.g.
[Ko99, Ko00, Ko03, HP, P, Ku11]. In particular, Harada-Proudfoot show that
every toric hyperK${\rm\ddot{a}}$hler manifold admits the residual
$S^{1}$-action and the equivariant cohomology of a toric
hyperK${\rm\ddot{a}}$hler manifold with $T^{n}\times S^{1}$-action is
determined by the half-space arrangements in $\mathbb{R}^{n}$. They also show
that the toric hyperK${\rm\ddot{a}}$hler manifolds with $T^{n}\times
S^{1}$-actions satisfy the GKM condition, i.e., its zero and one dimensional
orbits have the structure of a graph. Note that the $T^{n}$-action on a toric
hyperK${\rm\ddot{a}}$hler manifold does not satisfy the GKM condition.
Therefore, we can define the labeled graph (with legs) from toric
hyperK${\rm\ddot{a}}$hler manifolds with $T^{n}\times S^{1}$-actions. The GKM
graph of a toric hyperK${\rm\ddot{a}}$hler manifold is obtained from the one-
dimensional intersections of hyperplanes like the GKM graph of a symplectic
toric manifold is obtained from the one-skeleton of a moment-polytope. By
definition, the tangential representations of $T^{n}\times S^{1}$-actions on
the fixed points are isomorphic (up to automorphism on $T^{n}\times S^{1}$) to
the standard $T^{n}$-action on $T^{*}\mathbb{C}^{n}$ together with the scalar
multiplication of $S^{1}$ on the fiber, i.e.,

(1.1)
$\displaystyle(t_{1},\ldots,t_{n},r)\cdot(z_{1},\ldots,z_{n},w_{1},\ldots,w_{n})\mapsto(t_{1}z_{1},\ldots,t_{n}z_{n},rt_{1}^{-1}w_{1},\ldots,rt_{n}^{-1}w_{n}),$

where

$\displaystyle(t_{1},\ldots,t_{n})\in T^{n},\quad r\in
S^{1},\quad(z_{1},\ldots,z_{n},w_{1},\ldots,w_{n})\in
T^{*}\mathbb{C}^{n}(\simeq\mathbb{C}^{n}\times\mathbb{C}^{n}).$

We call the action defined by (1.1) is the standard $T^{n}\times S^{1}$-action
on $T^{*}\mathbb{C}^{n}$.

On the other hand, the cotangent bundle $T^{*}M$ of a $2n$-dimensional toric
manifold $M$ also has the $T^{n}\times S^{1}$-action. More precisely, because
$T$ acts on $M$ smoothly, each element $t\in T$ induces the diffeomorphism
$t:M\to M$, say $t:p\mapsto t\cdot p$ for $p\in M$. By taking its differential
$dt:TM\to TM$, we have the lift of the $T$-action on the tangent bundle $TM$.
Note that $(dt)_{p}:T_{p}M\to T_{t\cdot p}M$ is the linear isomorphism.
Because the cotangent bundle $T^{*}M$ is defined by the bundle over $M$ whose
fibres are $T^{*}_{p}M:={\rm Hom}(T_{p}M,\mathbb{R})$. Therefore, for an
element $f\in T^{*}_{p}M$, we can define the $t\in T$ action by $t\cdot
f:=f\circ(dt)_{p}^{-1}:T_{t\cdot p}M\to\mathbb{R}$. Together with the scalar
multiplication by $S^{1}$ on each fibre $T_{p}^{*}M$, we have the $T^{n}\times
S^{1}$-action on $T^{*}M$. It follows from the definition that this also
satisfies the GKM conditions and the tangential representation around every
fixed point in $T^{*}M$ is isomorphic to the standard $T^{n}\times
S^{1}$-action on $T^{*}\mathbb{C}^{n}$.

Note that $T^{*}M$ of a toric manifold $M$ is not a toric
hyperK${\rm\ddot{a}}$hler manifold except in the case when $M$ is a product of
some projective spaces, see [BD]. So the cotangent bundles of toric manifolds
and the toric hyperK${\rm\ddot{a}}$hler manifolds are different classes of
manifolds. However, it is known that their equivariant cohomologies are quite
similar. The equivariant cohomology $H_{T^{n}}^{*}(M)$ of a symplectic toric
manifold $M$ with $T^{n}$-action is isomorphic to the Stanley-Reisner ring of
the moment-polytope (see e.g. [BP, Lemma 7.4.34] for more general class of
manifolds with $T^{n}$-actions). Because there is an equivariant deformation
retract from $T^{*}M$ to $M$, we see that $H_{T^{n}}^{*}(T^{*}M)$ is also
isomorphic to the Stanley-Reisner ring of a polytope. On the other hand, by
Konno’s theorem [Ko99], the equivariant cohomology of a toric
hyperK${\rm\ddot{a}}$hler manifold with $T^{n}$-action is isomorphic to the
Stanley-Reisner ring of hyperplane arrangements. Therefore, these distinct
classes of manifolds have similar equivariant cohomology ring structures. So
it may be natural to ask whether we can unify these classes of manifolds. One
answer is that there exists an embedding from a symplectic toric manifold $M$
to a toric hyperK${\rm\ddot{a}}$hler manifold (see [BD, HP]). In this paper,
we answer this question from a different direction, namely, we unify the
equivariant cohomologies of these classes by using GKM graphs.

To achieve that, we introduce the class of GKM graphs whose axial functions
around vertices are modeled by the standard $T^{n}\times S^{1}$-action on
$T^{*}\mathbb{C}^{n}$, called a GKM graph locally modeled by $T^{n}\times
S^{1}$-action on $T^{*}\mathbb{C}^{n}$ or $T^{*}\mathbb{C}^{n}$-modeled GKM
graph for short in Definition 2.4. This GKM graph behaves like the hyperplane
arrangements but does not always come from the hyperplane arrangements, see
Section 3. We study the graph equivariant cohomology of
$T^{*}\mathbb{C}^{n}$-modeled GKM graphs. The first main theorem of this paper
is as follows (the technical notions will be introduced in Section 3 and
Section 4):

######  Theorem 1.1 (Theorem 4.1).

Let $\mathcal{G}$ be a $2n$-valent $T^{*}\mathbb{C}^{n}$-modeled GKM graph and
$\mathbf{L}=\\{L_{1},\ \cdots,\ L_{m}\\}$ be the set of all hyperplanes in
$\mathcal{G}$. Assume that $\mathcal{G}$ satisfies the following two
assumptions:

  1. (1)

For each $L\in\mathbf{L}$, there exist the unique pair of the halfspace $H$
and its opposite side $\overline{H}$ such that $H\cap\overline{H}=L$;

  2. (2)

For every subset $\mathbf{L}^{\prime}\subset\mathbf{L}$, its intersection
$\displaystyle\bigcap_{L\in\mathbf{L}^{\prime}}L$ is empty or connected.

Then the following ring isomorphism holds:

$\displaystyle H^{*}(\mathcal{G})\simeq{\mathbb{Z}}[\mathcal{G}].$

To prove this, we introduce an $x$-forgetful graph $\mathcal{\widetilde{G}}$
from a $T^{*}\mathbb{C}^{n}$-modeled GKM graph $\mathcal{G}$ in Section 5. An
$x$-forgetful graph $\mathcal{\widetilde{G}}$ is a labeled graph but not a GKM
graph, and it may be regarded as the combinatorial counterpart of the
$T^{n}$-actions on toric hyperK${\rm\ddot{a}}$hler manifolds and cotangent
bundles over toric manifolds. We define its graph equivariant cohomology
$H^{*}(\widetilde{\mathcal{G}})$, and prove its ring structure in Theorem 5.1.
In Section 6, we give a proof of Theorem 4.1 by using Theorem 5.1. In Section
7, we also study the $H^{*}(BT^{n})$-module structure of
$H^{*}(\widetilde{\mathcal{G}})$. As the second main result of this paper, in
Theorem 7.6, we exhibit an $H^{*}(BT^{n})$-module basis of
$H^{*}(\widetilde{\mathcal{G}})$ by using the shellablility of a simplicial
complex $\Delta_{\mathbf{L}}$ associated to $\mathbf{L}$. Dividing
$H^{*}(\widetilde{\mathcal{G}})$ by $H^{>0}(BT^{n})$, we also introduce
$H^{*}_{ord}(\widetilde{\mathcal{G}})$, which corresponds to the ordinary
cohomology of the usual equivariant cohomology. Then, we show that the
$H^{*}(BT^{n})$-module basis of $H^{*}(\widetilde{\mathcal{G}})$ induces a
$\mathbb{Z}$-module basis for $H^{*}_{ord}(\widetilde{\mathcal{G}})$. Finally,
in the case when $\widetilde{\mathcal{G}}$ corresponds to the line
arrangements in $\mathbb{R}^{2}$ (which corresponds geometrically to the
$8$-dimensional toric hyperK${\rm\ddot{a}}$hler manifolds), we describe the
structure constants of $H^{*}_{ord}(\widetilde{\mathcal{G}})$ with respect to
this basis.

##  2\. GKM graph locally modeled by $T^{n}\times S^{1}$-action on
$T^{*}\mathbb{C}^{n}$

This section aims to define a GKM graph with legs and its graph equivariant
cohomology. In particular, we introduce GKM graphs locally modeled by
$T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ as the special class of
GKM graphs with legs.

###  2.1. Notations

We first prepare some notations. In this paper $\Gamma$ is a connected graph
which possibly has legs, where a leg means an outgoing half-line from one
vertex (see the left graph in the Figure 1).

Figure 1. These are examples of regular graphs with legs and orientations. The
left $2$-valent graph has two legs, on the other hand the right $3$-valent
graph has no legs. Note that all edges have two orientations and all legs have
only one orientation.

We define a graph with legs more precisely. Let $\mathcal{V}$ be a set of
vertices, $E$ be a set of edges and $Leg$ be a set of legs in $\Gamma$, and
$\mathcal{E}=E\cup Leg$. The graph $\Gamma$ is denoted by

$\displaystyle\Gamma=(\mathcal{V},\ \mathcal{E}).$

In this paper, we assume that $\mathcal{V}$ and $\mathcal{E}$ are finite sets.
We also assume that $\Gamma$ is an oriented graph. For $\epsilon\in E$, we
denote by $i(\epsilon)$ and $t(\epsilon)$ the initial vertex and the terminal
vertex of $\epsilon$, respectively. We denote the opposite directed edge of
$\epsilon$ as $\overline{\epsilon}$, i.e.,
$i(\overline{\epsilon})=t(\epsilon)$ and $t(\overline{\epsilon})=i(\epsilon)$.
For $\ell\in Leg$, there is no terminal vertex but there exists an initial
vertex $i(\ell)$. Note that the leg in $\Gamma$ can be characterized by the
element $\epsilon$ in $\mathcal{E}$ such that there is no
$\overline{\epsilon}$. For a vertex $p\in\mathcal{V}$, we put the set of all
outgoing edges and legs from $p\in\mathcal{V}$ by

$\displaystyle\mathcal{E}_{p}=\\{\epsilon\in\mathcal{E}\ |\ i(\epsilon)=p\\}.$

Assume that $|\mathcal{E}_{p}|=m$ for all $p\in\mathcal{V}$, where the symbol
$|X|$ represents the cardinality of the finite set $X$. We call such a graph a
(regular) $m$-valent graph, see Figure 1.

Let $\Gamma=(\mathcal{V},\mathcal{E})$ be a graph with legs. We denote a
subgraph of $\Gamma$ by $G=(\mathcal{V}^{G},\ \mathcal{E}^{G})$, that is, $G$
satisfies $\mathcal{V}^{G}\subset\mathcal{V}$ and
$\mathcal{E}^{G}\subset\mathcal{E}$. We use the following symbols.

  * •

$\mathcal{E}^{G}_{p}$: the set of all outgoing edges and legs in
$\mathcal{E}^{G}$ from $p\in\mathcal{V}^{G}$.

  * •

$E^{G}\subset\mathcal{E}^{G}$ (resp. $E^{G}_{p}\subset\mathcal{E}^{G}_{p}$):
the set of all edges (resp. out going from $p$) in $G$, i.e., if $\epsilon\in
E^{G}$, then the both $i(\epsilon),t(\epsilon)\in\mathcal{V}^{G}$.

  * •

$Leg^{G}\subset\mathcal{E}^{G}$ (resp.
$(Leg)_{p}^{G}\subset\mathcal{E}^{G}_{p}$): the set of all legs (resp. out
going from $p$) in $G$.

Note the following remark about legs in $G$.

######  Remark 2.1.

Because $\Gamma$ is an oriented graph, we may consider the subgraph
$G=(\mathcal{V}^{G},\ \mathcal{E}^{G})$ of $\Gamma=(\mathcal{V},\mathcal{E})$
such that there exists a leg $\epsilon\in Leg^{G}$ in $G$ which is an edge
$\epsilon\in\mathcal{E}$ in $\Gamma$. In other words,
$t(\epsilon)\not\in\mathcal{V}^{G}$ but $t(\epsilon)\in\mathcal{V}$; or
$\overline{\epsilon}\not\in\mathcal{E}^{G}$ but
$\overline{\epsilon}\in\mathcal{E}$.

###  2.2. GKM graph with legs and its graph equivariant cohomology

In this section, we shall define a GKM graph ((possibly) with legs) and its
graph equivariant cohomology.

Let $\Gamma=(\mathcal{V},\ \mathcal{E})$ be an $m$-valent graph. We first
prepare the following notations. Let $\nabla=\\{\nabla_{\epsilon}\ |\
\epsilon\in E\\}$ be a collection of bijective maps

$\displaystyle\nabla_{\epsilon}:\mathcal{E}_{i(\epsilon)}\to\mathcal{E}_{t(\epsilon)}$

for all edges $\epsilon\in E$. A connection on $\Gamma$ is a set
$\nabla=\\{\nabla_{\epsilon}|\ \epsilon\in E\\}$ which satisfies the following
two conditions:

  * •

$\nabla_{\bar{\epsilon}}=\nabla_{\epsilon}^{-1}$;

  * •

$\nabla_{\epsilon}(\epsilon)=\bar{\epsilon}$.

We can easily check that an $m$-valent graph $\Gamma$ admits different
$((m-1)!)^{g}$ connections, where $g$ is the number of (unoriented) edges $E$.

Let $T^{n}$ be an $n$-dimensional torus. In particular, we often denote a
$1$-dimensional torus by $S^{1}$. If we do not emphasis the dimension of
$T^{n}$, then we denote it by $T$. Let $\mathfrak{t}$ be a Lie algebra of $T$,
$\mathfrak{t}_{{\mathbb{Z}}}$ be the lattice of $\mathfrak{t}$ and
$\mathfrak{t}^{*}$ (resp. $\mathfrak{t}_{{\mathbb{Z}}}^{*}$) be the dual of
$\mathfrak{t}$ (resp. $\mathfrak{t}_{{\mathbb{Z}}}$). The symbol ${\rm
Hom}(T,S^{1})$ represents a set of all homomorphisms from the torus $T$ to
$S^{1}$. It is well-known that ${\rm Hom}(T^{n},S^{1})\simeq\mathbb{Z}^{n}$.
Moreover, it may be regarded as $\mathfrak{t}^{*}_{{\mathbb{Z}}}$ and
$H^{1}(T)\simeq H^{2}(BT)$, where $BT$ is the classifying space of $T$. In
this paper, if we omit the coefficient of the cohomology, then it means the
cohomology with integer coefficients. Therefore, we have the identification

$\displaystyle{\rm Hom}(T,S^{1})\simeq\mathfrak{t}^{*}_{{\mathbb{Z}}}\simeq
H^{2}(BT).$

Define an axial function by the function

$\displaystyle\alpha:\mathcal{E}\longrightarrow H^{2}(BT)$

such that it satisfies the following three conditions:

  * •

$\alpha(\bar{\epsilon})=\pm\alpha(\epsilon)$ for all edges $\epsilon\in E$;

  * •

$\alpha(\mathcal{E}_{p})=\\{\alpha(\epsilon)\ |\
\epsilon\in\mathcal{E}_{p}\\}$ are pairwise linearly independent for all
$p\in\mathcal{V}$, that is, for every two distinct elements $\epsilon_{1},\
\epsilon_{2}\in\mathcal{E}_{p}$, $\alpha(\epsilon_{1}),\ \alpha(\epsilon_{2})$
are linearly independent in $H^{2}(BT)$;

  * •

there is a connection $\nabla$ which satisfies the following congruence
relation for all edges $\epsilon\in E$:

$\displaystyle\alpha(\epsilon^{\prime})-\alpha(\nabla_{\epsilon}(\epsilon^{\prime}))\equiv
0({\rm mod}\ \alpha(\epsilon))$

for all $\epsilon^{\prime}\in\mathcal{E}_{i(\epsilon)}$.

######  Definition 2.2 (GKM graph with legs).

Let $\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ be a collection of an $m$-valent
graph $\Gamma=(\mathcal{V},\ \mathcal{E})$, where the map

$\displaystyle\alpha:\mathcal{E}\longrightarrow H^{2}(BT^{n}),$

is an axial function ($n\leq m$), and $\nabla$ is a connection on $\Gamma$. We
call $\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ a GKM graph (with legs).

######  Remark 2.3.

Suppose that $\mathcal{E}_{p}$ satisfies the $3$-linearly independent
condition for all $p\in\mathcal{V}$, i.e., for every distinct three elements
$\epsilon_{1},\epsilon_{2},\epsilon_{3}\in\mathcal{E}_{p}$ the axial function
$\alpha(\epsilon_{1}),\alpha(\epsilon_{2}),\alpha(\epsilon_{3})$ are linearly
independent. Then, by the similar proof for the cases of GKM graphs without
legs in [GZ], if there exists a connection $\nabla$, then the connection
$\nabla$ is unique. In particular, if the GKM graph $\mathcal{G}$ satisfies
the $3$-linearly independent condition, then for any two edges (or legs)
$\epsilon,\epsilon^{\prime}$ in $E_{p}$ we can determine the $2$-valent GKM
subgraph which contains $\epsilon,\ \epsilon^{\prime}$. Hence, we often omit
the connection $\nabla$ in the GKM graph $\mathcal{G}$. Namely, we often
denote the GKM graph by $\mathcal{G}=(\Gamma,\alpha)$ if the connection is
obviously determined by the context.

Due to the theory of toric hyperK$\ddot{\rm a}$hler varieties (see [BD, HP,
Ko99]), the tangential representation on each fixed point is isomorphic to the
$T^{n}$-action on $T^{*}\mathbb{C}^{n}$ ($\simeq\mathbb{H}^{n}$, i.e., the
$n$-dimensional quaternionic space) which is defined by the strandard
$T^{n}$-action on $\mathbb{C}^{n}$, i.e.,

$\displaystyle(t_{1},\ldots,t_{n})\cdot(z_{1},\ldots,z_{n},w_{1},\ldots,w_{n}):=(t_{1}z_{1},\ldots,t_{n}z_{n},t_{1}^{-1}w_{1},\ldots,t_{n}^{-1}w_{n}),$

where $(t_{1},\ldots,t_{n})\in T^{n}$,
$z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ and $(w_{1},\ldots,w_{n})\in
T^{*}_{z}(\mathbb{C}^{n})$. On the other hand, Harada-Proudfoot [HP] found
that there exists the residual $S^{1}$-action on the toric hyperK$\ddot{\rm
a}$hler varieties and this action fits into the GKM theory. In this case, the
tangential representation on each fixed point may be regarded as the
$T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$, i.e.,

$\displaystyle(t_{1},\ldots,t_{n},r)\cdot(z_{1},\ldots,z_{n},w_{1},\ldots,w_{n}):=(t_{1}z_{1},\ldots,t_{n}z_{n},rt_{1}^{-1}w_{1},\ldots,rt_{n}^{-1}w_{n}),$

where $r\in S^{1}$. Therefore, the toric hyperK$\ddot{\rm a}$hler manifolds
with $T^{n}\times S^{1}$-actions induce the GKM graphs with legs whose axial
functions around every vertex are isomorphic to

$\displaystyle\\{e_{1}^{*},\ldots,e_{n}^{*},-e_{1}^{*}+x,\ldots,-e_{n}^{*}+x\\},$

where $e_{1}^{*},\ldots,e_{n}^{*}$ is a generator of a rank $n$ subspace in
$H^{2}(BT^{n}\times BS^{1})\simeq\mathbb{Z}^{n}\oplus\mathbb{Z}$ and $x$ is a
generator of $H^{2}(BS^{1})\simeq\mathbb{Z}$. By defining this abstractly, we
introduce the following notion:

######  Definition 2.4 (GKM graph locally modeled by $T^{n}\times
S^{1}$-action on $T^{*}\mathbb{C}^{n}$).

Let $\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ be a $2n$-valent GKM graph with
legs with an axial function

$\displaystyle\alpha:\mathcal{E}\longrightarrow H^{2}(BT^{n}\times
BS^{1})\simeq\mathfrak{t}^{*}_{{\mathbb{Z}}}\oplus{\mathbb{Z}}x,$

where $x$ is a generator of the dual of the Lie algebra of $S^{1}$. We call
$\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ a GKM graph modeled by the
$T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ (or simply a
$T^{*}\mathbb{C}^{n}$-modeled GKM graph) if it satisfies the following
conditions for all $p\in\mathcal{V}$:

  1. (1)

We can divide $\mathcal{E}_{p}$ into $\\{\epsilon_{1}^{+},\ \cdots,\
\epsilon_{n}^{+},\ \epsilon_{1}^{-},\ \cdots,\ \epsilon_{n}^{-}\\}$ such that

$\displaystyle\alpha(\epsilon_{j}^{+})+\alpha(\epsilon_{j}^{-})=x$

for all $j=1,\ \cdots,\ n$;

  2. (2)

The set $\\{\alpha(\epsilon_{j}^{+}),\ x\ |\ j=1,\cdots,n\\}$ spans
$\mathfrak{t}^{*}_{{\mathbb{Z}}}\oplus{\mathbb{Z}}x$, i.e.,

$\displaystyle\langle\alpha(\epsilon_{1}^{+}),\ \cdots,\
\alpha(\epsilon_{n}^{+}),\
x\rangle=\mathfrak{t}^{*}_{{\mathbb{Z}}}\oplus{\mathbb{Z}}x.$

We call $\\{\epsilon_{j}^{+},\ \epsilon_{j}^{-}\\}$ such that
$\alpha(\epsilon_{j}^{+})+\alpha(\epsilon_{j}^{-})=x$ a $1$-dimensional pair
in $\mathcal{E}_{p}$. Furthermore, we call an element $x$ a residual basis.

Figure 2 shows some examples of $T^{*}\mathbb{C}^{n}$-modeled GKM graphs.

$e_{2}^{*}$$e_{1}^{*}$$x-e_{2}^{*}$$x-e_{1}^{*}$$-e_{2}^{*}$$e_{2}^{*}+x$$e_{2}^{*}-e_{1}^{*}+x$$e_{1}^{*}-e_{2}^{*}$$-e_{1}^{*}$$e_{1}^{*}+x$$e_{1}^{*}-e_{2}^{*}+x$$e_{2}^{*}-e_{1}^{*}$$e_{2}^{*}+2x$$-e_{2}^{*}-x$$e_{1}^{*}$$x-e_{1}^{*}$$e_{2}^{*}+2x$$e_{1}^{*}+x$$-e_{1}^{*}$$e_{2}^{*}$$x-e_{2}^{*}$$e_{1}^{*}$$x-e_{1}^{*}$$e_{1}^{*}-x$$2x-e_{1}^{*}$$e_{2}^{*}+2x$
Figure 2. $T^{*}\mathbb{C}^{2}$-modeled GKM graphs, where $\langle{\rm
e_{1}^{*},\ e_{2}^{*}}\rangle\simeq(\mathfrak{t}^{2})^{*}_{{\mathbb{Z}}}$. In
the left and the right figures, we assume
$\alpha(\epsilon)=-\alpha(\overline{\epsilon})$ and omit some axial functions
which are automatically determined by the definition.

######  Remark 2.5.

Note that the axial function on $T^{*}\mathbb{C}^{n}$-modeled GKM graphs
satisfies the $3$-linearly independent condition for all vertices. Therefore,
the connection on a $T^{*}\mathbb{C}^{n}$-modeled GKM graph $\mathcal{G}$ is
uniquely determined and we may denote it by

$\displaystyle\mathcal{G}=(\Gamma,\alpha).$

We also have the following lemma for the $1$-dimensional pair
$\\{\epsilon^{+},\ \epsilon^{-}\\}$.

######  Lemma 2.6.

Let $\\{\epsilon^{+},\ \epsilon^{-}\\}$ be a $1$-dimensional pair in
$\mathcal{E}_{i(\epsilon^{\prime})}$ for some edge $\epsilon^{\prime}\in E$.
Then $\\{\nabla_{\epsilon^{\prime}}(\epsilon^{+}),\
\nabla_{\epsilon^{\prime}}(\epsilon^{-})\\}$ is also a $1$-dimensional pair in
$\mathcal{E}_{t(\epsilon^{\prime})}$.

###### Proof.

We first divide the edges and legs $\mathcal{E}_{i(\epsilon^{\prime})}$ by the
$1$-dimensional pairs as follows:

$\displaystyle\mathcal{E}_{i(\epsilon^{\prime})}=\\{\epsilon^{+}_{1},\
\epsilon^{-}_{1}\\}\cup\cdots\cup\\{\epsilon^{+}_{n},\ \epsilon^{-}_{n}\\}$

Because the axial function $\alpha$ satisfies the congruence relation, there
are integers $k^{+}_{j}$ and $k^{-}_{j}$, $j=1,\ldots,n$, such that

$\displaystyle\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{+}))-\alpha(\epsilon_{j}^{+})=k^{+}_{j}\alpha(\epsilon^{\prime}),$
$\displaystyle\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-}))-\alpha(\epsilon_{j}^{-})=k^{-}_{j}\alpha(\epsilon^{\prime}).$

Since $\\{\epsilon_{j}^{+},\ \epsilon_{j}^{-}\\}$ is a $1$-dimensional pair,
we also have

(2.1)
$\displaystyle(\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{+}))-\alpha(\epsilon_{j}^{+}))+(\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-}))-\alpha(\epsilon_{j}^{-}))$
$\displaystyle=$
$\displaystyle\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{+}))+\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-}))-x$
$\displaystyle=$
$\displaystyle(k_{j}^{+}+k_{j}^{-})\alpha(\epsilon^{\prime}).$

In order to show the statement, it is enough to show that the following
equation holds:

$\displaystyle k_{j}^{-}=-k_{j}^{+}.$

Suppose on the contrary that $k_{j}^{-}\not=-k_{j}^{+}$. Then, by (2.1), we
have

$\displaystyle\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-}))=-\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{+}))+x+(k_{j}^{+}+k_{j}^{-})\alpha(\epsilon^{\prime})\not=-\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{+}))+x.$

This implies that $\\{\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{+}),\
\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-})\\}$ is not a $1$-dimensional
pair. Therefore, there is another element
$\epsilon(\not=\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-}))$ in
$\mathcal{E}_{t(\epsilon^{\prime})}$ such that

$\displaystyle\alpha(\epsilon)=-\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{+}))+x.$

This gives that

$\displaystyle\alpha(\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-}))$
$\displaystyle=\alpha(\epsilon)+(k_{j}^{+}+k_{j}^{-})\alpha(\epsilon^{\prime}).$

However, since
$\\{\nabla_{\epsilon^{\prime}}(\epsilon_{j}^{-}),\epsilon,\overline{\epsilon^{\prime}}\\}\subset\mathcal{E}_{t(\epsilon^{\prime})}$,
this is a contradiction to the fact that $T^{*}\mathbb{C}^{n}$-modeled GKM
graph is always $3$-linearly independent. Hence, we must have
$k_{j}^{-}=-k_{j}^{+}$. This establishes the statement. ∎

Finally, in this section, we also define the notion of graph equivariant
cohomology. Let $\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ be a GKM graph (with
legs) such that $\alpha:\mathcal{E}\to H^{2}(BT)$. With the definition similar
to that of the GKM graph without legs, the graph equivariant cohomology is
defined as follows.

######  Definition 2.7 (graph equivariant cohomology).

The following ring is called a graph equivariant cohomology of $\mathcal{G}$:

$\displaystyle H^{*}(\mathcal{G})=\\{\varphi:\mathcal{V}\to H^{*}(BT)\ |\
\varphi(i(\epsilon))-\varphi(t(\epsilon))\equiv 0\ ({\rm mod}\
\alpha(\epsilon))\\},$

We call the relation $\varphi(i(\epsilon))-\varphi(t(\epsilon))\equiv 0\ ({\rm
mod}\ \alpha(\epsilon))$ a congruence relation of $\varphi$ on an edge
$\epsilon\in E$.

##  3\. Some notions of $T^{*}\mathbb{C}^{n}$-modeled GKM graphs

Let $\mathcal{G}=(\Gamma,\alpha,\nabla)$ be a $2n$-valent
$T^{*}\mathbb{C}^{n}$-modeled GKM graph such that $\alpha:\mathcal{E}\to
H^{2}(BT^{n})\oplus\mathbb{Z}x$, where $x$ is a residual basis. The goal of
this paper is to compute $H^{*}(\mathcal{G})$ for a certain class of
$T^{*}\mathbb{C}^{n}$-modeled GKM graphs. To do that we prepare some notions
and properties of $T^{*}\mathbb{C}^{n}$-modeled GKM graphs. We first introduce
the following notion.

We define a GKM subgraph $\mathcal{H}=(H,\alpha^{H},\nabla^{H})$ by a
$k$-valent subgraph $H$ of $\Gamma$ such that the axial function is defined by

$\displaystyle\alpha^{H}:=\alpha|_{\mathcal{E}^{H}},$

and the connection is defined by

$\displaystyle\nabla^{H}:=\\{\nabla_{\epsilon}|_{\mathcal{E}^{H}}\ |\
\epsilon\in E^{H}\\}.$

We also denote $\nabla^{H}_{\epsilon}:=\nabla_{\epsilon}|_{\mathcal{E}^{H}}$
for an edge $\epsilon\in E^{H}$.

###  3.1. Hyperplane

In this section, we introduce the notion of a hyperplane in $\mathcal{G}$ and
show a key property Lemma 3.3 which will be used to show the main theorem of
this paper.

######  Definition 3.1 (hyperplane).

Let $\mathcal{G}$ be a $T^{*}\mathbb{C}^{n}$-modeled GKM graph. Assume that a
GKM subgraph $\mathcal{L}=(L,\alpha^{L},\nabla^{L})$ of $\mathcal{G}$ is a
$(2n-2)$-valent subgraph of $\Gamma$ and it is a
$T^{*}\mathbb{C}^{n-1}$-modeled GKM graph with the residual basis $x$, i.e.,
there are $1$-dimensional pairs
$\\{\epsilon_{j}^{+},\epsilon_{j}^{-}\\}\subset\mathcal{E}_{p}^{L}$,
$j=1,\ldots,n-1$, on each vertex such that

$\displaystyle\alpha^{L}(\epsilon_{j}^{+})+\alpha^{L}(\epsilon_{j}^{-})=x.$

Such a GKM subgraph $\mathcal{L}$ is said to be a hyperplane if $L$ is a
maximal $(2n-2)$-valent connected subgraph in $\Gamma$, i.e., if $L^{\prime}$
is a $(2n-2)$-valent connected subgraph in $\Gamma$ such that $L\subset
L^{\prime}$ then $L=L^{\prime}$.

######  Example 3.2.

The following two figures show an example and a non-example of hyperplanes.

$e_{1}^{*}$$x-e_{1}^{*}$$-e_{1}^{*}$$e_{1}^{*}+x$$e_{2}^{*}$$e_{1}^{*}$$-e_{2}^{*}$$e_{1}^{*}-e_{2}^{*}$$-e_{1}^{*}$$e_{2}^{*}-e_{1}^{*}$
Figure 3. The left figure shows a hyperplane of the left GKM graph in Figure
2. The right figure is a $2$-valent GKM subgraph of the left GKM graph in
Figure 2 but it is not a hyperplane.

For the hyperplane $\mathcal{L}$ we have the following property (this may be
regarded as the analogue of the property of facets in a torus graph [MMP]).

######  Lemma 3.3.

Let $\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ be a $2n$-valent
$T^{*}\mathbb{C}^{n}$-modeled GKM graph. Take a vertex $p\in\mathcal{V}$.
Then, for every $1$-dimensional pair
$\\{\epsilon^{+},\epsilon^{-}\\}\subset\mathcal{E}_{p}$, there exists a unique
hyperplane $\mathcal{L}=(L,\alpha^{L},\nabla^{L})$ such that
$\mathcal{E}_{p}^{L}=\mathcal{E}_{p}\setminus\\{\epsilon^{+},\epsilon^{-}\\}$.

###### Proof.

We first prove the existence of the hyperplane which satisfies the statement.
Put
$\mathcal{E}_{p}^{L}=\mathcal{E}_{p}\setminus\\{\epsilon^{+},\epsilon^{-}\\}$.
Then we can write $\mathcal{E}_{p}^{L}=\\{\epsilon_{1}^{+},\ \cdots,\
\epsilon_{n-1}^{+},\ \epsilon_{1}^{-},\ \cdots,\ \epsilon_{n-1}^{-}\\}$ the
$(n-1)$ $1$-dimensional pairs in $\mathcal{E}_{p}$ which are different from
$\\{\epsilon^{+},\epsilon^{-}\\}$. Let

$\displaystyle R:=\langle\alpha(\epsilon_{1}^{+}),\ \cdots,\
\alpha(\epsilon_{n-1}^{+}),\ x\rangle.$

By Definition 2.4 (2) we may assume that
$\langle\alpha(\epsilon_{1}^{+}),\ldots,\alpha(\epsilon_{n-1}^{+})\rangle\subset\mathfrak{t}^{*}_{\mathbb{Z}}\oplus\mathbb{Z}x$
is a submodule of rank $(n-1)$. Thus we have

$\displaystyle R\simeq H^{2}(BT^{n-1})\oplus{\mathbb{Z}}x.$

Take an element $\epsilon\in\mathcal{E}_{p}^{L}$ which becomes an edge in
$\Gamma$, i.e., $i(\epsilon)=p$ and there exists $t(\epsilon)\in\mathcal{V}$.
In other words, $\epsilon\in\mathcal{E}^{L}_{p}\cap E_{p}$ (Note that this
will be $E_{p}^{L}$ of $L$). By Lemma 2.6, the subset
$\nabla_{\epsilon}(\mathcal{E}_{p}^{L})$ in $\mathcal{E}_{t(\epsilon)}$
consists of exactly $(n-1)$ $1$-dimensional pairs. Moreover, because $\alpha$
satisfies the congruence relation on the edge
$\epsilon\in\mathcal{E}^{L}_{p}\cap E_{p}$,
$\alpha(\nabla_{\epsilon}(\mathcal{E}_{p}^{L}))$ and $x$ span the same
subspace $R$ as above. This property holds for all edges
$\epsilon\in\mathcal{E}^{L}_{p}\cap E_{p}$. Hence, we can define the following
$(2n-2)$-valent subgraph in $\Gamma=(\mathcal{V},\mathcal{E})$:

$\displaystyle L_{1}=(\mathcal{V}^{L_{1}},\mathcal{E}^{L_{1}})$

such that

$\displaystyle\mathcal{V}^{L_{1}}:=\\{p,\ t(\epsilon)\ |\
\epsilon\in\mathcal{E}^{L}_{p}\cap E_{p}\\};$
$\displaystyle\mathcal{E}^{L_{1}}:=\bigcup_{\epsilon\in\mathcal{E}^{L}_{p}\cap
E_{p}}\nabla_{\epsilon}(\mathcal{E}_{p}^{L})\cup\mathcal{E}_{p}^{L}.$

If we restrict $\alpha$ and $\nabla$ onto $L_{1}$, then this becomes a
$(2n-2)$-valent $T^{*}\mathbb{C}^{n-1}$-modeled GKM subgraph, say
$\mathcal{L}_{1}$, in $\mathcal{G}$. If $L_{1}$ is maximal, i.e., if there is
a $(2n-2)$-valent graph $L^{\prime}$ such that $L_{1}\subset L^{\prime}$ then
$L_{1}=L^{\prime}$, then $\mathcal{L}_{1}$ is a hyperplane. Assume that
$L_{1}$ is not maximal. In this case, for every vertex
$q\in\mathcal{V}^{L_{1}}$ and every edge
$\epsilon\in\mathcal{E}_{q}^{L_{1}}\cap E_{q}$, we can apply the similar
method stated as above. Then we can construct the $(2n-2)$-valent
$T^{*}\mathbb{C}^{n-1}$-modeled GKM subgraph $\mathcal{L}_{2}$ which contains
$\mathcal{L}_{1}$. If $\mathcal{L}_{2}$ is maximal, then this is a hyperplane
which we want to have. Otherwise, by repeating similar arguments, we get the
hyperplane $\mathcal{L}$ which contains $\mathcal{E}_{p}^{L}$.

Suppose that there are two hyperplanes $\mathcal{L}=(L,\alpha^{L},\nabla^{L})$
and
$\mathcal{L}^{\prime}=(L^{\prime},\alpha^{L^{\prime}},\nabla^{L^{\prime}})$
such that $\mathcal{E}_{p}^{L}=\mathcal{E}_{p}^{L^{\prime}}$. Because
$E_{p}^{L}=E_{p}^{L^{\prime}}$ and two connections are restricted from the
connection $\nabla$ of $\mathcal{G}$, we see that the following two subgraphs
are the same graph:

$\displaystyle\bigcup_{\epsilon\in
E_{p}^{L}}\nabla_{\epsilon}(\mathcal{E}_{p}^{L})\cup\mathcal{E}_{p}^{L}=\bigcup_{\epsilon\in
E_{p}^{L^{\prime}}}\nabla_{\epsilon}(\mathcal{E}_{p}^{L^{\prime}})\cup\mathcal{E}_{p}^{L^{\prime}}.$

By iterating this construction along all edges in $L$ and $L^{\prime}$,
finally, we know that $L=L^{\prime}$. Therefore, such a hyperplane is unique.
∎

###  3.2. Pre-halfspace and its Thom class

In this section, we introduce a pre-halfspace and its Thom class.

Take a subgraph $H=(\mathcal{V}^{H},\mathcal{E}^{H})$ of $\Gamma$ such that
$|\mathcal{E}^{H}_{p}|=2n-1$ or $2n$ for all $p\in\mathcal{V}^{H}$. We assume
that there always exists a vertex $p\in\mathcal{V}^{H}$ with
$|\mathcal{E}_{p}^{H}|=2n-1$. Moreover, we assume that $H$ is closed under the
connection $\nabla$ of $\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$, that is,

(C1):

$\nabla_{\epsilon}^{H}:=\nabla_{\epsilon}|_{\mathcal{E}_{i(\epsilon)}^{H}}:\mathcal{E}^{H}_{i(\epsilon)}\to\mathcal{E}^{H}_{t(\epsilon)}$
is bijective, if
$|\mathcal{E}^{H}_{i(\epsilon)}|=|\mathcal{E}^{H}_{t(\epsilon)}|=2n-1$ or
$2n$;

(C2):

$\nabla_{\epsilon}^{H}:\mathcal{E}^{H}_{i(\epsilon)}\to\mathcal{E}^{H}_{t(\epsilon)}$
is injective, if
$|\mathcal{E}^{H}_{i(\epsilon)}|=2n-1<|\mathcal{E}^{H}_{t(\epsilon)}|=2n$.

In addition, we also assume that if
$|\mathcal{E}^{H}_{i(\epsilon)}|(=2n-1)<|\mathcal{E}^{H}_{t(\epsilon)}|(=2n)$
then $\nabla_{\epsilon}^{H}$ satisfies the following congruence relation for
$\\{n^{H}(i(\epsilon))\\}=\mathcal{E}^{\Gamma}_{i(\epsilon)}-\mathcal{E}^{H}_{i(\epsilon)}$
(we call such an $n^{H}(p)$ a normal edge or a normal leg of $H$ at $p$):

(3.1) $\displaystyle\alpha(n^{H}(p))-x\equiv 0\ ({\rm mod}\
\alpha(\epsilon)).$

Now we may define the pre-halfspace.

######  Definition 3.4 (pre-halfspace).

Let $\mathcal{H}:=(H,\alpha^{H},\nabla^{H})$ be the triple as above, i.e.,

  * •

$H=(\mathcal{V}^{H},\mathcal{E}^{H})$ is a subgraph $\Gamma$ such that
$|\mathcal{E}^{H}_{p}|=2n-1$ or $2n$ for all $p\in\mathcal{V}^{H}$, where
there exists a vertex $p\in\mathcal{V}^{H}$ with $|\mathcal{E}^{H}_{p}|=2n-1$;

  * •

$\alpha^{H}:=\alpha|_{\mathcal{E}^{H}}:\mathcal{E}^{H}\to
H^{2}(BT^{n})\oplus\mathbb{Z}x$ is the axial function restricted onto $H$;

  * •

$\nabla^{H}:=\\{\nabla_{\epsilon}^{H}\ |\ \epsilon\in E^{H}\\}$ is the
restrected connection of $\nabla$ onto $H$ which satisfies the conditions
(C1), (C2), (3.1) as above,

then we call $\mathcal{H}$ a pre-halfspace of $\mathcal{G}=(\Gamma,\ \alpha,\
\nabla)$.

######  Example 3.5.

The following left figure (Figure 5) shows an example of pre-halfspace of the
left graph in Figure 2. On the other hand, the following right figure (Figure
5) is an abstract subgraph of the left graph in Figure 2. However, this is not
closed under the connection $\nabla$, because the congruence relation does not
hold on the diagonal edge. Therefore, this is not a pre-halfspace.

$e_{2}^{*}$$e_{1}^{*}$$x-e_{1}^{*}$$-e_{2}^{*}$$e_{2}^{*}+x$$e_{2}^{*}-e_{1}^{*}+x$$e_{1}^{*}-e_{2}^{*}$$-e_{1}^{*}$$e_{1}^{*}+x$$e_{2}^{*}-e_{1}^{*}$
Figure 4.

$e_{2}^{*}$$e_{1}^{*}$$x-e_{1}^{*}$$-e_{2}^{*}$$e_{2}^{*}-e_{1}^{*}+x$$e_{1}^{*}-e_{2}^{*}$$-e_{1}^{*}$$e_{1}^{*}+x$$e_{2}^{*}-e_{1}^{*}$
Figure 5.

Let $\mathcal{H}=(H,\ \alpha^{H},\ \nabla^{H})$ be a pre-halfspace of a
$T^{*}\mathbb{C}^{n}$-modeled GKM graph $\mathcal{G}=(\Gamma,\ \alpha,\
\nabla)$. We can define the notion of a Thom class for the pre-halfspace (also
see [MMP]).

######  Definition 3.6 (Thom class).

A Thom class of $\mathcal{H}$ is defined by the map $\tau_{H}:\mathcal{V}\to
H^{2}(BT^{n})\oplus\mathbb{Z}x$ such that

$\displaystyle\tau_{H}(p)=\left\\{\begin{array}[]{ll}0&{\rm if}\
p\not\in\mathcal{V}^{H}\\\ x&{\rm if}\ |\mathcal{E}^{H}_{p}|=2n\\\
\alpha(n^{H}(p))&{\rm if}\ |\mathcal{E}^{H}_{p}|=2n-1,\end{array}\right.$

We also call a vertex $p\in\mathcal{V}$ with $\tau_{H}(p)=0$ (resp.
$\tau_{H}(p)=x$, $\tau_{H}(p)=\alpha(n^{H}(p))$) an exterior (resp. interior,
boundary) vertex of $H$.

For the Thom class $\tau_{H}$ of a pre-halfspace $H$, we have the following
lemma.

######  Lemma 3.7.

The Thom class $\tau_{H}$ of a pre-halfspace $\mathcal{H}$ is an element of
$H^{*}(\mathcal{G})$.

###### Proof.

Take an edge $\epsilon\in E$. We claim that $\tau_{H}$ satisfies the
congruence relation on $\epsilon$, that is,
$\tau_{H}(i(\epsilon))-\tau_{H}(t(\epsilon))\equiv 0$ $({\rm mod}\
\alpha(\epsilon))$.

We first assume that $i(\epsilon),\ t(\epsilon)\not\in\mathcal{V}^{H}$ or
$|\mathcal{E}_{i(\epsilon)}^{H}|=2n=|\mathcal{E}_{t(\epsilon)}^{H}|$. Namely,
both of $i(\epsilon)$ and $t(\epsilon)$ are exterior vertices or interior
vertices of $H$. Then, by definition of the Thom class,

$\displaystyle\tau_{H}(i(\epsilon))-\tau_{H}(t(\epsilon))=0\equiv 0\ {\rm
mod}\ \alpha(\epsilon).$

So the congruence relation holds for these cases.

We next consider the other cases, i.e., the case when both of $i(\epsilon)$
and $t(\epsilon)$ are boundary vertices or the case when $i(\epsilon)$ is a
boundary vertex but $t(\epsilon)$ is an exterior or an interior vertex. Put
$p=i(\epsilon)$. Assume that $t(\epsilon)(=:q)$ is also a boundary vertex.
Because the pre-halfspace is closed by the connection $\nabla$, we have that
$\nabla_{\epsilon}(n^{H}(p))=n^{H}(q)$. By the definition of Thom classes, we
have that

$\displaystyle\tau_{H}(i(\epsilon))-\tau_{H}(t(\epsilon))$
$\displaystyle=\alpha(n^{H}(p))-\alpha(n^{H}(q))$
$\displaystyle=\alpha(n^{H}(p))-\alpha(\nabla_{\epsilon}(n^{H}(p)))\equiv 0\
{\rm mod}\ \alpha(\epsilon).$

Assume that $t(\epsilon)$ is an exterior vertex, Then $\epsilon=n^{H}(p)$ and

$\displaystyle\tau_{H}(i(\epsilon))-\tau_{H}(t(\epsilon))=\alpha(n^{H}(p))-0\equiv
0\ {\rm mod}\ \alpha(\epsilon)=\alpha(n^{H}(p)).$

Assume that $t(\epsilon)(=:q)$ is an interior vertex. Namely, $\tau_{H}(q)=x$.
In this case, we have

$\displaystyle\tau_{H}(p)-\tau_{H}(q)=\alpha(n^{H}(p))-x.$

It follows from (3.1) that

$\displaystyle\alpha(n^{H}(p))-x\equiv 0\ {\rm mod}\ \alpha(\epsilon).$

So the congruence relation also holds for these cases. Consequently, we have
$\tau_{H}\in H^{*}(\mathcal{G})$. ∎

######  Example 3.8.

The following figure (Figure 6) shows an example of the Thom class of the pre-
halfspace in Figure 5.

$x-e_{2}^{*}$$x$$e_{1}^{*}-e_{2}^{*}+x$ Figure 6.

###  3.3. Opposite side of a pre-halfspace

Next we define the opposite side of the pre-halfspace. In order to define it,
we need to prove Lemma 3.9

In order to prove it, we define a boundary of a pre-halfspace. Let $H$ be a
pre-halfspace. By the definition of a pre-halfspace, there is a vertex
$p\in\mathcal{V}^{H}$ such that $|\mathcal{E}_{p}^{H}|=2n-1$. Then
$\mathcal{E}_{p}^{H}$ has $(n-1)$ $1$-dimensional pairs, say
$\\{\epsilon_{1}^{+},\ \cdots,\ \epsilon_{n-1}^{+},\ \epsilon_{1}^{-},\
\cdots,\ \epsilon_{n-1}^{-}\\}$. Because of Lemma 3.3, there exists a unique
hyperplane $\mathcal{L}=(L,\alpha^{L},\nabla^{L})$ in $\mathcal{G}$ such that

$\displaystyle\alpha^{L}:=\alpha|_{\mathcal{E}^{L}},\quad\nabla^{L}:=\\{\nabla_{\epsilon}|_{\mathcal{E}_{i(\epsilon)}^{L}}\
|\ \epsilon\in E^{L}\\}.$

Moreover, $\mathcal{L}$ satisfies that

$\displaystyle p\in\mathcal{V}^{L}$

and

$\displaystyle\mathcal{E}_{p}^{L}=\\{\epsilon_{1}^{+},\ \cdots,\
\epsilon_{n-1}^{+},\ \epsilon_{1}^{-},\ \cdots,\ \epsilon_{n-1}^{-}\\}.$

We call the union of all such hyperplanes $\mathcal{L}$ a boundary of the pre-
halfspace $\mathcal{H}$, and we denote it by $\partial\mathcal{H}=(\partial
H,\alpha^{\partial H},\nabla^{\partial H})$. Note that a boundary of
$\mathcal{H}$ may not be connected (see Figure 8). To define the opposite side
of $\mathcal{H}$, we need the following lemma:

######  Lemma 3.9.

Let $\mathcal{H}=(H,\alpha^{H},\nabla^{H})$ be a pre-halfspace in
$\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ and $x$ be a residual basis. Then
there is a unique pre-halfspace $\mathcal{I}=(I,\alpha^{I},\nabla^{I})$ such
that

  * •

$H\cup I=\Gamma$;

  * •

$\tau_{H}+\tau_{I}=\chi\in H^{*}(\mathcal{G})$,

where $\chi$ is an element of $H^{*}(\mathcal{G})$ defined by $\chi(p)=x$ for
all $p\in\mathcal{V}$.

###### Proof.

Set $H=(\mathcal{V}^{H},\ \mathcal{E}^{H})$ and
$\hat{I}=(\mathcal{V}^{\Gamma}-\mathcal{V}^{H},\
\mathcal{E}^{\Gamma}-\mathcal{E}^{H})$. Define

$\displaystyle I=\hat{I}\cup\partial H.$

Namely,

$\displaystyle I=(\mathcal{V}^{I},\
\mathcal{E}^{I})=((\mathcal{V}^{\Gamma}-\mathcal{V}^{H})\cup\mathcal{V}^{\partial
H},\ (\mathcal{E}^{\Gamma}-\mathcal{E}^{H})\cup\mathcal{E}^{\partial H}).$

Then we can easily see that $H\cup I=\Gamma$ and $H\cap I=\partial H$.

We next prove $I$ is a pre-halfspace. Take $p\in\mathcal{V}^{I}$. If
$p\in\mathcal{V}^{\hat{I}}=\mathcal{V}^{\Gamma}-\mathcal{V}^{H}$, then
$\mathcal{E}_{p}^{H}=\emptyset$; therefore,
$\mathcal{E}_{p}^{I}=\mathcal{E}_{p}^{\Gamma}-\emptyset=\mathcal{E}_{p}^{\Gamma}$
and $|\mathcal{E}_{p}^{I}|=|\mathcal{E}_{p}^{\Gamma}|=2n$. If
$p\in\mathcal{V}^{\partial H}$, then
$\mathcal{E}_{p}^{I}=\mathcal{E}_{p}^{\partial H}\cup\\{n_{H}(p)\\}$, that is,
$|\mathcal{E}_{p}^{I}|=2n-1$. Here $n_{H}(p)$ is a normal edge (leg) of $H$ on
$p$. Therefore, for an edge $\epsilon\in E^{I}$ and the restricted connection
$\nabla_{\epsilon}^{I}:=\nabla_{\epsilon}|_{\mathcal{E}_{i(\epsilon)}^{I}}$,
it follows from the definition of $\nabla$ on $\Gamma$ that we have

  * •

$\nabla_{\epsilon}^{I}:\mathcal{E}_{i(\epsilon)}^{I}\to\mathcal{E}_{t(\epsilon)}^{I}$
is bijective, if
$|\mathcal{E}_{i(\epsilon)}^{I}|=|\mathcal{E}_{t(\epsilon)}^{I}|$,

  * •

$\nabla_{\epsilon}^{I}:\mathcal{E}_{i(\epsilon)}^{I}\to\mathcal{E}_{t(\epsilon)}^{I}$
is injective, if
$|\mathcal{E}_{i(\epsilon)}^{I}|=2n-1<|\mathcal{E}_{t(\epsilon)}^{I}|=2n$.

Take an edge $\epsilon\in E^{I}$ in $I$ such that
$|\mathcal{E}_{i(\epsilon)}^{I}|=2n-1<|\mathcal{E}_{t(\epsilon)}^{I}|=2n$. Put
$i(\epsilon)=p\in\mathcal{V}^{\partial H}$ and
$n_{H}(p)=\epsilon^{+}(=\epsilon)$. Then the normal edge (leg) of $I$ on $p$
can be taken as $\epsilon^{-}=n_{I}(p)$, where $\\{\epsilon^{+},\
\epsilon^{-}\\}=\mathcal{E}_{p}-\mathcal{E}^{\partial H}_{p}$ is a
$1$-dimensional pair in $\mathcal{E}_{p}$. So we have the following equation:

(3.3)
$\displaystyle\alpha(n_{I}(p))+\alpha(n_{H}(p))=\alpha(\epsilon^{-})+\alpha(\epsilon^{+})=(-\alpha(\epsilon^{+})+x)+\alpha(\epsilon^{+})=x.$

Therefore, we have that

$\displaystyle\alpha(n_{I}(p))-x=$ $\displaystyle-\alpha(n_{H}(p))$
$\displaystyle=$ $\displaystyle-\alpha(\epsilon)\equiv 0\ ({\rm mod}\
\alpha(n_{H}(p))=\alpha(\epsilon)).$

Consequently, we have $\mathcal{I}=(I,\alpha^{I},\nabla^{I})$ is a pre-
halfspace such that $H\cup I=\Gamma$, where
$\alpha^{I}:=\alpha|_{\mathcal{E}^{I}}$ and
$\nabla^{I}:=\\{\nabla_{\epsilon}|_{\mathcal{E}^{I}}\ |\ \epsilon\in
E^{I}\\}$. Moreover, we have that $\tau_{H}+\tau_{I}=\chi$, because of the
above equation and the definition of the Thom class of the pre-halfspace.

We finally claim the uniqueness of $I$. By two conditions $H\cup I=\Gamma$,
$\tau_{H}+\tau_{I}=\chi$ and the definition of the Thom class, we see
$\partial I=\partial H$ and $I=(\mathcal{V}^{\Gamma}-\mathcal{V}^{H},\
\mathcal{E}^{\Gamma}-\mathcal{E}^{H})\cup\partial I$. From Lemma 3.3, the
boundary $\partial H=\partial I$ is uniquely determined (though it may not be
connected). So we know the uniqueness of $I$. ∎

We call $\mathcal{I}$ in Lemma 3.9 an opposite side of $\mathcal{H}$ and
denote it by
$\overline{\mathcal{H}}=(\overline{H},\alpha^{\overline{H}},\nabla^{\overline{H}})$.
Note that

$\displaystyle H\cap\overline{H}=\partial H$

by the proof of Lemma 3.9.

###  3.4. Halfspace and Ring ${\mathbb{Z}}[\mathcal{G}]$

Under the above preparations, we may define the halfspace.

######  Definition 3.10 (halfspace).

A pre-halfspace $\mathcal{H}$ is said to be a halfspace, if $H$ is a connected
subgraph and its opposite side is also connected.

$0$$e_{2}^{*}$$e_{2}^{*}-e_{1}^{*}$$x$$x-e_{2}^{*}$$e_{1}^{*}-e_{2}^{*}+x$
Figure 7. The above figures are a halfspace and its opposite side of the left
GKM graph in Figure 2. The labels on vertices mean the values of their Thom
classes on vertices, where $0$ means the value of the Thom class $\tau_{H}$ on
the exterior vertex of a halfspace $H$. We call such a vertex a fake vertex
when we consider the Thom class $\tau_{H}$ of $H$. Note that the boundary
$\partial H=H\cap\overline{H}$ is connected.
$e_{2}^{*}+x$$e_{2}^{*}+x$$0$$0$$0$$x$$x$$x$$-e_{2}^{*}$$-e_{2}^{*}$$x$$x$$x$$0$$0$$0$
Figure 8. The above figures are a halfspace and its opposite side of the right
GKM graph in Figure 2, where $0$’s are the fake vertices. Note that the
boundary $\partial H=H\cap\overline{H}$ is not connected.
$e_{1}^{*}$$x-e_{1}^{*}$$-e_{1}^{*}$$e_{1}^{*}+x$$e_{1}^{*}$$-e_{1}^{*}$$x-e_{1}^{*}$$e_{1}^{*}+x$
Figure 9. The middle graph (edge) $\mathcal{H}$ is a pre-halfspace of the left
GKM graph. However, its opposite side is the right graph (two legs). This is
not connected; therefore, $\mathcal{H}$ is not a halfspace.

Let $\mathbf{H}$ be the set of all halfspaces in $\mathcal{G}$. Because the
graph $\Gamma$ is finite and the opposite side of the halfspace is also a
halfspace, we may write the set of all halfspaces by

$\displaystyle\mathbf{H}=\\{H_{1},\ \cdots,\ H_{m},\ \overline{H_{1}},\
\cdots,\ \overline{H_{m}}\\}.$

Put

$\displaystyle{\mathbb{Z}}[X,\ \mathbf{H}]={\mathbb{Z}}[X,\ H_{1},\ \cdots,\
H_{m},\ \overline{H_{1}},\ \cdots,\ \overline{H_{m}}]$

where ${\mathbb{Z}}[X,\ H_{1},\ \cdots,\ H_{m},\ \overline{H_{1}},\ \cdots,\
\overline{H_{m}}]$ is a polynomial ring which is generated by $X$ and all
elements in $\mathbf{H}$, and put

$\displaystyle\mathcal{I}=\Big{\langle}H_{i}+\overline{H_{i}}-X,\
\prod_{H\in\mathbf{H}^{\prime}}H\ \Big{|}\ i=1,\ \cdots,\ m,\
\mathbf{H}^{\prime}\in\mathbf{I}(\mathbf{H})\Big{\rangle}$

which is the ideal in ${\mathbb{Z}}[X,\ \mathbf{H}]$ generated by
$H_{i}+\overline{H_{i}}-X$ $(i=1,\ \cdots,\ m)$ and the product

$\displaystyle\prod_{H\in\mathbf{H}^{\prime}\in\mathbf{I}(\mathbf{H})}H,$

where

$\displaystyle\displaystyle\mathbf{I}(\mathbf{H})=\\{\mathbf{H}^{\prime}\subset\mathbf{H}\
|\ \bigcap_{H\in\mathbf{H}^{\prime}}H=\emptyset\\}.$

We define the following ring ${\mathbb{Z}}[\mathcal{G}]$:

$\displaystyle{\mathbb{Z}}[\mathcal{G}]:={\mathbb{Z}}[X,\
\mathbf{H}]\Big{/}\mathcal{I}.$

From the next section, we shall prove this ring $\mathbb{Z}[\mathcal{G}]$ is
isomorphic to the graph equivariant cohomology ring $H^{*}(\mathcal{G})$ under
some conditions.

##  4\. Ring structure of the graph equivariant cohomology of a
$T^{*}\mathbb{C}^{n}$-modeled GKM graph

The first goal of this paper is to prove the following theorem.

######  Theorem 4.1.

Let $\mathcal{G}$ be a $2n$-valent $T^{*}\mathbb{C}^{n}$-modeled GKM graph and
$\mathbf{L}=\\{L_{1},\ \cdots,\ L_{m}\\}$ be the set of all hyperplanes in
$\mathcal{G}$. Assume that $\mathcal{G}$ satisfies the following two
assumptions:

  1. (1)

For each $L\in\mathbf{L}$, there exist the unique pair of the halfspace $H$
and its opposite side $\overline{H}$ such that $H\cap\overline{H}=L$;

  2. (2)

For every subset $\mathbf{L}^{\prime}\subset\mathbf{L}$, its intersection
$\displaystyle\bigcap_{L\in\mathbf{L}^{\prime}}L$ is empty or connected.

Then the following ring isomorphism holds:

$\displaystyle H^{*}(\mathcal{G})\simeq{\mathbb{Z}}[\mathcal{G}].$

Henceforth in this section the $T^{*}\mathbb{C}^{n}$-modeled GKM graph
$\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ satisfies assumptions $(1)$, $(2)$ of
Theorem 4.1. For example, the left GKM graph in Figure 2 satisfies these
assumptions; however, the right GKM graph does not satisfy the assumption (1)
(also see Figure 8). We also note that the following example satisfies the
assumptions in Theorem 4.1.

######  Example 4.2.

The GKM graph in Figure 10 can be obtained from the cotangent bundle of a
$4$-dimensional toric manifold with five fixed points. Note that this can not
be realized as a hyperplane arrangement in $\mathbb{R}^{2}$, because there
must be $8$ intersection points (i.e., $8$ vertices) if all straight lines
(five lines) extend to infinity but there are only $5$ vertices in Figure 10.
Therefore, there is no corresponding toric hyperK${\rm\ddot{a}}$hler manifold
because of the fundamental theorem of toric hyperK${\rm\ddot{a}}$hler
manifolds in [BD].

$e_{2}^{*}$$e_{1}^{*}$$e_{2}^{*}+e_{1}^{*}$$e_{2}^{*}+e_{1}^{*}$$e_{1}^{*}$
Figure 10. The $T^{*}\mathbb{C}^{2}$-modeled GKM graph defined from the
cotangent bundle of a toric manifold. Note that we assume that the axial
functions satisfy $\alpha(e)=-\alpha(\overline{e})$ for all edges. We omit the
axial functions on legs because it is automatically determined by the
definition.

Let $\chi:\mathcal{V}\to H^{2}(BT)\oplus\mathbb{Z}x$ be the function such that
$\chi(p)=x$ for all $p\in\mathcal{V}$, and $\tau_{H}$ be the Thom class of the
halfspace $H$. In order to prove Theorem 4.1, we will prove that the following
map is an isomorphism:

$\displaystyle\Psi:{\mathbb{Z}}[\mathcal{G}]\to H^{*}(\mathcal{G})$

where this map is the induced homomorphism from $\Psi(H):=\tau_{H}$ and
$\Psi(X):=\chi$.

We first claim that the map $\Psi$ is well-defined (also see the definition of
$\mathbb{Z}[\mathcal{G}]$). By Lemma 3.9, we have

$\displaystyle\tau_{H}+\tau_{\overline{H}}=\chi.$

Let $\mathbf{H}$ be the set of all halfspaces in $\mathcal{G}$. If a subset
$\mathbf{H}^{\prime}\subset\mathbf{H}$ satisfies that
$\displaystyle\bigcap_{H\in\mathbf{H}^{\prime}}H=\emptyset$, then it follows
from the definition of the Thom class that

$\displaystyle\prod_{H\in\mathbf{H}^{\prime}}\tau_{H}=0.$

Therefore, the map $\Psi$ is a well-defined homomorphism.

From the next section, we start to prove the bijectivity of $\Psi$. The proof
will be divided into two steps:

  1. (I)

To study an equivariant graph cohomology of an $x$-forgetful graph
$\mathcal{\widetilde{G}}$ and to prove
$H^{*}(\widetilde{\mathcal{G}})\simeq{\mathbb{Z}}[\widetilde{\mathcal{G}}]$;

  2. (II)

To prove $\Psi$ is surjective and injective.

In the first step, we will use the technique of [MMP] (or [MP]) which was used
to show the ring structure of the graph equivariant cohomology of a certain
GKM graph called a torus graph. In the second step, we will use the technique
of [HP] which was applied to show the ring structure of the equivariant
cohomology of a toric hyperK${\rm\ddot{a}}$hler variety (also referred to as
hypertoric variety).

##  5\. An $x$-forgetful graph $\mathcal{\widetilde{G}}$

Let $\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$ be a
$T^{*}\mathbb{C}^{n}$-modeled graph. We assume that $\mathcal{G}$ satisfies
the conditions in Theorem 4.1. In this section, as a preparation to prove
Theorem 4.1, we introduce an $x$-forgetful graph $\mathcal{\widetilde{G}}$ and
its graph equivariant cohomology $H^{*}(\widetilde{\mathcal{G}})$, and prove
the ring structure of $H^{*}(\widetilde{\mathcal{G}})$.

###  5.1. $x$-forgetful graph $\mathcal{\widetilde{G}}$ and its graph
equivariant cohomology

For every $\mathcal{G}$, we may define an $x$-forgetful graph
$\mathcal{\widetilde{G}}=(\Gamma,\ \widetilde{\alpha},\ \nabla)$ as follows:
$\Gamma$ and $\nabla$ is the same graph and connection with $\mathcal{G}$, but
the function $\widetilde{\alpha}$ is defined as

$\displaystyle\widetilde{\alpha}=F\circ\alpha:\mathcal{E}\to H^{2}(BT^{n})$

where $F:H^{2}(BT^{n})\oplus{\mathbb{Z}}x\to
H^{2}(BT^{n})(\simeq(\mathfrak{t}^{n})^{*}_{{\mathbb{Z}}})$ is the the
$x$-forgetful map. We call $\widetilde{\alpha}$ an $x$-forgetful axial
function.

$e_{2}^{*}$$e_{1}^{*}$$-e_{2}^{*}$$-e_{1}^{*}$$-e_{2}^{*}$$e_{2}^{*}$$e_{2}^{*}-e_{1}^{*}$$e_{1}^{*}-e_{2}^{*}$$-e_{2}^{*}$$e_{1}^{*}$$e_{1}^{*}-e_{2}^{*}$$e_{2}^{*}-e_{1}^{*}$
Figure 11. An example of the $x$-forgetful graph for the left GKM graph in
Figure 2.

Moreover we define a graph equivariant cohomology of $\mathcal{\widetilde{G}}$
as follows:

$\displaystyle H^{*}(\mathcal{\widetilde{G}})=\\{f:\mathcal{V}\to
H^{*}(BT^{n})\ |\ f(i(\epsilon))-f(t(\epsilon))\equiv 0\ ({\rm mod}\
\widetilde{\alpha}(\epsilon))\\}.$

Let $L\in\mathbf{L}$ be a hyperplane in $\mathcal{G}$. Fix the halfsapce $H$
such that $\partial H=L$. Define the Thom class of $L$ by

$\displaystyle\tau_{L}=F\circ\tau_{H}:\mathcal{V}\to H^{2}(BT^{n}),$

where $F$ is the $x$-forgetful map. Note that for the opposite side
$\overline{H}$ of $H$, the following relation:

$\displaystyle F\circ\tau_{\overline{H}}=-\tau_{L};$

therefore, the Thom class of $L$ depends on the choice of a halfspace $H$ with
$\partial H=L$. So we fix $\\{H_{1},\ \cdots,\ H_{m}\\}$ in the set of all
halfspaces $\mathbf{H}=\\{H_{1},\ \cdots,\ H_{m},\ \overline{H}_{1},\ \cdots,\
\overline{H}_{m}\\}$. By the assumption (1) of Theorem 4.1, there is a one to
one corresponding between $H$ and $L=H\cap\overline{H}$. Therefore, we may put
the set of all hyperplanes by $\mathbf{L}=\\{L_{1},\ \cdots,\ L_{m}\\}$ where
$L_{i}=H_{i}\cap\overline{H}_{i}$ for all $i=1,\ \cdots,\ m$. Let
$\mathcal{V}^{L}$ be the set of all vertices on $L$. Then, we have

$\displaystyle\tau_{L}(p)=\left\\{\begin{array}[]{ll}0&p\not\in\mathcal{V}^{L}\\\
\widetilde{\alpha}(n_{H}(p))&p\in\mathcal{V}^{L}\end{array}\right.$

by the definitions of $\tau_{H}$ and the $x$-forgetful map $F$, where
$n_{H}(p)$ is a normal edge (or leg) of $H$ on $p$. Since $\tau_{H}\in
H^{*}(\mathcal{G})$ (see Lemma 3.7), it is easy to check that

$\displaystyle\tau_{L}\in H^{*}(\mathcal{\widetilde{G}}).$

###  5.2. The ring structure of $H^{*}(\mathcal{\widetilde{G}})$

Next we define the following ring:

$\displaystyle{\mathbb{Z}}[\mathcal{\widetilde{G}}]={\mathbb{Z}}[L_{1},\
\cdots,\ L_{m}]\Big{/}\Big{\langle}\prod_{L\in\mathbf{L}^{\prime}}L\ \Big{|}\
\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\Big{\rangle},$

where
$\displaystyle\mathbf{I}(\mathbf{L})=\\{\mathbf{L}^{\prime}\subset\mathbf{L}\
|\ \bigcap_{L\in\mathbf{L}^{\prime}}L=\emptyset\\}$ and
$\langle\prod_{L\in\mathbf{L}^{\prime}}L\ |\
\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\rangle$ is an ideal which is
generated by the product $\prod_{L\in\mathbf{L}^{\prime}}L$ for all
$\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})$.

The goal of this section (the first step (I) of the proof of Theorem 4.1) is
to prove the following theorem:

######  Theorem 5.1.

Let $\mathcal{G}$ be a $2n$-valent $T^{*}\mathbb{C}^{n}$-modeled GKM graph and
$\mathbf{L}=\\{L_{1},\ \cdots,\ L_{m}\\}$ be the set of all hyperplanes in
$\mathcal{G}$. Assume that $\mathcal{G}$ satisfies the two assumptions in
Theorem 4.1. If $\widetilde{\mathcal{G}}$ is the $x$-forgetful graph, then the
following ring isomorphism holds:

$\displaystyle
H^{*}(\mathcal{\widetilde{G}})\simeq{\mathbb{Z}}[\mathcal{\widetilde{G}}].$

Define the induced homomorphism

$\displaystyle\Psi^{\prime}:{\mathbb{Z}}[\mathcal{\widetilde{G}}]\to
H^{*}(\mathcal{\widetilde{G}})$

by $\Psi^{\prime}(L)=\tau_{L}$. Obviously, $\Psi^{\prime}$ is a well-defined
homomorphism. In order to show Theorem 5.1, it is enough to prove that this
homomorphism is bijective.

###  5.3. The localization map and the injectivity of $\Psi^{\prime}$

We first prove the injectivity of $\Psi^{\prime}$. In order to prove it, we
introduce the map $\rho$ which is the analogue of the localization of the
equivariant cohomology of a $T$-manifold to its fixed points.

Let us define the following ring:

$\displaystyle{\mathbb{Z}}[\mathcal{\widetilde{G}}]_{p}={\mathbb{Z}}[L_{1},\
\cdots,\ L_{m}]/\langle L\ |\ p\not\in\mathcal{V}^{L}\rangle,$

where $\langle L\ |\ p\not\in\mathcal{V}^{L}\rangle$ is an ideal which is
generated by $L$ such that $p\not\in\mathcal{V}^{L}$. As a beginning, we prove
the following lemma.

######  Lemma 5.2.

For the $x$-forgetful graph $\mathcal{\widetilde{G}}=(\Gamma,\
\widetilde{\alpha},\ \nabla)$, we have

$\displaystyle
I_{p}:{\mathbb{Z}}[\mathcal{\widetilde{G}}]_{p}\simeq{\mathbb{Z}}[L\ |\
p\in\mathcal{V}^{L}]={\mathbb{Z}}[L_{p,1},\ \cdots,\
L_{p,n}]\stackrel{{\scriptstyle\iota_{p}}}{{\simeq}}H^{*}(BT^{n}),$

where the last isomorphism $\iota_{p}$ is defined by
$\iota_{p}:L\mapsto\tau_{L}(p)$.

###### Proof.

By the definition of ${\mathbb{Z}}[\mathcal{\widetilde{G}}]_{p}$, the first
equivalence ${\mathbb{Z}}[\mathcal{\widetilde{G}}]_{p}\simeq{\mathbb{Z}}[L\ |\
p\in\mathcal{V}^{L}]$ is obvious. We claim ${\mathbb{Z}}[L\ |\
p\in\mathcal{V}^{L}]={\mathbb{Z}}[L_{p,1},\ \cdots,\
L_{p,n}]\stackrel{{\scriptstyle\iota_{p}}}{{\simeq}}H_{T^{n}}^{*}(pt)$.

Because $\Gamma$ is a $2n$-valent graph, we may put

$\displaystyle\mathcal{E}_{p}=\\{\epsilon_{1}^{+}(p),\ \cdots,\
\epsilon_{n}^{+}(p),\ \epsilon_{1}^{-}(p),\ \cdots,\ \epsilon_{n}^{-}(p)\\}$

for all $p\in\mathcal{V}$. There is a unique $L_{i}$ such that

$\displaystyle\tau_{L_{i}}(p)=\widetilde{\alpha}(\epsilon_{i}^{+}(p))=-\widetilde{\alpha}(\epsilon_{i}^{-}(p))$

for all $i=1,\ \cdots,\ n$ by Lemma 3.3. Hence, we have

$\displaystyle{\mathbb{Z}}[L\ |\ p\in\mathcal{V}^{L}]={\mathbb{Z}}[L_{p,1},\
\cdots,\ L_{p,n}].$

Next, by the definition of the axial function of a
$T^{*}\mathbb{C}^{n}$-modeled GKM graph,

$\displaystyle\mathbb{Z}\alpha(\epsilon_{1}^{+}(p))\oplus\cdots\oplus\mathbb{Z}\alpha(\epsilon_{n}^{+}(p))\oplus\mathbb{Z}x\simeq
H^{2}(BT^{n})\oplus{\mathbb{Z}}x.$

Hence, because $\widetilde{\alpha}:=F\circ\alpha$ is defined by the
$x$-forgetful map $F:H^{2}(BT^{n})\oplus{\mathbb{Z}}x\to H^{2}(BT^{n})$, we
have that

$\displaystyle{\mathbb{Z}}[\widetilde{\alpha}(\epsilon_{1}^{+}(p)),\ \cdots,\
\widetilde{\alpha}(\epsilon_{n}^{+}(p))]\simeq H^{*}(BT^{n}).$

Therefore, $\iota_{p}$ is an isomorphism. ∎

Next we shall define a localization map
$\rho:{\mathbb{Z}}[\mathcal{\widetilde{G}}]\to\bigoplus_{p\in\mathcal{V}}{\mathbb{Z}}[\mathcal{\widetilde{G}}]_{p}$
and prove that it is injective in Lemma 5.3. Since the set
$\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})$ satisfies that
$\cap_{L\in\mathbf{L}^{\prime}}L=\emptyset$, for every $p\in\mathcal{V}$ there
is an $L\in\mathbf{L}^{\prime}$ such that $p\not\in\mathcal{V}^{L}$.
Therefore, there exists the following relation for two ideals in
$\mathbb{Z}[L_{1},\ldots,L_{m}]$:

$\displaystyle\langle L\ |\
p\not\in\mathcal{V}^{L}\rangle\supset\langle\prod_{L\in\mathbf{L}^{\prime}}L\
|\ \mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\rangle.$

Hence, the following natural homomorphism is well-defined:

$\displaystyle\rho_{p}:{\mathbb{Z}}[\mathcal{\widetilde{G}}]:={\mathbb{Z}}[L_{1},\
\cdots,\ L_{m}]\Big{/}\Big{\langle}\prod_{L\in\mathbf{L}^{\prime}}L\ \Big{|}\
\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\Big{\rangle}\longrightarrow{\mathbb{Z}}[\mathcal{\widetilde{G}}]_{p}:={\mathbb{Z}}[L_{1},\
\cdots,\ L_{m}]/\langle L\ |\ p\not\in\mathcal{V}^{L}\rangle.$

For this projection $\rho_{p}$, we can easily show that its kernel is as
follows:

$\displaystyle{\rm Ker}\ \rho_{p}=\langle L\ |\
p\not\in\mathcal{V}^{L}\rangle/\langle\prod_{L\in\mathbf{L}^{\prime}}L\ |\
\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\rangle.$

Now we may define the homomorphism $\rho$ as follows:

$\displaystyle\rho=\bigoplus_{p\in\mathcal{V}}\rho_{p}:{\mathbb{Z}}[\mathcal{\widetilde{G}}]\longrightarrow\bigoplus_{p\in\mathcal{V}}{\mathbb{Z}}[\mathcal{\widetilde{G}}]_{p},$

such that

$\displaystyle\rho(Y)=\bigoplus_{p\in\mathcal{V}}\rho_{p}(Y)$

for $Y\in{\mathbb{Z}}[\mathcal{\widetilde{G}}]$. We call $\rho$ a localization
map. The following lemma holds.

######  Lemma 5.3.

$\rho$ is injective.

###### Proof.

Obviously we have

$\displaystyle{\rm Ker}\ \rho=\bigcap_{p\in\mathcal{V}}{\rm Ker}\
\rho_{p}=\left(\bigcap_{p\in\mathcal{V}}\ \langle L\ |\
p\not\in\mathcal{V}^{L}\rangle\right)\Bigg{/}\left\langle\prod_{L\in\mathbf{L}^{\prime}}L\
\Bigg{|}\ \mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\right\rangle.$

Hence, to prove $\rho$ is injective, it is enough to show that ${\rm Ker}\
\rho=\\{0\\}$, i.e., we shall prove the following relation:

(5.2) $\displaystyle\bigcap_{p\in\mathcal{V}}\ \langle L\ |\
p\not\in\mathcal{V}^{L}\rangle\subset\left\langle\prod_{L\in\mathbf{L}^{\prime}}L\
|\
\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\right\rangle(\subset{\mathbb{Z}}[L_{1},\
\cdots,\ L_{m}]).$

Take a non-zero polynomial

$\displaystyle A$ $\displaystyle=$
$\displaystyle\sum_{a_{1},\cdots,a_{m}\in{\mathbb{N}}\cup\\{0\\}}k(a_{1},\
\cdots,\ a_{m})L_{1}^{a_{1}}\cdots L_{m}^{a_{m}}$ $\displaystyle\in$
$\displaystyle\bigcap_{p\in\mathcal{V}}\ \langle L\ |\
p\not\in\mathcal{V}^{L}\rangle\subset{\mathbb{Z}}[L_{1},\ \cdots,\ L_{m}],$

where we only consider the case when $k(a_{1},\ \cdots,\
a_{m})\in{\mathbb{Z}}-\\{0\\}$. Because $A$ is an element of the monomial
ideal $\langle L\ |\ p\not\in\mathcal{V}^{L}\rangle$ for all
$p\in\mathcal{V}$, we have that for each term

$\displaystyle k(a_{1},\ \cdots,\ a_{m})L_{1}^{a_{1}}\cdots
L_{m}^{a_{m}}\in\langle L\ |\ p\not\in\mathcal{V}^{L}\rangle.$

This shows that for each term $k(a_{1},\ \cdots,\ a_{m})L_{1}^{a_{1}}\cdots
L_{m}^{a_{m}}$ of a non-zero element $A$ there exists $r(=r(p))\in\\{1,\
\cdots,\ m\\}$ such that $p\not\in\mathcal{V}^{L_{r}}$ and $a_{r}\not=0$.
Because this satisfies for all $p\in\mathcal{V}$, we have that each term can
be written by

$\displaystyle k(a_{1},\ \cdots,\ a_{m})L_{1}^{a_{1}}\cdots
L_{m}^{a_{m}}=B\prod_{p\in\mathcal{V}}L_{r(p)}^{a_{r(p)}},$

where $B$ is some monomial in ${\mathbb{Z}}[L_{1},\ \cdots,\ L_{m}]$ and
$a_{r(p)}\not=0$. Since $p\not\in\mathcal{V}^{L_{r(p)}}$, we have that

$\displaystyle\bigcap_{p\in\mathcal{V}}L_{r(p)}=\emptyset.$

This shows that for each term of $A$

$\displaystyle k(a_{1},\ \cdots,\ a_{m})L_{1}^{a_{1}}\cdots
L_{m}^{a_{m}}=B\prod_{p\in\mathcal{V}}L_{r(p)}^{a_{r(p)}}\in\langle\prod_{L\in\mathbf{L}^{\prime}}L\
|\ \mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\rangle.$

Therefore, $A\in\langle\prod_{L\in\mathbf{L}^{\prime}}L\ |\
\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\rangle$. This establishes the
relation (5.2). ∎

By using Lemma 5.2 and 5.3, we can prove the following lemma for the
homomorphism $\Psi^{\prime}:\mathbb{Z}[\widetilde{\mathcal{G}}]\to
H^{*}(\widetilde{\mathcal{G}})$ which is defined from
$\Psi^{\prime}(L):=\tau_{L}$.

######  Lemma 5.4.

$\Psi^{\prime}$ is injective.

###### Proof.

We first define

$\displaystyle\rho^{\prime}:H^{*}(\mathcal{\widetilde{G}})\to\bigoplus_{p\in\mathcal{V}}H^{*}(BT^{n})$

by the homomorphism

$\displaystyle\rho^{\prime}(f)=\bigoplus_{p\in\mathcal{V}}f(p).$

Then it is easy to check that the following diagram is commutative:

where $I_{p}:\mathbb{Z}[\widetilde{\mathcal{G}}]_{p}\to H^{*}(BT^{n})$ is the
isomorphism defined by $I_{p}(L):=\tau_{L}(p)$ in Lemma 5.2. Because of Lemma
5.3, $\rho$ is injective. Therefore, the composition map
$\oplus_{p}I_{p}\circ\rho$ is injective. Because of the commutativity of the
diagram, $\rho^{\prime}\circ\Psi^{\prime}=\oplus_{p}I_{p}\circ\rho$ is also
injective. Consequently, $\Psi^{\prime}$ is injective. ∎

###  5.4. The surjectivity of $\Psi^{\prime}$

We next prove the surjectivity of $\Psi^{\prime}$. In order to prove it, we
will define an ideal $I(K)$ of $H^{*}(BT^{n})$, where $K$ is the non-empty
intersection of some hyperplanes, say $K=L_{1}\cap\cdots\cap
L_{k}(\not=\emptyset)$. Note that the graph $K$ is connected because of the
assumption (2) of Theorem 4.1. Because $L_{1},\ldots,L_{k}$ defines
hyperplanes $\mathcal{L}_{1},\ldots,\mathcal{L}_{k}$ (respectively) of
$\mathcal{G}=(\Gamma,\ \alpha,\ \nabla)$, the subgraph $K$ is also defines a
$(2n-2k)$-valent ($T^{*}\mathbb{C}^{n-k}$-modeled) GKM subgraph of
$\mathcal{G}$, say $\mathcal{K}:=(K,\ \alpha^{K},\nabla^{K})$. Now we may
define its $x$-forgetful graph, i.e., for
$\widetilde{\alpha}^{K}:=F\circ\alpha^{K}$, the labeled graph

$\displaystyle\widetilde{\mathcal{K}}:=(K,\
\widetilde{\alpha}^{K},\nabla^{K}).$

We define an ideal $I(K)$ (in $H^{*}(BT^{n})$) on $K$ as follows:

$\displaystyle
I(K)=\langle\widetilde{\alpha}^{K}(\epsilon)(=\widetilde{\alpha}(\epsilon))\
|\ \epsilon\in\mathcal{E}^{K}\rangle,$

that is, this ideal is generated by all $x$-forgetful axial functions of edges
and legs in $K$. The following lemma, which will be used to prove the
surjectivity of $\Psi^{\prime}$, holds for $I(K)$.

######  Lemma 5.5.

Let $f$ be an element in $H^{*}(\mathcal{\widetilde{G}})$. If $f(p)\not\in
I(K)$ for some $p\in\mathcal{V}^{K}$, then $f(q)\not\in I(K)$ for all
$q\in\mathcal{V}^{K}$.

###### Proof.

Let $K:=(\mathcal{V}^{K},\mathcal{E}^{K})$. For $f\in
H^{*}(\widetilde{\mathcal{G}})$, we assume that $f(p)\not\in I(K)$ for some
$p\in\mathcal{V}^{K}$. We also assume that there exists a vertex
$q\in\mathcal{V}^{K}$ such that $f(q)\in I(K)$. Since $K$ is connected, there
is a path in $K$ from $q$ to $p$, which consists of edges

$\displaystyle qr_{1},\ r_{1}r_{2},\ \cdots,\ r_{s-1}r_{s},\ r_{s}p\in
E^{K}\subset\mathcal{E}^{K}.$

Because of the congruence relations in $H^{*}(\widetilde{\mathcal{G}})$, there
are $A_{1},\ldots,A_{s+1}\in H^{*}(BT^{n})$ such that

$\displaystyle f(q)-f(p)$ $\displaystyle=$
$\displaystyle(f(q)-f(r_{1}))+(f(r_{1})-f(r_{2}))+\cdots+(f(r_{s-1})-f(r_{s}))+(f(r_{s})-f(p))$
$\displaystyle=$ $\displaystyle
A_{1}\widetilde{\alpha}(qr_{1})+A_{2}\widetilde{\alpha}(r_{1}r_{2})\cdots+A_{s}\widetilde{\alpha}(r_{s-1}r_{s})+A_{s+1}\widetilde{\alpha}(r_{s}p).$

Therefore, by the definition of $I(K)$, we have

$\displaystyle f(q)-f(p)\in I(K).$

However, since $f(q),\ A_{1}\widetilde{\alpha}(qr_{1}),\cdots,\
A_{s+1}\widetilde{\alpha}(r_{s}p)\in I(K)$, we have $f(p)\in I(K)$. This gives
a contradiction. This established that if $f(p)\not\in I(K)$ then $f(q)\not\in
I(K)$ for all $q\in\mathcal{V}^{K}$. ∎

By using this lemma, we can prove the surjectivity of
$\Psi^{\prime}:\mathbb{Z}[\widetilde{\mathcal{G}}]\to
H^{*}(\widetilde{\mathcal{G}})$.

######  Lemma 5.6.

$\Psi^{\prime}$ is surjective.

###### Proof.

Let $f\in H^{*}(\widetilde{\mathcal{G}})$. For some $p\in\mathcal{V}$, we
assume that $f(p)\in H^{*}(BT^{n})$ has a non-zero constant term
$k\in{\mathbb{Z}}-\\{0\\}$, i.e.,

$\displaystyle f(p)=k+g(p)$

where $g(p)\in H^{>0}(BT^{n})\cup\\{0\\}$. Note that
$H^{*}(BT^{n})\simeq\mathbb{Z}[x_{1},\ldots,x_{n}]$, where $\deg x_{i}=2$ for
all $i=1,\ldots n$. Because $f\in H^{*}(\mathcal{\widetilde{G}})$ satisfies
the congruence relation, there exists $g\in
H^{>0}(\mathcal{\widetilde{G}})\cup\\{0\\}$ such that for all
$q\in\mathcal{V}$ we may write

$\displaystyle f(q)=k+g(q),$

where $H^{>0}(\mathcal{\widetilde{G}})\cup\\{0\\}$ is the set of $g\in
H^{*}(\mathcal{\widetilde{G}})$ whose constant term is $0$, i.e., for all
$p\in\mathcal{V}$ the constant term of the polynomial
$g(p)\in\mathbb{Z}[x_{1},\ldots,x_{n}]$ is $0$. This shows that for all $f\in
H^{*}(\widetilde{\mathcal{G}})$ there exists the constant term $k$ and $g\in
H^{>0}(\mathcal{\widetilde{G}})\cup\\{0\\}$ such that

$\displaystyle f=k+g.$

Therefore, we can take
$k\in{\mathbb{Z}}\subset{\mathbb{Z}}[\mathcal{\widetilde{G}}]$ such that

$\displaystyle f=\Psi^{\prime}(k)+g.$

Take $g=f-\Psi^{\prime}(k)$. Then $g(p)\in H^{>0}(BT^{n})\cup\\{0\\}$ for all
$p\in\mathcal{V}$. Now we may put

$\displaystyle Z(g)=\\{p\in\mathcal{V}\ |\ g(p)=0\\}.$

We first assume that $Z(g)=\emptyset$. Then $g(p)\not=0$ for all
$p\in\mathcal{V}$. Note that by Lemma 5.2 we have

$\displaystyle g(p)(\not=0)\in H^{*}(BT^{n})={\mathbb{Z}}[\tau_{L_{p,1}}(p),\
\cdots,\ \tau_{L_{p,n}}(p)],$

where $L_{p,i}$, $i=1,\ldots,n$, are the hyperplanes such that
$p\in\mathcal{V}^{L_{p,i}}$. This also shows that for the fixed vertex
$p\in\mathcal{V}$, we may take an element

$\displaystyle A\in{\mathbb{Z}}[\mathcal{\widetilde{G}}]$

such that

$\displaystyle\Psi^{\prime}(A)(p)=g(p).$

Because $g-\Psi^{\prime}(A)\in H^{*}(\widetilde{\mathcal{G}})$ and
$g(p)-\Psi^{\prime}(A)(p)=0$, we have that

$\displaystyle p\in Z(g-\Psi^{\prime}(A)).$

Next, by taking $h=g-\Psi^{\prime}(A)=f-\Psi^{\prime}(k+A)$, we may assume
that $Z(h)\not=\emptyset$. Take $p\in\mathcal{V}\backslash Z(h)$, i.e.,
$h(p)\not=0$. Let $a\tau_{L_{1}}^{a_{p,1}}\cdots\tau_{L_{n}}^{a_{p,n}}(p)$ be
a monomial appearing in $h(p)$, where $a$ is a non-zero integer,
$p\in\mathcal{V}^{L_{p,i}}$ and $a_{i}\geq 0$ ($i=1,\ \cdots,\ n$). Since
$h(p)\in H^{>0}(BT^{n})$, we may assume that

$\displaystyle a_{1},\ \cdots,\ a_{b}\not=0,\quad a_{b+1}=\cdots=a_{n}=0.$

Put $K=\cap_{i=1}^{b}L_{p,i}$. Then we have

$\displaystyle h(p)\not\in I(K)=\langle\widetilde{\alpha}^{K}(\epsilon)\ |\
\epsilon\in\mathcal{E}^{K}\rangle\subset H^{*}(BT^{n})$

because $h(p)$ contains the non-zero monomial
$a\tau_{L_{p,1}}^{a_{1}}\cdots\tau_{L_{p,b}}^{a_{b}}(p)$ such that
$\tau_{L_{p,i}}(p)$ ($i=1,\ \cdots,\ b$) is defined by the axial function of
the normal edge or leg of $K$ on $p$ (which are not the edges or legs in
$\mathcal{E}^{K}$). Therefore, by Lemma 5.5, we have that for all
$q\in\mathcal{V}^{K}$,

$\displaystyle h(q)\not\in I(K).$

In particular, $h(q)\not=0$ for all $q\in\mathcal{V}^{K}$. Let
$r\not\in\mathcal{V}^{K}$. Because $K=L_{p,1}\cap\cdots\cap L_{p,b}$, we see
that

$\displaystyle a\tau_{L_{p,1}}^{a_{1}}\cdots\tau_{L_{p,b}}^{a_{b}}(r)=0.$

Therefore, if we put

$\displaystyle
h^{\prime}=h-a\tau_{L_{p,1}}^{a_{1}}\cdots\tau_{L_{p,b}}^{a_{b}}=h-\Psi^{\prime}(aL_{p,1}^{a_{1}}\cdots
L_{p,b}^{a_{b}})=f-\Psi^{\prime}(k+A+aL_{p,1}^{a_{1}}\cdots L_{p,b}^{a_{b}}),$

then $h^{\prime}(r)=h(r)$ for all $r\not\in\mathcal{V}^{K}$. Namely,
$h(q)\not=0$ for all $q\in\mathcal{V}^{K}$ and $h^{\prime}(r)=h(r)$ for all
$r\not\in\mathcal{V}^{K}$. This shows that

$\displaystyle Z(h^{\prime})\supset Z(h).$

Note that by the definition of $h^{\prime}$, the number of monomials in
$h^{\prime}(p)$ is strictly smaller than that in $h(p)$. If $h^{\prime}(p)=0$,
then we have $Z(h^{\prime})\supsetneq Z(h).$ If $h^{\prime}(p)\not=0$, then we
may apply the same argument as above for $h^{\prime}\in
H^{*}(\widetilde{\mathcal{G}})$ and the vertex $p\in\mathcal{V}$ again because
$Z(h^{\prime})\not=\emptyset$. Then we have that there exists hyperplanes
$L_{p,i_{1}},\ldots,L_{p,i_{c}}$ in $\\{L_{p,1},\ldots,L_{p,n}\\}$ and a non-
zero integer $a^{\prime}$ such that

$\displaystyle
h^{\prime\prime}=h^{\prime}-\Psi^{\prime}(a^{\prime}L_{p,i_{1}}^{a^{\prime}_{1}}\cdots
L_{p,i_{c}}^{a^{\prime}_{c}})$

which satisfies that

$\displaystyle Z(h^{\prime\prime})\supset Z(h^{\prime})$

and the number of monomials in $h^{\prime\prime}(p)$ is strictly smaller than
that in $h^{\prime}(p)$, where $a^{\prime}_{1},\ldots,a^{\prime}_{c}$ are
positive integers. If $h^{\prime\prime}(p)\not=0$, then we repeat the same
argument again. Because the number of monomials in $h(p)$ is strictly smaller
than smaller in each step, finally we have an element

$\displaystyle B\in{\mathbb{Z}}[\mathcal{\widetilde{G}}]$

such that

$\displaystyle Z(h-\Psi^{\prime}(B))\supsetneq Z(h).$

Moreover repeating this procedure, we can find an element
$C\in{\mathbb{Z}}[\mathcal{\widetilde{G}}]$ such that
$Z(h-\Psi^{\prime}(C))=\mathcal{V}$. This shows that

$\displaystyle h-\Psi^{\prime}(C)=f-\Psi^{\prime}(k+A+C)=0.$

Therefore, for all $f\in H^{*}(\widetilde{\mathcal{G}})$ there exists an
element $k+A+C\in\mathbb{Z}[\widetilde{\mathcal{G}}]$ such that
$f=\Psi^{\prime}(k+A+C)$. This establishes that $\Psi^{\prime}$ is surjective.
∎

Consequently $\Psi^{\prime}$ is an isomorphic map by Lemma 5.4 and 5.6, and we
have

$\displaystyle
H^{*}(\mathcal{\widetilde{G}})\simeq{\mathbb{Z}}[\mathcal{\widetilde{G}}].$

This establishes Theorem 5.1.

######  Remark 5.7.

From the above argument, we know that the assumption (2) of Theorem 4.1 is not
needed to prove the “injectivity” of $\Psi^{\prime}$; however, it is needed to
prove the “surjectivity” of $\Psi^{\prime}$. Hence, the assumption (2) of
Theorem 4.1 means that $H^{*}(\mathcal{\widetilde{G}})$ (resp.
$H^{*}(\mathcal{G})$) is generated by elements of
$H^{2}(\mathcal{\widetilde{G}})$ (resp. $H^{2}(\mathcal{G})$), that is,
$\tau_{L}\in H^{2}(\mathcal{\widetilde{G}})$ (resp. $\tau_{H},\ \chi\in
H^{2}(\mathcal{G})$). For example, the Figure 12 shows the
$T^{*}\mathbb{C}^{2}$-modeled GKM graph which does not satisfy the assumption
(2) of Theorem 4.1 and its $x$-forgetful graph. In this case, we need a
generator which is not in $H^{2}(\widetilde{\mathcal{G}})$.

$e_{2}^{*}$$e_{1}^{*}$$x-e_{1}^{*}$$x-e_{2}^{*}$$e_{2}^{*}$$e_{1}^{*}$$x-e_{1}^{*}$$x-e_{2}^{*}$$e_{2}^{*}$$e_{1}^{*}$$-e_{1}^{*}$$-e_{2}^{*}$$e_{2}^{*}$$e_{1}^{*}$$-e_{1}^{*}$$-e_{2}^{*}$

Figure 12. An example of the $T^{*}\mathbb{C}^{2}$-modeled graph and its
$x$-forgetful graph which does not satisfy the assumption (2) in Theorem 4.1.
Geometrically, this graph can be defined from $T^{*}S^{4}$ with the
$T^{2}\times S^{1}$-action

.

##  6\. Proof of Theorem 4.1

In this section, we prove Theorem 4.1. We first recall the statement of
Theorem 4.1. Let $\mathcal{G}$ be a $2n$-valent $T^{*}\mathbb{C}^{n}$-modeled
GKM graph and $\mathbf{L}=\\{L_{1},\ \cdots,\ L_{m}\\}$ be the set of all
hyperplanes in $\mathcal{G}$. Assume the following two assumptions for
$\mathcal{G}$:

  1. (1)

For each $L\in\mathbf{L}$, there exist the unique pair of the halfspace $H$
and its opposite side $\overline{H}$ such that $H\cap\overline{H}=L$;

  2. (2)

For every subset $\mathbf{L}^{\prime}\subset\mathbf{L}$, its intersection
$\displaystyle\bigcap_{L\in\mathbf{L}^{\prime}}L$ is empty or connected.

Then, we will prove the following isomorphism:

$\displaystyle{\mathbb{Z}}[\mathcal{G}]\simeq H^{*}(\mathcal{G}).$

Recall the ring homomorphism in Section 4

$\displaystyle\Psi:{\mathbb{Z}}[\mathcal{G}]\to H^{*}(\mathcal{G})$

is defined by

(6.1) $\displaystyle\Psi(X)=\chi,\quad\Psi(H)=\tau_{H}.$

To prove Theorem 4.1, we claim that $\Psi$ is an isomorphism.

By the assumption (1) of Theorem 4.1, we can put the set of all halfspaces in
$\mathcal{G}$ by

$\displaystyle\mathbf{H}=\\{H_{1},\ \cdots,\ H_{m},\ \overline{H_{1}},\
\cdots,\ \overline{H_{m}}\\},$

where $L_{i}=H_{i}\cap\overline{H_{i}}$. We prepare the following diagram:

(6.8)

where the maps in the diagram is defined as follows:

  * •

$\hat{\pi}$ is the natural projection;

  * •

$\Psi$ is defined by (6.1) as before;

  * •

$\phi^{\prime}$ is the surjective homomorphism induced from

$\displaystyle\phi^{\prime}(X)=X,\quad\phi^{\prime}(H_{i})=H_{i},\quad\phi^{\prime}(\overline{H_{i}})=X-H_{i},\
i=1,\ldots,m;$

  * •

$\pi$ is the homomorphism induced from

$\displaystyle\pi(X)=\chi,\quad\pi(H_{i})=\tau_{H_{i}},\ i=1,\ldots,m;$

  * •

$\widetilde{F}$ is the homomorphism defined by

$\displaystyle\widetilde{F}(f)(p):=F\circ f(p)$

for $f\in H^{*}(\mathcal{G})$ and $p\in\mathcal{V}$, where
$F:H^{*}(BT^{n}\times BS^{1})\to H^{*}(BT^{n})$ is the $x$-forgetful map for
the fixed generator $x$ of $H^{2}(BS^{1})\simeq{\mathbb{Z}}x$;

  * •

$\phi$ is the surjective homomorphism induced from

$\displaystyle\phi(X)=0,\quad\phi(H_{i})=L_{i},\ i=1,\ldots,m;$

  * •

$\pi^{\prime}$ is the homomorphism induced from

$\displaystyle\pi^{\prime}(L_{i})=\tau_{L_{i}},\ i=1,\ldots,m.$

It easily follows from the definitions of homomorphisms as above and Lemma 3.9
that the top diagram is commutative. By Section 5.1, we may choose
$H_{1},\ldots,H_{m}$ as $\tau_{L_{i}}=F\circ\tau_{H_{i}}$ for $i=1,\ldots,m$.
Therefore, we may assume that the bottom diagram is also commutative.
Therefore, this diagram is commutative.

By the proof of Theorem 5.1, i.e., $\mathbb{Z}[\widetilde{\mathcal{G}}]\simeq
H^{*}(\widetilde{\mathcal{G}})$, we have that $\pi^{\prime}$ is surjective.
This shows that $\pi^{\prime}\circ\phi=\widetilde{F}\circ\pi$ is surjective;
therefore, $\widetilde{F}$ is also surjective.

###  6.1. Surjectivity of $\Psi$

We first prove the surjectivity of $\Psi$. By the commutativity of the top
diagram, it is enough to prove that the homomorphism

$\displaystyle\pi:\mathbb{Z}[X,H_{1},\ldots,H_{m}]\to H^{*}(\mathcal{G})$

is surjective (see Lemma 6.4). To do that, we will prove the following three
lemmas.

The following first lemma is about the kernel of
$\widetilde{F}:H^{*}(\mathcal{G})\to H^{*}(\widetilde{\mathcal{G}})$.

######  Lemma 6.1.

Let $\chi$ be the element in $H^{*}(\mathcal{G})$ such that $\chi(p)=x$ for
all $p\in\mathcal{V}$, where $x$ is the residual basis. Then we have ${\rm
Ker}\ \widetilde{F}=\langle\chi\rangle$, i.e., the ideal generated by $\chi$.

###### Proof.

Let $f\in{\rm Ker}\ \widetilde{F}$. By the definition of $\widetilde{F}$, we
have $\widetilde{F}(f)(p)=F\circ f(p)=0$ for all $p\in\mathcal{V}$. Since
$F:H^{*}(BT^{n}\times BS^{1})={\mathbb{Z}}[\alpha_{1},\ \cdots,\ \alpha_{n},\
x]\to{\mathbb{Z}}[\beta_{1},\ \cdots,\ \beta_{n}]=H^{*}(BT^{n})$ is defined by
$F(x)=0$ and $F(\alpha_{i})=\beta_{i}$ for all $i=1,\ \cdots,\ n$, we have

$\displaystyle f(p)\in{\rm Ker}\ F=\langle x\rangle\subset H^{*}(BT^{n}\times
BS^{1}).$

Therefore, for every $p\in\mathcal{V}$, there exists a polynomial $g(p)\in
H^{*}(BT^{n}\times BS^{1})$ such that

$\displaystyle f(p)=g(p)x.$

Because $f\in H^{*}(\mathcal{G})$, it satisfies the congruence relation

$\displaystyle
f(i(\epsilon))-f(t(\epsilon))=g(i(\epsilon))x-g(t(\epsilon))x=(g(i(\epsilon))-g(t(\epsilon)))x\equiv
0\ ({\rm mod}\ \alpha(\epsilon))$

for every edge $\epsilon$. Because $x$ is the residual basis, by definition of
$T^{*}\mathbb{C}^{n}$-modeled GKM graph (see Definition 2.4), we see that
$\alpha(\epsilon)\not=x$ for every edge $\epsilon\in E$. Hence, because the
polynomial ring is an integral domain, we have

$\displaystyle g(i(\epsilon))-g(t(\epsilon))\equiv 0\ ({\rm mod}\
\alpha(\epsilon))$

for every edge $\epsilon$. This implies that $g\in H^{*}(\mathcal{G})$.
Therefore for all $f\in{\rm Ker}\ \widetilde{F}$, there exists an element
$g\in H^{*}(\mathcal{G})$ such that $f=g\chi$. Hence, ${\rm Ker}\
\widetilde{F}\subset\langle\chi\rangle$. On the other hand, we can easily
check that ${\rm Ker}\ \widetilde{F}\supset\langle\chi\rangle$. This
establishes that ${\rm Ker}\ \widetilde{F}=\langle\chi\rangle$. ∎

The following second lemma is about the degree-wise decomposition of an
element in $H^{*}(\mathcal{G})$.

######  Lemma 6.2.

For every $f\in H^{*}(\mathcal{G})$, there exists a non-negative integer $l$
and an element $f_{2i}\in H^{2i}(\mathcal{G})$ for each $0\leq i\leq l$ which
satisfy

$\displaystyle f=f_{0}+f_{2}+\cdots+f_{2l},$

where $H^{2i}(\mathcal{G})$ consists of the element, say $h_{2i}$, which
satisfies $h_{2i}(p)\in H^{2i}(BT^{n}\times BS^{1})$ for all
$p\in\mathcal{V}$.

###### Proof.

Since $f(p)\in H^{*}(BT^{n}\times BS^{1})$, for every $p\in\mathcal{V}$ there
exists a non-negative integer $l(p)$ and an element $f_{2i}(p)\in
H^{2i}(BT^{n}\times BS^{1})$ such that

$\displaystyle f(p)=f_{0}(p)+\cdots+f_{2l(p)}(p).$

If we take the maximal integer $l=\max\\{l(p)\ |\ p\in\mathcal{V}\\}$, then we
may write

$\displaystyle f(p)=f_{0}(p)+\cdots+f_{2l}(p).$

for all $p\in\mathcal{V}$. Therefore, we can define the map
$f_{2i}:\mathcal{V}\to H^{2i}(BT^{n}\times BS^{1})$ by $p\mapsto f_{2i}(p)$
for all $0\leq i\leq l$. We claim that $f_{2i}\in H^{*}(\mathcal{G})$. Because
$f$ satisfies the congruence relation for all edges $\epsilon$, we see that

(6.9) $\displaystyle
f(i(\epsilon))-f(t(\epsilon))=(f_{0}(i(\epsilon))-f_{0}(t(\epsilon)))+\cdots+(f_{2l}(i(\epsilon))-f_{2l}(t(\epsilon)))=A\alpha(\epsilon)$

for some $A\in H^{*}(BT^{n}\times BS^{1})$. Moreover, there is a monomial
$A_{2i}\in H^{2i}(BT^{n}\times BS^{1})$ for each $0\leq i\leq l-1$ such that

$\displaystyle A=A_{0}+\cdots+A_{2l-2}.$

Comparing the same degree monomials of both sides in (6.9), we have

$\displaystyle
f_{2i}(i(\epsilon))-f_{2i}(t(\epsilon))=A_{2i-2}\alpha(\epsilon)\equiv
0\mod\alpha(\epsilon).$

Because this relation satisfies for all $\epsilon\in E$, we have $f_{2i}\in
H^{*}(\mathcal{G})$ for all $i=0,\ \cdots,\ l$. This establishes the
statement. ∎

We call each $f_{2i}$ in Lemma 6.2 a $2i$ degree homogeneous term of $f$ for
$i=0,\ \cdots,\ l$. We denote ${\rm deg}\ f_{2i}=2i$. Of course, $f_{2i}\in
H^{2i}(\mathcal{G})$.

The following third lemma is about the map
$\pi:\mathbb{Z}[X,H_{1},\ldots,H_{m}]\to H^{*}(\mathcal{G})$. This will be a
technical part to show that $\pi$ is surjective (Lemma 6.4).

######  Lemma 6.3.

Assume that there exists an element $f\in H^{*}(\mathcal{G})$ such that
$f\not\in{\rm Im}\ \pi$. Then there are
$A\in{\mathbb{Z}}[X,H_{1},\cdots,H_{m}]$ and some integer $j_{k}$ such that

$\displaystyle\pi(A)-f=\chi\sum_{k}g_{2j_{k}},$

where $g_{2j_{k}}\in H^{2j_{k}}(\mathcal{G})$ but $g_{2j_{k}}\not\in{\rm Im}\
\pi$ with $j_{0}<j_{1}<\cdots<j_{k}<\cdots$.

###### Proof.

Assume $f\not\in{\rm Im}\ \pi$. Recall that the following two homomorphisms in
(6.8) are surjective by the assumption (1) of Theorem 4.1 and Theorem 5.1:

$\displaystyle\phi:\mathbb{Z}[X,H_{1},\ldots,H_{m}]\longrightarrow\mathbb{Z}[L_{1},\ldots,L_{m}];$
$\displaystyle\pi^{\prime}:\mathbb{Z}[L_{1},\ldots,L_{m}]\longrightarrow
H^{*}(\widetilde{\mathcal{G}}).$

Therefore, there exists a non-zero polynomial

$\displaystyle B\in{\mathbb{Z}}[X,H_{1},\cdots,H_{m}]$

such that for $\widetilde{F}:H^{*}(\mathcal{G})\to
H^{*}(\widetilde{\mathcal{G}})$,

$\displaystyle\widetilde{F}(f)=\pi^{\prime}\circ\phi(B).$

Because $\pi^{\prime}\circ\phi=\widetilde{F}\circ\pi$ in the diagram (6.8), we
have

$\displaystyle\pi^{\prime}\circ\phi(B)=\widetilde{F}\circ\pi(B)=\widetilde{F}(f).$

Hence $\pi(B)-f\in{\rm Ker}\ \widetilde{F}$. Because of Lemma 6.1, i.e., ${\rm
Ker}\widetilde{F}=\langle\chi\rangle$, there is a $g^{\prime}\in
H^{*}(\mathcal{G})$ such that

(6.10) $\displaystyle\pi(B)-f=g^{\prime}\chi.$

Since $f\not\in{\rm Im}\ \pi$ and $\pi(X)=\chi$, we have

$\displaystyle g^{\prime}\not\in{\rm Im}\ \pi.$

Because of Lemma 6.2, this element $g^{\prime}$ can be divided into

$\displaystyle g^{\prime}=g_{0}+\cdots+g_{2l},$

where $g_{2i}$ is a $2i$ degree homogeneous term, for $0\leq i\leq l$. If
$g_{2i}\in{\rm Im}\ \pi$, then $g^{\prime}-g_{2i}\not\in{\rm Im}\ \pi$.
Therefore, $g^{\prime}$ can be divided into two terms
$(0\not=)g=\sum_{k}g_{2j_{k}}$ for all $g_{2j_{k}}\not\in{\rm Im}\ \pi$ and
$h=\sum_{k^{\prime}}g_{2i_{k^{\prime}}}$ for all $g_{2i_{k^{\prime}}}\in{\rm
Im}\ \pi$ such that

$\displaystyle g^{\prime}=g+h.$

Since

$\displaystyle g^{\prime}\chi=g\chi+h\chi=g\chi+\pi(CX)$

for some $C\in{\mathbb{Z}}[X,H_{1},\cdots,H_{m}]$, together with (6.10), we
see that there is an element $A=B-CX\in{\mathbb{Z}}[X,H_{1},\cdots,H_{m}]$
such that $\pi(A)-f=g\chi$. ∎

Now we may prove Lemma 6.4.

######  Lemma 6.4.

The homomorphism $\pi:{\mathbb{Z}}[X,H_{1},\cdots,H_{m}]\to
H^{*}(\mathcal{G})$ is surjective.

###### Proof.

By Lemma 6.3, it is enough to show that every homogeneous term of $f\in
H^{*}(\mathcal{G})$ is an element of ${\rm Im}\ \pi$.

Assume that $H^{*}(\mathcal{G})\backslash{\rm Im}\ \pi\not=\emptyset$. Let $f$
be a minimal degree homogeneous element in $H^{*}(\mathcal{G})\backslash{\rm
Im}\ \pi$. Because of Lemma 6.3, there exists a polynomial
$A\in{\mathbb{Z}}[X,H_{1},\cdots,H_{m}]$ and an element $g\in
H^{*}(\mathcal{G})\backslash{\rm Im}\ \pi$ such that

$\displaystyle f=\pi(A)-g\chi.$

By using Lemma 6.3 again, we also have that $g$ is a sum of homogeneous
elements in $H^{*}(\mathcal{G})\backslash{\rm Im}\ \pi$.

We claim that $\pi(A)(\in{\rm Im}\ \pi)$ and $g\chi(\in
H^{*}(\mathcal{G})\backslash{\rm Im}\ \pi)$ are also homogeneous elements in
$H^{*}(\mathcal{G})$ whose degrees are the same with the degree of $f$. Assume
that $\pi(A)=\sum_{k}h_{2i_{k}}$ and $g\chi=\sum_{k}g_{2j_{k}}\chi$, where
$h_{2i_{k}}\in H^{2i_{k}}(\mathcal{G})\cap{\rm Im}\ \pi$ for all
$i_{0}<i_{1}<\cdots$ and $g_{2j_{k}}\in H^{2j_{k}}(\mathcal{G})\setminus{\rm
Im}\ \pi$ for all $j_{0}<j_{1}<\cdots$. Because $f$ is a minimal homogeneous
element in $H^{*}(\mathcal{G})\backslash{\rm Im}\ \pi$, we see that $\deg
g_{2j_{0}}\chi=\deg f$ and $\deg h_{2i_{0}}\geq\deg f$; moreover, the higher
terms of $\pi(A)\in{\rm Im}\ \pi$ and $g\chi\not\in{\rm Im}\ \pi$ are the
same, i.e., they must be $0$. Hence, both of $\pi(A)$ and $g$ are also
homogeneous elements.

However, in this case, we have

$\displaystyle\deg g=\deg g\chi-\deg\chi=\deg f-2<\deg f.$

This gives a contradiction to that $f$ is a minimal homogeneous element in
$H^{*}(\mathcal{G})\backslash{\rm Im}\ \pi$. Hence, there does not exist any
homogeneous elements in $H^{*}(\mathcal{G})\backslash{\rm Im}\ \pi$.
Consequently, by Lemma 6.3, we have that $H^{*}(\mathcal{G})\backslash{\rm
Im}\ \pi=\emptyset$, i.e., $\pi$ is surjective. ∎

Therefore, by the commutativity of the top diagram in (6.8), the following
lemma holds:

######  Lemma 6.5.

$\Psi$ is surjective.

###  6.2. Injectivity of $\Psi$

Finally, in this section, we will prove the injectivity of $\Psi$. In this
section we use the following notation:

$\displaystyle I_{j}=\\{1,\ \cdots,\ l\\}-\\{j\\}$

for $j=1,\ \cdots,\ l$. We first prove Lemma 6.7. In order to prove Lemma 6.7,
we prepare the following lemma.

######  Lemma 6.6.

Assume that $\cap_{k=1}^{l}L_{k}=\emptyset$ and
$L_{k}=H_{k}\cap\overline{H_{k}}$ $(k=1,\ \cdots,\ l)$. Then for all
$j=1,\ldots,l$, one of the following holds:

  * •

$H_{j}\cap(\cap_{k\in I_{j}}L_{k})=\emptyset$;

  * •

$\overline{H_{j}}\cap(\cap_{k\in I_{j}}L_{k})=\emptyset$.

###### Proof.

Assume $\cap_{k=1}^{l}L_{k}=\emptyset$. For $j\in\\{1,\ \cdots,\ l\\}$, if the
following relation holds:

$\displaystyle L_{j}\cap(\cap_{k\in I_{j}}L_{k})=\cap_{k\in
I_{j}}L_{k}=\emptyset,$

then it follows from $\cap_{k\in I_{j}}L_{k}=\emptyset$ that for each $H_{j}$
and $\overline{H_{j}}$ we have

$\displaystyle H_{j}\cap(\cap_{k\in
I_{j}}L_{k})=\overline{H_{j}}\cap(\cap_{k\in I_{j}}L_{k})=\emptyset.$

So we may take $j\in\\{1,\ \cdots,\ l\\}$ such that

$\displaystyle\bigcap_{k\in I_{j}}L_{k}\not=\emptyset.$

In this case, there exists a vertex $p\in\mathcal{V}^{\cap_{k\in
I_{j}}L_{k}}$. Since $L_{j}\cap(\cap_{k\in
I_{j}}L_{k})=\cap_{k=1}^{l}L_{k}=\emptyset$, we have that
$p\not\in\mathcal{V}^{L_{i}}$. Therefore, for all vertices
$p\in\mathcal{V}^{\cap_{k\in I_{j}}L_{k}}$, the following equation holds:

$\displaystyle\tau_{H_{j}}(p)=\left\\{\begin{array}[]{cc}0&({\rm if}\
p\not\in\mathcal{V}^{H_{j}})\\\ x&({\rm if}\
p\in\mathcal{V}^{H_{j}})\end{array}\right.$

where $\tau_{H_{j}}$ is the Thom class of $H_{j}$.

If there are two vertices $p,q\in\mathcal{V}^{\cap_{k\in I_{j}}L_{k}}$ such
that

$\displaystyle\tau_{H_{j}}(p)=0;\quad\tau_{H_{j}}(q)=x.$

By the assumption (2) in Theorem 4.1, there exists a path from $p$ to $q$ in
$\cap_{k\in I_{j}}L_{k}$, i.e., we may take the following sequence in
$E^{\cap_{k\in I_{j}}L_{k}}$:

$\displaystyle\epsilon_{1},\cdots,\epsilon_{s}\in E^{\cap_{k\in I_{j}}L_{k}}$

such that $i(\epsilon_{1})=p$ and $t(\epsilon_{s})=q$. By the definition of
the $T^{*}\mathbb{C}^{n}$-modeled GKM graph, the axial function satisfies that
$\alpha(\epsilon)\not=x$ for all $\epsilon\in E$. Moreover, $\tau_{H_{j}}$
satisfies the congruence relation. Therefore, there exists an edge
$\epsilon\in\\{\epsilon_{1},\ldots,\epsilon_{s}\\}$ such that
$r=i(\epsilon_{t})$ satisfies that

$\displaystyle\tau_{H_{j}}(r)\not=0,\ x.$

By the definition of the Thom class of the halfspace $H_{j}$, the vertex
$r\in\partial H_{j}=L_{j}$. However, this gives that $r\in
L_{j}\cap(\cap_{k\in I_{j}}L_{k})=\cap_{k=1}^{l}L_{k}$. This gives a
contradiction to that $\cap_{k=1}^{l}L_{k}=\emptyset$.

Therefore, we may assume $\tau_{H_{j}}(p)=0$ (resp. $x$) for all
$p\in\mathcal{V}^{\cap_{k\in I_{j}}L_{k}}$. Then, by definition of the
halfspace, we have $H_{j}\cap\mathcal{V}^{\cap_{k\in I_{j}}L_{k}}=\emptyset$
(resp. $\overline{H_{j}}\cap\mathcal{V}^{\cap_{k\in I_{j}}L_{k}}=\emptyset$).
This establishes the statement of this lemma. ∎

From Lemma 6.6, we have the following key fact.

######  Lemma 6.7.

Assume the $T^{*}\mathbb{C}^{n}$-modeled GKM graph $\mathcal{G}$ satisfies two
assumptions (1), (2) of Theorem 4.1. If $\cap_{k=1}^{l}L_{k}=\emptyset$ and
$L_{k}=H_{k}\cap\overline{H_{k}}$ $(k=1,\ \cdots,\ l)$, then we can take a
halfspace $H_{k}$ such that $\cap_{k=1}^{l}H_{k}=\emptyset$.

###### Proof.

If $\cap_{k=1}^{l}L_{k}=\emptyset$ and $L_{k}=H_{k}\cap\overline{H_{k}}$
$(k=1,\ \cdots,\ l)$, we can take $H_{j}$ as $H_{j}\cap(\cap_{k\in
I_{j}}L_{k})=\emptyset$ for all $j=1,\ \cdots,\ l$ from Lemma 6.6. Now we may
set

$\displaystyle\mathbf{H}^{\prime}=\left\\{H_{1},\ \cdots,\ H_{l}\ |\
H_{j}\cap(\cap_{k\in I_{j}}L_{k})=\emptyset,\ j=1,\ldots,l\right\\}.$

We claim that $\cap_{H\in\mathbf{H}^{\prime}}H=\cap_{j=1}^{l}H_{j}=\emptyset$.
If there exists a vertex $p\in\mathcal{V}^{\cap_{j=1}^{l}H_{j}}$, it follows
from the assumption $\cap_{k=1}^{l}L_{k}=\emptyset$ that we have
$\tau_{H_{j}}(p)=x$ for all $j=1,\ldots,l$; therefore,

$\displaystyle\prod_{j=1}^{l}\tau_{H_{j}}(p)=x^{l}.$

Because $\prod_{j=1}^{l}\tau_{H_{j}}\in H^{*}(\mathcal{G})$,
$\prod_{j=1}^{l}\tau_{H_{j}}$ satisfies the congruence relations for all edges
$\epsilon\in E$. By definition of $T^{*}\mathbb{C}^{n}$-modeled GKM graph, the
axial function satisfies $\alpha(\epsilon)\not=x$ for all edges $\epsilon\in
E$. This shows that for all edge $\epsilon\in E_{p}$ the following equation
holds:

$\displaystyle\prod_{j=1}^{l}\tau_{H_{j}}(t(\epsilon))=x^{l}.$

Because the graph $\Gamma$ is connected, we can apply the same argument for
all vertices; therefore, we have

$\displaystyle\prod_{j=1}^{l}\tau_{H_{j}}(q)=x^{l}$

for all $q\in\mathcal{V}$. This shows that
$\mathcal{V}=\mathcal{V}^{\cap_{j=1}^{l}H_{j}}$. However, by definition of the
halfspace, it is obvious that
$\mathcal{V}\not=\mathcal{V}^{\cap_{j=1}^{l}H_{j}}$ and this gives a
contradiction. Hence, we have
$\cap_{j=1}^{l}H_{j}=\cap_{H\in\mathbf{H}^{\prime}}H=\emptyset$. ∎

We next will prove Lemma 6.10. In order to prove it, we prepare some notations
and two lemmas: Lemma 6.8 and 6.9.

Let $\widetilde{\pi}:{\mathbb{Z}}[X,\ H_{1},\ \cdots,\
H_{m}]\to{\mathbb{Z}}[\mathcal{G}]$ be the natural homomorphism such that
$\widetilde{\pi}(X)=X$, $\widetilde{\pi}(H_{i})=H_{i}$ for $i=1,\ \cdots,\ m$.
Because $\overline{H_{i}}=X-H_{i}$ in ${\mathbb{Z}}[\mathcal{G}]$, we have

$\displaystyle\widetilde{\pi}\circ\phi^{\prime}=\hat{\pi}:{\mathbb{Z}}[X,\
H_{1},\
\cdots,H_{m},\overline{H}_{1},\cdots,\overline{H_{m}}]\to{\mathbb{Z}}[\mathcal{G}].$

Since $\hat{\pi}$ is surjective, $\widetilde{\pi}$ is also surjective.
Moreover we have

$\displaystyle\Psi\circ\widetilde{\pi}=\pi:{\mathbb{Z}}[X,\ H_{1},\ \cdots,\
H_{m}]\to H^{*}(\mathcal{G})$

by definitions of $\Psi$ and $\pi$. Hence we have the following commutative
diagram:

Define the following ideal in
${\mathbb{Z}}[X,H_{1},\cdots,H_{m},\overline{H}_{1},\cdots,\overline{H_{m}}]$:

$\displaystyle\mathcal{I}=\Big{\langle}H_{i}+\overline{H_{i}}-X,\
\prod_{H\in\mathbf{H}^{\prime}}H\ \Big{|}\ i=1,\ \cdots,\ m,\
\mathbf{H}^{\prime}\in\mathbf{I}(\mathbf{H})\Big{\rangle},$

where $\mathbf{I}(\mathbf{H})=\\{\mathbf{H}^{\prime}\subset\mathbf{H}\ |\
\cap_{H\in\mathbf{H}^{\prime}}H=\emptyset\\}$. For this ideal, the following
property holds.

######  Lemma 6.8.

For the ideal
$\mathcal{I}\subset{\mathbb{Z}}[X,H_{1},\cdots,H_{m},\overline{H}_{1},\cdots,\overline{H_{m}}]$,
the following two properties hold:

  1. (i)

${\rm Ker}\ \widetilde{\pi}=\phi^{\prime}(\mathcal{I})$;

  2. (ii)

${\rm Ker}\ \pi^{\prime}=\phi\circ\phi^{\prime}(\mathcal{I})$.

###### Proof.

Since $\hat{\pi}$ is the natural projection, it follows from the definition of
$\mathbb{Z}[\mathcal{G}]$ that

$\displaystyle\mathcal{I}={\rm Ker}\ \hat{\pi}.$

So, by the commutativity of the diagram, we have that

$\displaystyle\widetilde{\pi}(\phi^{\prime}(\mathcal{I}))=\hat{\pi}(\mathcal{I})=\hat{\pi}({\rm
Ker}\ \hat{\pi})=\\{0\\}.$

Hence $\phi^{\prime}(\mathcal{I})\subset{\rm Ker}\ \widetilde{\pi}$. Let $A$
be an element in ${\rm Ker}\ \widetilde{\pi}$. Because $\phi^{\prime}$ is
surjective, there is an element $B\in{\mathbb{Z}}[X,\ H_{1},\
\cdots,H_{m},\overline{H}_{1},\cdots,\overline{H_{m}}]$ such that
$\phi^{\prime}(B)=A$. By the commutativity of the diagram, we also have

$\displaystyle\hat{\pi}(B)=\widetilde{\pi}\circ\phi^{\prime}(B)=\widetilde{\pi}(A)=0.$

So $B\in{\rm Ker}\ \hat{\pi}=\mathcal{I}$. Hence
$A=\phi^{\prime}(B)\in\phi^{\prime}(\mathcal{I})$, that is, ${\rm Ker}\
\widetilde{\pi}\subset\phi^{\prime}(\mathcal{I})$. Therefore, we establish the
first property: ${\rm Ker}\ \widetilde{\pi}=\phi^{\prime}(\mathcal{I})$.

By Theorem 5.1, we know

$\displaystyle{\rm Ker}\ \pi^{\prime}=\langle\prod_{L\in\mathbf{L}^{\prime}}L\
|\ \mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\rangle,$

where $\mathbf{I}(\mathbf{L})=\\{\mathbf{L}^{\prime}\subset\mathbf{L}\ |\
\cap_{L\in\mathbf{L}^{\prime}}L=\emptyset\\}$. Take a generator
$\prod_{L\in\mathbf{L}^{\prime}}L\in{\rm Ker}\ \pi^{\prime}$. From Lemma 6.7,
for $\mathbf{L}^{\prime}=\\{L_{1},\ \cdots,\
L_{l}\\}\in\mathbf{I}(\mathbf{L})$, there exists a set of halfspaces
$\mathbf{H}^{\prime}=\\{H_{1},\ \cdots,\ H_{l}\\}\in\mathbf{I}(\mathbf{H})$
such that $H_{k}\cap\overline{H_{k}}=L_{k}$. By the definition of the ideal
$\mathcal{I}$, a product $\prod_{k=1}^{l}H_{k}$ is one of the generators of
$\mathcal{I}$. Moreover, by the definitions of $\phi^{\prime}$ and $\phi$, we
see that

$\displaystyle\phi\circ\phi^{\prime}(\mathcal{I})\ni\phi\circ\phi^{\prime}(\prod_{k=1}^{l}H_{k})=\pm\prod_{k=1}^{l}L_{k}.$

Because this satisfies for all generators $\prod_{L\in\mathbf{L}^{\prime}}L$
in ${\rm Ker}\ \pi^{\prime}$, we have that

$\displaystyle{\rm Ker}\
\pi^{\prime}\subset\phi\circ\phi^{\prime}(\mathcal{I}).$

On the other hand, because $\phi^{\prime}(H+\overline{H}-X)=0$ and
$\phi\circ\phi^{\prime}(\prod_{H\in\mathbf{H}^{\prime}}H)=\pm\prod_{L\in\mathbf{L}^{\prime}}L\in{\rm
Ker}\ \pi^{\prime}$, for all $A\in\mathcal{I}$ we have

$\displaystyle\pi^{\prime}\circ\phi\circ\phi^{\prime}(A)=\\{0\\}.$

So we have ${\rm Ker}\
\pi^{\prime}\supset\phi\circ\phi^{\prime}(\mathcal{I})$. Therefore we conclude
the second property: ${\rm Ker}\
\pi^{\prime}=\phi\circ\phi^{\prime}(\mathcal{I})$. ∎

In order to prove Lemma 6.10, we also prepare the following technical lemma
for general polynomial rings.

######  Lemma 6.9.

Let $\mathcal{I}\subset{\mathbb{Z}}[x_{1},\ \cdots,\ x_{l}]$ be an ideal
generated by homogeneous polynomials, that is, $\mathcal{I}=\langle p_{1},\
\cdots,\ p_{m}\rangle$ where $p_{i}$ is a homogeneous polynomial of
${\mathbb{Z}}[x_{1},\ \cdots,\ x_{l}]$ such that ${\rm deg}\ p_{i}\leq{\rm
deg}\ p_{j}$ for $i<j$. For every element $A\in\mathcal{I}$, if we denote
$A=A_{1}+\cdots+A_{n}$, where $A_{i}$ is a homogeneous term ($i=1,\ \cdots,\
n$) and ${\rm deg}\ A_{i}<{\rm deg}\ A_{j}$ for $i<j$, then
$A_{i}\in\mathcal{I}$ for all $i=1,\ \cdots,\ n$.

###### Proof.

Because $A\in\mathcal{I}=\langle p_{1},\ \cdots,\ p_{m}\rangle$, there exists
$X_{k}\in{\mathbb{Z}}[x_{1},\ \cdots,\ x_{l}]$, $k=1,\ldots,m$, such that

$\displaystyle A=X_{1}p_{1}+\cdots+X_{m}p_{m}.$

Then we can put $X_{k}=X_{k1}+X_{k2}+\cdots+X_{ks_{k}}$ where $X_{ki}$ is a
homogeneous term ($i=1,\ \cdots,\ s_{k}$) and ${\rm deg}\ X_{ki}<{\rm deg}\
X_{kj}$ for $i<j$. Hence, by changing the order of monomials, we may rewrite

$\displaystyle A$
$\displaystyle=(X_{11}+\cdots+X_{1s_{1}})p_{1}+\cdots+(X_{m1}+\cdots+X_{ms_{m}})p_{m}$
$\displaystyle=X_{11}p_{1}+\cdots X_{ms_{m}}p_{m}$

as

$\displaystyle A=A_{1}+\cdots+A_{n},$

where $\deg A_{i}<\deg A_{j}$ if $i<j$. Because $A_{i}$ is a homogeneous term,
we have

$\displaystyle A_{i}=\sum_{j\in\mathbf{D}_{i}}X_{jh_{j}}p_{j}$

where $\mathbf{D}_{i}=\\{j\ |\ \deg X_{jh_{j}}+\deg p_{j}=\deg A_{i}\\}$.
Therefore, $A_{i}\in\mathcal{I}$ for all $i=1,\ \cdots,\ n$. ∎

Using two lemmas as above, we have the following lemma.

######  Lemma 6.10.

${\rm Ker}\ \widetilde{\pi}={\rm Ker}\ \pi$.

###### Proof.

By Lemma 6.8 (i), ${\rm Ker}\ \widetilde{\pi}=\phi^{\prime}(\mathcal{I})$.
Therefore, by using the commutativity of the diagram and ${\rm Ker}\
\hat{\pi}=\mathcal{I}$, we have

$\displaystyle\pi({\rm Ker}\
\widetilde{\pi})=\pi\circ\phi^{\prime}(\mathcal{I})=\Psi\circ\hat{\pi}(\mathcal{I})=0.$

Hence,

$\displaystyle{\rm Ker}\ \widetilde{\pi}=\phi^{\prime}(\mathcal{I})\subset{\rm
Ker}\ \pi.$

We claim that ${\rm Ker}\ \pi\subset{\rm Ker}\
\widetilde{\pi}(\subset{\mathbb{Z}}[X,\ H_{1},\ \cdots,\ H_{m}])$. Assume that
${\rm Ker}\ \pi\backslash\phi^{\prime}(\mathcal{I})\not=\emptyset$. Let
$A\in{\rm Ker}\ \pi\backslash\phi^{\prime}(\mathcal{I})\subset{\mathbb{Z}}[X,\
H_{1},\ \cdots,\ H_{m}]$ be a minimal degree homogeneous polynomial. By the
previous diagram,

$\displaystyle\pi^{\prime}\circ\phi(A)=\widetilde{F}\circ\pi(A)=0.$

Hence, by Lemma 6.8 (ii),

$\displaystyle\phi(A)\in{\rm Ker}\
\pi^{\prime}=\phi\circ\phi^{\prime}(\mathcal{I}).$

Therefore, we can take $B\in\phi^{\prime}(\mathcal{I})(\subset{\rm Ker}\ \pi)$
such that $\phi(A)=\phi(B)$. Because $\phi^{\prime}(\mathcal{I})$ is an ideal
in $\mathbb{Z}[X,H_{1},\ldots,H_{m}]$ and $A$ is a homogeneous polynomial, it
follows from Lemma 6.9 that we may also take $B$ as the homogeneous polynomial
in $\phi^{\prime}(\mathcal{I})$ such that

$\displaystyle\deg A=\deg B.$

By the definition of $\phi$, it is easy to check that ${\rm Ker}\ \phi=\langle
X\rangle\subset\mathbb{Z}[X,H_{1},\ldots,H_{m}]$; therefore, we have

$\displaystyle A-B\in{\rm Ker}\ \phi=\langle X\rangle.$

This means that there exists a polynomial $C\in{\mathbb{Z}}[X,\ H_{1},\
\cdots,\ H_{m}]$ such that

$\displaystyle A-B=CX.$

Because $A=B+CX\not\in\phi^{\prime}(\mathcal{I})$ and
$B\in\phi^{\prime}(\mathcal{I})$, we have
$CX\not\in\phi^{\prime}(\mathcal{I})$. Moreover, because we take $\deg A=\deg
B$, $CX$ is also a homogeneous polynomial with $\deg A=\deg B=\deg CX$.
Therefore, because $A,\ B\in{\rm Ker}\ \pi$, we have $CX$ is a homogeneous
polynomial in ${\rm Ker}\ \pi\setminus\phi^{\prime}(\mathcal{I})$. Then, we
have $\deg A=\deg CX=\deg C+\deg X=\deg C+2$. Moreover, because $A$ is a
minimal homogeneous polynomial in ${\rm Ker}\
\pi\setminus\phi^{\prime}(\mathcal{I})$, we have that

$\displaystyle C\in\phi^{\prime}(\mathcal{I})(\subset{\rm Ker}\ \pi).$

However, because $\phi^{\prime}(\mathcal{I})$ is an ideal in
$\mathbb{Z}[X,H_{1},\ldots,H_{m}]$, we see that

$\displaystyle CX\in\phi^{\prime}(\mathcal{I}).$

This shows that $A=B+CX\in\phi^{\prime}(\mathcal{I})$ and this gives the
contradiction to that there is an element in ${\rm Ker}\
\pi\backslash\phi^{\prime}(\mathcal{I})$. Hence, we have ${\rm Ker}\
\pi\backslash\phi^{\prime}(\mathcal{I})=\emptyset$, that is, ${\rm Ker}\
\pi=\phi^{\prime}(\mathcal{I})={\rm Ker}\ \widetilde{\pi}$ by Lemma 6.8 (i). ∎

So we can prove the injectivity of $\Psi$.

######  Lemma 6.11.

$\Psi$ is injective.

###### Proof.

Let $A$ be in ${\rm Ker}\ \Psi$. Since $\widetilde{\pi}$ is surjective, there
is an element $B\in{\mathbb{Z}}[X,\ H_{1},\ \cdots,\ H_{m}]$ such that
$\widetilde{\pi}(B)=A$. So we have
$\pi(B)=\Psi\circ\widetilde{\pi}(B)=\Psi(A)=0$. Hence $B\in{\rm Ker}\ \pi={\rm
Ker}\ \widetilde{\pi}$ by Lemma 6.10. Therefore, we have
$A=\widetilde{\pi}(B)=0$. This concludes that $\Psi$ is injective. ∎

Because of Lemma 6.5 and 6.11, we have that $\Psi$ is the isomorphism.
Consequently the proof of Theorem 4.1 is complete, that is, we get

$\displaystyle H^{*}(\mathcal{G})\simeq{\mathbb{Z}}[\mathcal{G}].$

##  7\. Generators of $\mathbb{Z}[\widetilde{\mathcal{G}}]$ as
$H^{*}(BT^{n})$-module

Let $\mathcal{G}=(\Gamma,\alpha,\nabla)$ be a $2n$-valent
$T^{*}\mathbb{C}^{n}$-modeled GKM graph and
$\mathbf{L}=\\{L_{1},\ldots,L_{m}\\}$ be the set of all hyperplanes in
$\mathcal{G}$. Assume that $\mathcal{G}$ satisfies the two assumptions of
Theorem 4.1 so that $H^{*}(\mathcal{G})\simeq\mathbb{Z}[\mathcal{G}]$.

###  7.1. Simplicial complex associated to $\mathbf{L}$

.

Let $\mathbf{L}=\\{L_{1},\ldots,L_{m}\\}$. Let $\Delta_{\mathbf{L}}$ denote
the simplicial complex associated to $\mathbf{L}$ defined as follows. There is
a vertex $v_{i}$ in $\Delta_{\mathbf{L}}$ corresponding to the hyperplane
$L_{i}$ such that whenever $L_{i_{1}}\cap\cdots\cap L_{i_{k}}\neq\emptyset$ in
$\mathcal{G}$, the vertices $\\{v_{i_{1}},\ldots,v_{i_{k}}\\}$ span a simplex
in $\Delta_{\mathbf{L}}$. In particular, for $1\leq i\leq d$, let
$\sigma_{i}=\langle v_{i_{1}},\ldots,v_{i_{n}}\rangle$ be the
$(n-1)$-dimensional simplex of $\Delta_{\mathbf{L}}$ corresponding to a vertex
${\bf p}_{\sigma_{i}}:=L_{i_{1}}\cap\cdots\cap L_{i_{n}}$ of $\mathcal{G}$.

Note that $\Delta_{\mathbf{L}}$ is pure i.e., all maximal faces (also called
facets) are of the same dimension $n-1$. Let $\Delta_{\mathbf{L}}(n-1)$ denote
the set of facets of $\Delta_{\mathbf{L}}$. Then
$d=|\Delta_{\mathbf{L}}(n-1)|$ which is also equal to $|\mathcal{V}^{\Gamma}|$
the number of vertices of $\Gamma$. For simplices $\tau$ and $\sigma$ in
$\Delta_{\mathbf{L}}$, by $\tau\preceq\sigma$ we mean that $\tau$ is a face of
$\sigma$.

We say that $\Delta_{\mathbf{L}}$ is a shellable simplicial complex if the
following holds: There is an ordering
$\sigma_{1},\sigma_{2},\ldots,\sigma_{d}$ of $\Delta_{\mathbf{L}}(n-1)$ such
that if $\Delta_{j}$ denotes the subcomplex generated by
$\sigma_{1},\ldots,\sigma_{j}$ for each $1\leq j\leq d$, then
$\Delta_{i}\setminus\Delta_{i-1}$ has a unique minimal face $\mu_{i}$ for each
$2\leq i\leq d$. We further let $\mu_{1}:=\emptyset$ to be the unique minimal
face of $\Delta_{1}\setminus\Delta_{0}$ where $\Delta_{0}=\emptyset$ (see [S,
Section 2.1 p.79]). (Also see [BH, Definition 5.1.11] for other equivalent
definitions of shellability of a pure simplicial complex.)

######  Example 7.1.

In the case when $n=1$, $\mathcal{G}$ is a $2$-valent
$T^{*}\mathbb{C}$-modelled GKM graph and the set of hyperplanes
$\mathbf{L}=\\{L_{1},\ldots,L_{m}\\}$ of $\mathcal{G}$ coincides with the set
of vertices of $\mathcal{G}$ (see Figure 13). Here $\mathcal{G}$ corresponds
to the hyperplane arrangement of a $4$-dimensional toric
hyperK$\ddot{\mathrm{a}}$hler manifold. Furthermore, since $L_{i}\cap
L_{j}=\emptyset$ for every $i\neq j$, $1\leq i,j\leq m$, the associated
simplicial complex $\Delta_{\mathbf{L}}$ is a pure $0$-dimensional simplicial
complex consisting of a vertex (or $0$-dimensional simplex) $v_{i}$
corresponding to $L_{i}$ for every $1\leq i\leq m$. Then $\Delta_{\mathbf{L}}$
is seen to be trivially shellable for any ordering of the set of vertices
$\\{v_{1},\ldots,v_{m}\\}$ which are also the facets of $\Delta_{\mathbf{L}}$
in this case.

$L_{1}$$L_{2}$$L_{3}$$L_{4}$$L_{5}$$e_{1}^{*}$$x-e_{1}^{*}$ Figure 13. The
$T^{*}\mathbb{C}$-modeled GKM graph $\mathcal{G}$ defined from a five vertices
(hyperplanes) arrangement on the line. In this case, the hyperplane $L_{i}$ in
$\mathcal{G}$ corresponds to the $0$-dimensional simplex $v_{i}$ in
$\Delta_{\mathbf{L}}$ for $i=1,\ldots,5$. This is defined from a
$4$-dimensional toric hyperK$\ddot{\mathrm{a}}$hler manifold. Note that we
omit the axial functions which are automatically determined.

######  Example 7.2.

Consider the simplicial complex $\Delta_{\mathbf{L}}$ corresponding to the
$T^{*}\mathbb{C}^{2}$-modelled GKM graph given in Figure 10. We label the set
of $5$ hyperplanes as $L_{1},\ldots,L_{5}$ where $L_{i}\cap
L_{i+1}\neq\emptyset$ for $1\leq i\leq 4$ and $L_{5}\cap L_{1}\neq\emptyset$
(see Figure 14). Then $\Delta_{\mathbf{L}}$ consists respectively of the
corresponding vertices $v_{1},\ldots,v_{5}$ and the $1$-simplices
$[v_{1},v_{2}],[v_{2},v_{3}],[v_{3},v_{4}],[v_{4},v_{5}],[v_{5},v_{1}]$ which
are its facets. For the ordering

$\sigma_{1}=[v_{1},v_{2}],\sigma_{2}=[v_{2},v_{3}],\sigma_{3}=[v_{3},v_{4}],\sigma_{4}=[v_{4},v_{5}],\sigma_{5}=[v_{5},v_{1}]$

of $\Delta_{\mathbf{L}}(1)$, we see that
$\Delta_{1}\setminus\Delta_{0}=\\{\emptyset,\\{v_{1}\\},\\{v_{2}\\},[v_{1},v_{2}]\\}$
has the minimal element $\mu_{1}=\emptyset$,
$\Delta_{2}\setminus\Delta_{1}=\\{\\{v_{3}\\},[v_{2},v_{3}]\\}$ has the
minimal element $\mu_{2}=\\{v_{3}\\}$,
$\Delta_{3}\setminus\Delta_{2}=\\{\\{v_{4}\\},[v_{3},v_{4}]\\}$ has the
minimal element $\mu_{3}=\\{v_{4}\\}$,
$\Delta_{4}\setminus\Delta_{3}=\\{\\{v_{5}\\},[v_{4},v_{5}]\\}$ has the
minimal element $\mu_{4}=\\{v_{5}\\}$ and
$\Delta_{5}\setminus\Delta_{4}=\\{[v_{5},v_{1}]\\}$ has the minimal element
$\mu_{5}=[v_{5},v_{1}]$. Thus $\Delta_{\mathbf{L}}$ is a shellable simplicial
complex.

$p_{2}$$p_{5}$$p_{1}$$p_{4}$$p_{3}$$L_{1}$$L_{2}$$L_{5}$$L_{3}$$L_{4}$ Figure
14. The GKM graph $\mathcal{G}$ in Figure 10. In this case, the hyperplane
$L_{i}$ (resp. vertex $p_{i}$) in $\mathcal{G}$ corresponds to the $0$ (resp.
$1$)-dimensional simplex $v_{i}$ (resp. $\sigma_{i}$) in $\Delta_{\mathbf{L}}$
for $i=1,\ldots,5$.

In subsection 7.5, we shall show the shellability of $\Delta_{\mathbf{L}}$ for
a $T^{*}\mathbb{C}^{2}$-modelled GKM graph $\mathcal{G}$ that is induced from
the $8$-dimensional toric hyperK$\ddot{\mathrm{a}}$hler manifold.

For $\gamma\in\Delta_{\mathbf{L}}$, let $j$ be the smallest $1\leq j\leq d$
such that $\gamma\preceq\sigma_{j}$. Then
$\gamma\in\Delta_{j}\setminus\Delta_{j-1}$. Thus it follows that
$\mu_{j}\preceq\gamma\preceq\sigma_{j}$. Hence there exists a unique $j$,
$1\leq j\leq d$ such that $\gamma\in[\mu_{j},\sigma_{j}]$ where
$[\mu_{j},\sigma_{j}]:=\\{\gamma~{}~{}\mid~{}~{}\mu_{j}\preceq\gamma\preceq\sigma_{j}\\}$.
In other words we can write

(7.1)
$\Delta_{\mathbf{L}}=[\mu_{1},\sigma_{1}]\sqcup\cdots\sqcup[\mu_{d},\sigma_{d}].$

If a simplicial complex $\Delta_{\mathbf{L}}$ satisfies (7.1) then it is
called partitionable (see [S, p.80, Section 2.1]). In particular, shellable
simplicial complexes are partitionable.

Moreover,

(7.2) $\mu_{i}\preceq\sigma_{j}\Rightarrow j\geq i.$

Let

$\mathcal{E}_{{\bf
p}_{\sigma_{i}}}=\\{\epsilon^{+}_{i_{1}},\ldots,\epsilon^{+}_{i_{n}},\epsilon^{-}_{i_{1}},\ldots,\epsilon^{-}_{i_{n}}\\}$

for $1\leq i\leq d$. Recall from Definition 2.4 that the set
$\\{\alpha(\epsilon_{i_{j}}^{+}),x\mid j=1,\ldots,n\\}$ spans
$\mathfrak{t}_{\mathbb{Z}}^{*}\oplus\mathbb{Z}x$ i.e.,

(7.3)
$\langle\alpha(\epsilon_{i_{1}}^{+}),\ldots,\alpha(\epsilon_{i_{n}}^{+}),x\rangle=\mathfrak{t}_{\mathbb{Z}}^{*}\oplus\mathbb{Z}x.$

Let $\widetilde{\mathcal{G}}$ denote the $x$-forgetful graph associated to
$\mathcal{G}$ where
$\widetilde{\mathcal{G}}=(\Gamma,\widetilde{\alpha},\nabla)$ having $\Gamma$
and $\nabla$ same as $\mathcal{G}$ and $\widetilde{\alpha}$ is the
$x$-forgetful axial function defined as
$\widetilde{\alpha}=F\circ\alpha:\mathcal{E}\longrightarrow H^{2}(BT^{n})$
where $F:H^{2}(BT^{n})\bigoplus\mathbb{Z}x\longrightarrow H^{2}(BT^{n})$ is
the $x$-forgetful map (see Section 5.2). Recall that

$\displaystyle\mathbb{Z}[\widetilde{\mathcal{G}}]:=\frac{\mathbb{Z}[L_{1},\ldots,L_{m}]}{\langle\prod_{L\in\mathbf{L}^{\prime}}L\mid\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})\rangle}$

where
$\displaystyle\mathbf{I}(\mathbf{L})=\\{\mathbf{L}^{\prime}\subseteq\mathbf{L}\mid\bigcap_{L\in\mathbf{L}^{\prime}}L=\emptyset\\}$.

Let $x_{\gamma}$ denote the monomial

$x_{\gamma}:=\prod_{j=1}^{p}L_{i_{j}}$

where $\gamma=\langle
v_{i_{1}},\ldots,v_{i_{p}}\rangle\in\Delta_{\mathbf{L}}$.

###  7.2. The characteristic function associated to the hyperplane $L$

######  Definition 7.3.

Let $L$ be a connected $(2n-2)$-valent hyperplane in $\mathcal{G}$. Let $H$
and $\overline{H}$ be the unique halfspace such that $L=H\cap\overline{H}$.

For $p\in\mathcal{V}^{L}$,

$\mathcal{E}^{L}_{p}=\\{\epsilon_{1}^{+},\ldots,\epsilon_{n-1}^{+},\epsilon_{1}^{-},\ldots,\epsilon_{n-1}^{-}\\}$

is the $(n-1)$-pairs and

$\mathcal{E}^{\Gamma}_{p}=\\{\epsilon_{1}^{+},\ldots,\epsilon_{n}^{+},\epsilon_{1}^{-},\ldots,\epsilon_{n}^{-}\\}$

is the $n$-pairs so that $n_{H}(p)=\epsilon_{n}^{+}$ and
$n_{\overline{H}}(p)=\epsilon_{n}^{-}$.

By (7.3), the axial functions
$\widetilde{\alpha}(\epsilon^{+}_{1}),\widetilde{\alpha}(\epsilon^{+}_{2}),\ldots,\widetilde{\alpha}(\epsilon^{+}_{n})$
form a basis for $(\mathfrak{t}^{n}_{\mathbb{Z}})^{*}$. The characteristic
function associated to $L$ is defined as the unique element
$\lambda(L)\in\mathfrak{t}^{n}_{\mathbb{Z}}$ such that

$\displaystyle\langle\widetilde{\alpha}(\epsilon^{+}_{i}),\lambda(L)\rangle=\delta_{i,n}.$

######  Lemma 7.4.

The definition of $\lambda(L)$ is independent of the choice of a vertex
$p\in\mathcal{V}^{L}$.

###### Proof.

Let $\epsilon=pq\in\mathcal{E}^{L}$, in particular let
$\epsilon=\epsilon_{j}^{+}$ for some $1\leq j\leq n-1$. Here $i(\epsilon)=p$
and $t(\epsilon)=q$. Then under the connection ${\nabla}_{\epsilon}$,
$\mathcal{E}_{p}^{\Gamma}$, the set of edges around $p$, maps bijectively onto
$\mathcal{E}_{q}^{\Gamma}$, the set of edges around $q$. Since the hyperplane
$L$ is closed under the connection $\nabla$, $\mathcal{E}_{p}^{L}$ maps
bijectively onto $\mathcal{E}_{q}^{L}$. Moreover, since a halfspace $H$ is
closed under $\nabla$, it follows that
${\nabla_{\epsilon}}(n_{H}(p))=n_{H}(q)$ so that
$\widetilde{\alpha}({\nabla_{\epsilon}}(n_{H}(p)))\equiv\widetilde{\alpha}(n_{H}(q))\mod\widetilde{\alpha}(\epsilon)$.
Moreover, since $\epsilon\in\mathcal{E}^{L}$ and
$\nabla_{\epsilon}(\epsilon)=\overline{\epsilon}$ by definition of
$\nabla_{\epsilon}$ it follows that the elements
$\widetilde{\alpha}(\epsilon^{+}_{1}),\ldots,\widetilde{\alpha}(\epsilon_{n-1}^{+})$
and
$\widetilde{\alpha}({\nabla}_{\epsilon}(\epsilon^{+}_{1})),\ldots,\widetilde{\alpha}({\nabla}_{\epsilon}(\epsilon_{n-1}^{+}))$
span the same subspace of $(\mathfrak{t}^{n}_{\mathbb{Z}})^{*}$. Further,
since $\epsilon\in\mathcal{E}^{L}$ and
$\langle\widetilde{\alpha}(\epsilon),\lambda(L)\rangle=0$, by the congruence
relation we have

$\langle\widetilde{\alpha}({\nabla}_{\epsilon}(\epsilon_{n}^{+})),\lambda(L)\rangle=\langle\widetilde{\alpha}(\epsilon_{n}^{+}),\lambda(L)\rangle.$

Thus for the $n$-pairs
$\mathcal{E}^{\Gamma}_{q}=\\{\nabla_{\epsilon}(\epsilon_{1}^{+}),\ldots,\nabla_{\epsilon}(\epsilon_{n}^{+}),\nabla_{\epsilon}(\epsilon_{1}^{-}),\ldots,\nabla_{\epsilon}(\epsilon_{n}^{-})\\}$
we have

$\langle\lambda(L),\widetilde{\alpha}(\nabla_{\epsilon}(\epsilon_{i}^{+}))\rangle=\delta_{i,n}.$

Hence without loss of generality we could have started with the vertex
$q\in\mathcal{V}^{L}$ to define $\lambda(L)$. Moreover, since $L$ is
connected, by repeating the above procedure for an edge $\epsilon^{\prime}$
such that $i(\epsilon^{\prime})=q$, it follows that the definition of
$\lambda(L)$ is independent of the choice of $p\in\mathcal{V}^{L}$. ∎

###  7.3. The $H^{*}(BT^{n})$-algebra structure of
$H^{*}(\widetilde{\mathcal{G}})$

Since $\displaystyle
H^{*}(\widetilde{\mathcal{G}})\subset\bigoplus_{p\in\mathcal{V}}H^{*}_{T}(p)\simeq\bigoplus_{p\in\mathcal{V}}H^{*}(BT^{n})$,
the ring $H^{*}(\widetilde{\mathcal{G}})$ may be regarded as the
$H^{*}(BT^{n})$-submodule of
$\displaystyle\bigoplus_{p\in\mathcal{V}}H^{*}(BT^{n})$. In Theorem 7.6 of
this section, which is the second main theorem of this paper, we determine
module generators of $H^{*}(\widetilde{\mathcal{G}})$ as a
$H^{*}(BT^{n})$-module. For this purpose, we begin with the following lemma
(also see [MMP] for the corresponding statement on torus graphs).

######  Lemma 7.5.

(i) The $H^{*}(BT^{n})$-module structure on $H^{*}(\widetilde{\mathcal{G}})$
is obtained from the following map from $H^{2}(BT^{n})$ to
$H^{*}(\widetilde{\mathcal{G}})$:

$\displaystyle H^{2}(BT^{n})\ni u\mapsto\displaystyle\sum_{i=1}^{m}\langle
u,\lambda(L_{i})\rangle\cdot\tau_{L_{i}}\in H^{*}(\widetilde{\mathcal{G}}).$

Moreover,
$\Psi^{\prime}:H^{*}(\widetilde{\mathcal{G}})\to\mathbb{Z}[\widetilde{\mathcal{G}}]$
is an isomorphism of $H^{*}(BT^{n})$-algebras where the algebra structure on
$\mathbb{Z}[\widetilde{\mathcal{G}}]$ is obtained by sending $u\in
H^{2}(BT^{n})$ to the element $\displaystyle\sum_{i=1}^{m}\langle
u,\lambda(L_{i})\rangle\cdot L_{i}\in\mathbb{Z}[\widetilde{\mathcal{G}}]$.

(ii) We therefore have the following presentation for
$H^{*}(\widetilde{\mathcal{G}})$ as an $H^{*}(BT^{n})$-algebra:

$H^{*}(\widetilde{\mathcal{G}})\simeq\frac{H^{*}(BT^{n})[L_{1},\ldots,L_{m}]}{\langle\prod_{L\in\mathbf{L}^{\prime}}L\mid\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})~{};~{}\sum_{i=1}^{m}\langle
u,\lambda(L_{i})\rangle\cdot L_{i}-u,~{}~{}\forall~{}u\in
H^{2}(BT^{n})\rangle}$

###### Proof.

Let $p\in\mathcal{V}^{L}$ where $p=L_{i_{1}}\cap\cdots\cap L_{i_{n}}$. Then by
Section 5.1 we have

(7.4) $\sum_{i=1}^{m}\langle
u,\lambda(L_{i})\rangle\cdot\tau_{L_{i}}(p)=\sum_{j=1}^{n}\langle
u,\lambda(L_{i_{j}})\rangle\cdot\widetilde{\alpha}(n_{H_{i_{j}}}(p)).$

Note that $n_{H_{i_{j}}}(p)=\epsilon_{j}^{+}(p)$ for $1\leq j\leq n$ so that
$\widetilde{\alpha}(n_{H_{i_{j}}}(p))$ for $1\leq j\leq n$ form a basis of
$\mathfrak{t}^{*}_{\mathbb{Z}}$. Since
$\lambda(L_{i_{j}})\in\mathfrak{t}_{\mathbb{Z}}$ for $1\leq j\leq n$ is the
corresponding dual basis, the right hand side of (7.4) is nothing but $u$.
Thus

(7.5) $(\sum_{i=1}^{m}\langle
u,\lambda(L_{i})\rangle\cdot\tau_{L_{i}})(p)=u~{}~{}\mbox{for
every}~{}~{}p\in\mathcal{V}.$

Since $p\in\mathcal{V}$ was arbitrary from Section 5.3 it follows that the
$H^{*}(BT^{n})$-algebra structure defined above is canonical corresponding to
the diagonal inclusion of $H^{*}(BT^{n})$ in
$\bigoplus_{i=1}^{d}H^{*}(BT^{n})=(H^{*}(BT^{n}))^{d}$.

Finally, by definition of $\Psi^{\prime}$ in Section 5.2, we also have that
the $H^{*}(BT^{n})$-algebra structure on $\mathbb{Z}[\widetilde{\mathcal{G}}]$
is obtained as in the statement. ∎

The following theorem is the second main theorem in this paper.

######  Theorem 7.6.

Let $\Delta_{\mathbf{L}}$ be the simplicial complex defined by the hyperplanes
$\\{L_{1},\ldots,L_{m}\\}$ of $\widetilde{\mathcal{G}}$. Suppose that
$\Delta_{\mathbf{L}}$ is a shellable simplicial complex with respect to
ordering $\sigma_{1},\ldots,\sigma_{d}$ of $\Delta_{\mathbf{L}}(n)$. In
particular, $\Delta_{\mathbf{L}}$ is partitionable with partition (7.1) and
(7.2) holds. Then the following statements hold:

(i) For $\gamma\in\Delta_{\mathbf{L}}$, let
$x_{\gamma}={L_{j_{1}}}\cdots{L_{j_{p}}}\in\mathbb{Z}[\widetilde{\mathcal{G}}]$
where $\gamma=\langle v_{j_{1}},\ldots,v_{j_{p}}\rangle$. Then there exists an
element $u\in H^{2}(BT^{n})$ such that

$\displaystyle L_{j_{1}}\cdot x_{\gamma}=-\sum_{k}\langle
u,\lambda(L_{j_{k}})\rangle\cdot x_{\gamma_{k}}+u\cdot x_{\gamma}$

where $k$ runs through $1\leq k\leq m$ such that
$k\not\in\\{j_{1},\ldots,j_{n}\\}$ and $\gamma_{k}=\langle
v_{k},v_{j_{1}},\ldots,v_{j_{p}}\rangle$.

(ii) Let $\eta\preceq\gamma\preceq\theta$ be simplices in
$\Delta_{\mathbf{L}}$. Then we can write

$x_{\gamma}=\sum_{k}c_{k}\cdot x_{\eta_{k}}+c\cdot x_{\eta}$

for $c_{k},c\in H^{*}(BT^{n})$ and $\eta_{k}\npreceq\theta$.

(iii) The monomials $x_{\mu_{i}}$ for $1\leq i\leq d$ form a basis of
$\mathbb{Z}[\widetilde{\mathcal{G}}]$ as $H^{*}(BT^{n})$-module.

(iv) Let $f\in\mathbb{Z}[\widetilde{\mathcal{G}}]$ and

(7.6) $f=\sum_{j=1}^{d}a_{j}\cdot x_{\mu_{j}}$

for unique $a_{j}\in H^{*}(BT^{n})$. Let $i=i(f)$ be the smallest $1\leq i\leq
d$ such that $a_{i}\neq 0$. Then we can determine the coefficients $a_{j}$,
$j\geq i$ iteratively as follows: We have
$a_{i}=\frac{\rho_{\mathbf{p}_{\sigma_{i}}(f)}}{\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{i}})}$.
Suppose $a_{i},a_{i+1},\ldots,a_{j-1}$ are determined by induction then

$\displaystyle
a_{j}=\frac{\rho_{\mathbf{p}_{\sigma_{j}}}(f-\sum_{k=i}^{j-1}a_{k}\cdot
x_{\mu_{k}})}{\rho_{\mathbf{p}_{\sigma_{j}}}(x_{\mu_{j}})}.$

###### Proof.

(i) Let $\sigma=\langle v_{j_{1}},\ldots,v_{j_{n}}\rangle$ be an
$(n-1)$-simplex containing $\gamma$ and let ${\bf
p}_{\sigma}=L_{j_{1}}\cap\cdots\cap L_{j_{n}}$ be the corresponding vertex in
$\widetilde{\mathcal{G}}$. Consider
$\lambda(L_{j_{1}})=e_{j_{1}}\in\mathfrak{t}^{n}_{\mathbb{Z}}$ which is dual
to $u:=\widetilde{\alpha}(\epsilon^{+}_{j_{1}})$. In this case, by Lemma 7.5,
we have that

$\displaystyle u=\sum_{i=1}^{m}\langle u,\lambda(L_{i})\rangle\cdot L_{i}.$

Since $\langle u,e_{j_{i}}\rangle=\delta_{i,1}$ for $i=1,\ldots,n$, we have
the relation

(7.7) $L_{j_{1}}=-\sum_{k}\langle u,\lambda(L_{k})\rangle\cdot L_{k}+u$

as an $H^{*}(BT^{n})$-module, where the sum on the right hand side is over
those $1\leq k\leq m$ such that $k\notin\\{j_{1},\ldots,j_{n}\\}$. Multiplying
(7.7) by $x_{\gamma}$ we get

(7.8) $L_{j_{1}}\cdot x_{\gamma}=-\sum\langle u,\lambda(L_{k})\rangle\cdot
x_{\gamma_{k}}+u\cdot x_{\gamma}$

where $\gamma_{k}\in\Delta_{\mathbf{L}}$ is spanned by the vertices
$v_{k},v_{j_{1}},\ldots,v_{j_{p}}$ and $x_{\gamma_{k}}=L_{k}\cdot
L_{j_{1}}\cdots L_{j_{p}}$. This proves (i)

(ii) Let $\eta\prec\gamma\preceq\theta$ be simplices in $\Delta_{\mathbf{L}}$.
Let $\gamma=\langle v_{j_{1}},\ldots,v_{j_{p}}\rangle$, $\eta=\langle
v_{j_{1}},\ldots,v_{j_{r}}\rangle\in\Delta_{\mathbf{L}}$ and $\theta=\langle
v_{j_{1}},\ldots,v_{j_{l}}\rangle$ and $r<p\leq l$. Because the following
argument can also apply for any such $l$, we can assume that $l=n$ i.e.,
$\theta$ is $(n-1)$-dimensional. Thus
$\lambda(L_{j_{1}}),\ldots,\lambda(L_{j_{n}})$ is the basis of
$\mathfrak{t}_{\mathbb{Z}}$ dual to the basis
$\widetilde{\alpha}(\epsilon_{j_{1}}),\ldots,\widetilde{\alpha}(\epsilon_{j_{n}})$
of $\mathfrak{t}^{*}_{\mathbb{Z}}$. Let
$u:=\widetilde{\alpha}(\epsilon_{j_{p}})\in H^{*}(BT^{n})$. Thus, with the
similar reason to obtain (7.7) as in the proof of (i), we have the following
relation in $\mathbb{Z}[\widetilde{\mathcal{G}}]$:

(7.9) $L_{j_{p}}+\sum_{k}\langle u,\lambda(L_{k})\rangle\cdot L_{k}-u=0$

where $k$ runs through $\\{1,\ldots,m\\}\setminus\\{j_{1},\ldots,j_{n}\\}$ in
the equation. Multiplying (7.9) by ${L_{j_{1}}}\cdots{L_{j_{p-1}}}$, we get

(7.10) $x_{\gamma}+\sum_{k}\langle u,\lambda(L_{k})\rangle\cdot
x_{\eta_{k}}-u\cdot x_{\gamma^{\prime}}=0$

where $\eta_{k}=\langle v_{k},v_{j_{1}},\ldots,v_{j_{p-1}}\rangle$,
$\gamma^{\prime}=\langle
v_{j_{1}},\ldots,v_{j_{p-1}}\rangle\in\Delta_{\mathbf{L}}$. Note that
$\eta_{k}\npreceq\theta$ since $k\notin\\{j_{1},\ldots,j_{n}\\}$. Also
$\eta\preceq\eta_{k}$ and $\eta\preceq\gamma^{\prime}$, since $r\leq p-1$.
Therefore, $\eta\preceq\gamma^{\prime}\prec\gamma\preceq\theta$. Now,
proceeding by downward induction on $p$ and repeating the above arguments for
$\gamma^{\prime}$ we arrive at (ii).

(iii) By the ring structure of $\mathbb{Z}[\widetilde{\mathcal{G}}]$ defined
in Section 5.2, for every element in $\mathbb{Z}[\widetilde{\mathcal{G}}]$ can
be written by the sum of $x_{\gamma}$’s for $\gamma\in\Delta_{\mathbf{L}}$
with $H^{*}(BT^{n})$-coefficients. Therefore, for every
$\gamma\in\Delta_{\mathbf{L}}$, it suffices to show that $x_{\gamma}$ lies in
the $H^{*}(BT^{n})$-submodule of $\mathbb{Z}[\widetilde{\mathcal{G}}]$ spanned
by $x_{\mu_{i}}$ for $1\leq i\leq d$, where $\mu_{i}$ is the minimal face
which appears in
$\Delta_{\mathbf{L}}=[\mu_{1},\sigma_{1}]\sqcup\cdots\sqcup[\mu_{d},\sigma_{d}]$.
Since $\Delta_{\mathbf{L}}$ is a shellable simplicial complex, for every
$\gamma\in\Delta_{\mathbf{L}}$ there exists the unique $1\leq i\leq d$ such
that $\mu_{i}\preceq\gamma\preceq\sigma_{i}$ (see (7.2)). We prove (iii) by
downward induction on $i$.

If $\gamma\in[\mu_{d},\sigma_{d}]$, we are done since
$\mu_{d}=\sigma_{d}=\gamma$, i.e., $x_{\gamma}=x_{\mu{d}}$ and hence lies in
the $H^{*}(BT^{n})$-span of $x_{\mu_{d}}$. Assume that for every
$\gamma\in[\mu_{i+1},\sigma_{i+1}]\sqcup\cdots\sqcup[\mu_{d},\sigma_{d}]$,
$x_{\gamma}\in H^{*}(BT^{n})x_{\mu_{i+1}}\oplus\cdots\oplus
H^{*}(BT^{n})x_{\mu_{d}}$. If $\gamma\in[\mu_{i},\sigma_{i}]$, then by (ii) we
can write

(7.11) $x_{\gamma}=\sum_{\mu_{i}\prec\gamma_{j}\nprec\sigma_{i}}c_{j}\cdot
x_{\gamma_{j}}+c\cdot x_{\mu_{i}}$

for $c_{j},c\in H^{*}(BT^{n})$. Now there is the unique $r$ such that
$\mu_{r}\preceq\gamma_{j}\preceq\sigma_{r}$. This implies by (7.2) that $r>i$.

Thus by induction assumption $x_{\gamma_{j}}$ lies in the $H^{*}(BT^{n})$-span
of $x_{\mu_{q}}$ for $q\geq r$. This together with (7.11) implies that
$x_{\gamma}$ lies in the $H^{*}(BT^{n})$-span of $x_{\mu_{q}}$ for $q\geq i$.

It remains now to show that $x_{\mu_{i}}$ for $1\leq i\leq d$ are linearly
independent. Suppose that there exist $a_{i}\in H^{*}(BT^{n})$ for $1\leq
i\leq d$ such that

(7.12) $\sum_{i=1}^{d}a_{i}\cdot x_{\mu_{i}}=0$

in $\mathbb{Z}[\widetilde{\mathcal{G}}]$. Let $i\in\\{1,\ldots,n\\}$ be the
smallest integer such that $a_{i}\neq 0$.

Recall that $\sigma_{i}=\langle v_{i_{1}},\ldots,v_{i_{n}}\rangle$ where
$\mathbf{p}_{\sigma_{i}}=L_{i_{1}}\cap\cdots\cap L_{i_{n}}$ in $\mathcal{G}$.
Consider the localization map
$\rho=(\rho_{\mathbf{p}_{\sigma_{j}}})_{j=1}^{d}$ defined in Section 5.3. By
(7.2) and the definition of $\rho_{\mathbf{p}_{\sigma_{i}}}$ it follows that
$\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{j}})=0$ for $j>i$ (since
$\mu_{j}\npreceq\sigma_{i}$ there exists $L_{k}$ in $\mathbf{L}$ such that the
corresponding vertex $v_{k}\in\mu_{j}$ and $v_{k}\notin\sigma_{i}$ in
$\Delta_{\mathbf{L}}$. Thus $\rho_{\mathbf{p}_{\sigma_{i}}}(L_{k})=0$ in
$\mathbb{Z}[\widetilde{\mathcal{G}}]_{\mathbf{p}_{\sigma_{i}}}$). Thus
applying $\rho_{\mathbf{p}_{\sigma_{i}}}$ on (7.12) we get

$\rho_{\mathbf{p}_{\sigma_{i}}}(\sum_{j=1}^{n}a_{j}\cdot
x_{\mu_{j}})=\rho_{\mathbf{p}_{\sigma_{i}}}(\sum_{j\geq i}a_{j}\cdot
x_{\mu_{j}})=\rho_{\mathbf{p}_{\sigma_{i}}}(a_{i}\cdot
x_{\mu_{i}})=\rho_{\mathbf{p}_{\sigma_{i}}}(a_{i})\cdot\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{i}})=0$

in the integral domain
$\mathbb{Z}[\widetilde{\mathcal{G}}]_{\mathbf{p}_{\sigma_{i}}}\simeq\mathbb{Z}[L_{i_{1}},\ldots,L_{i_{n}}]$.
Since ${\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{i}})}$ is the monomial
$L_{i_{j_{1}}},\ldots,L_{i_{j_{p}}}$, where $\mu_{i}=\langle
v_{i_{j_{1}}},\ldots,v_{i_{j_{p}}}\rangle$, and hence a non-zero element of
$\mathbb{Z}[L_{i_{1}},\ldots,L_{i_{n}}]$, we get that
$\rho_{\mathbf{p}_{\sigma_{i}}}(a_{i})=0$. Moreover, $\rho$ can be seen to be
the diagonal embedding when restricted to the subalgebra $H^{*}(BT^{n})$ of
$\mathbb{Z}[\widetilde{\mathcal{G}}]$ ($u\in H^{*}(BT^{n})$ is equal to
$\displaystyle\sum_{j=1}^{m}\langle u,\lambda({L_{j}})\rangle\cdot
L_{j}\in\mathbb{Z}[\widetilde{\mathcal{G}}]$ maps to
$\displaystyle\sum_{j=1}^{n}\langle u,\lambda({L_{i_{j}}})\rangle\cdot
L_{i_{j}}\in\mathbb{Z}[L_{i_{1}},\ldots,L_{i_{n}}]$ which is identified with
$\displaystyle\sum_{j=1}^{n}\langle
u,\lambda({L_{i_{j}}})\rangle\cdot\tau_{L_{i_{j}}}$ in $H^{*}(BT^{n})$ (see
Lemma 5.2) which is equal to $u$ see (7.5)) and $\rho$ is injective by Lemma
5.3, which implies that $a_{i}=0$. This contradicts our original assumption
that $a_{i}\neq 0$. Thus we cannot have a relation of the type (7.12) in
$\mathbb{Z}[\widetilde{\mathcal{G}}]$ unless $a_{i}=0$ in $H^{*}(BT^{n})$ for
each $1\leq i\leq d$. Hence we conclude that $x_{\mu_{i}}$ for $1\leq i\leq d$
are linearly independent in $\mathbb{Z}[\widetilde{\mathcal{G}}]$. This proves
(iii).

(iv) As for the proof of linear independence of $x_{\mu_{i}}$ $1\leq i\leq d$
our idea is to again use the localization map
$\displaystyle\rho=(\rho_{\mathbf{p}_{\sigma_{i}}}):\mathbb{Z}[\widetilde{\mathcal{G}}]\longrightarrow\bigoplus_{i=1}^{d}\mathbb{Z}[\widetilde{\mathcal{G}}]_{\mathbf{p}_{\sigma_{i}}}(\simeq\bigoplus_{i=1}^{d}H^{*}(BT^{n}))$
defined in Section 5.3. We know that $\rho$ is injective and by (7.2) and the
definition of $\rho_{\mathbf{p}_{\sigma_{i}}}$ we have
$\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{j}})=0$ for $j>i$. Also $\rho$ is the
diagonal map when restricted to the subalgebra $H^{*}(BT^{n})$ of
$\mathbb{Z}[\widetilde{\mathcal{G}}]$. In particular, this implies that
$\rho_{\mathbf{p}_{\sigma_{j}}}(a_{k})=a_{k}$ for $1\leq j,k\leq d$.

Applying $\rho_{\mathbf{p}_{\sigma_{i}}}$ on (7.6) we get

$\rho_{\mathbf{p}_{\sigma_{i}}}(f)=\rho_{\mathbf{p}_{\sigma_{i}}}(\sum_{j=1}^{d}a_{j}\cdot
x_{\mu_{j}})=\rho_{\mathbf{p}_{\sigma_{i}}}(\sum_{j\geq i}a_{j}\cdot
x_{\mu_{j}})=a_{i}\cdot\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{i}})$

in the unique factorization domain $\mathbb{Z}[L_{i_{1}},\ldots,L_{i_{n}}]$
where $\mathbf{p}_{\sigma_{i}}=L_{i_{1}}\cap\cdots\cap L_{i_{n}}$. Thus
$\rho_{\mathbf{p}_{\sigma_{i}}}(f)$ is divisible by the irreducible elements
$L_{i_{j_{1}}},\ldots,L_{i_{j_{p}}}$ and hence by the monomial
$\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{i}})=L_{i_{j_{1}}}\cdots
L_{i_{j_{p}}}$ in $\mathbb{Z}[L_{i_{1}},\ldots,L_{i_{n}}]$. Thus
$a_{i}=\frac{\rho_{\mathbf{p}_{\sigma_{i}}}(f)}{\rho_{\mathbf{p}_{\sigma_{i}}}(x_{\mu_{i}})}\in\mathbb{Z}[L_{i_{1}},\ldots,L_{i_{n}}]$.
Now, let

$f_{1}:=f-a_{i(f)}\cdot x_{\mu_{i(f)}}\in\mathbb{Z}[\widetilde{\mathcal{G}}].$

Then $\displaystyle f_{1}=\sum_{j>i(f)}a_{j}\cdot x_{\mu_{j}}.$ Moreover, now
putting $i=i(f_{1})$ and repeating the above argument given for determining
$a_{i(f)}$ we get

$a_{i(f_{1})}=\frac{\rho_{\mathbf{p}_{\sigma_{i(f_{1})}}}(f_{1})}{\rho_{\mathbf{p}_{\sigma_{i(f_{1})}}}(x_{\mu_{i(f_{1})}})}$

in $\mathbb{Z}[\widetilde{\mathcal{G}}]_{\mathbf{p}_{\sigma_{i(f_{1})}}}\simeq
H^{*}(BT^{n})$. Proceeding similarly after $k$ steps we get

$\displaystyle f_{k}=f-\sum_{i(f_{1})\leq j<i(f_{k})}a_{j}\cdot
x_{\mu_{j}}=\sum_{j>i(f_{k-1})}a_{j}\cdot x_{\mu_{j}}.$

Putting $i=i(f_{k})$ to be the smallest index in
$\\{i(f_{k-1}),i(f_{k-1})+1,\ldots,d\\}$ such that $a_{i(f_{k})}\neq 0$ and
following similar arguments as above we get

$a_{i(f_{k})}=\frac{\rho_{\mathbf{p}_{\sigma_{i(f_{k})}}}(f_{k})}{\rho_{\mathbf{p}_{\sigma_{i(f_{k})}}}(x_{\mu_{i(f_{k})}})}$

in $\mathbb{Z}[\widetilde{\mathcal{G}}]_{\mathbf{p}_{\sigma_{i(f_{k})}}}\simeq
H^{*}(BT^{n})$. This proves (iv).

∎

###  7.4. The ordinary cohomology ring of $\widetilde{\mathcal{G}}$ and an
example

In geometry, if the equivariant cohomology $H_{T}^{*}(M;\mathbb{Z})$ has the
structure of a free $H^{*}(BT)$-algebra, then we can compute its ordinary
cohomology $H^{*}(M;\mathbb{Z})$ by
$H_{T}^{*}(M;\mathbb{Z})\otimes_{H^{*}({BT})}\mathbb{Z}$. By Theorem 7.6, we
know that $H^{*}(\widetilde{\mathcal{G}})$ is a free $H^{*}(BT)$-algebra of
rank $d$. So we may define the “ordinary” cohomology of
$\widetilde{\mathcal{G}}$ by
$H^{*}(\widetilde{\mathcal{G}})\otimes_{H^{*}({BT})}\mathbb{Z}$; we denote it
by $H^{*}_{{ord}}(\widetilde{\mathcal{G}})$. The precise computation of
$H^{*}_{{ord}}(\widetilde{\mathcal{G}})$ is given as the following corollary.

######  Corollary 7.7.

(i) The following is the presentation

$H_{ord}^{*}(\widetilde{\mathcal{G}})\simeq\frac{\mathbb{Z}[L_{1},\ldots,L_{m}]}{\langle\prod_{L\in\mathbf{L}^{\prime}}L\mid\mathbf{L}^{\prime}\in\mathbf{I}(\mathbf{L})~{};~{}\sum_{i=1}^{m}\langle
u,\lambda(L_{i})\rangle\cdot L_{i},~{}~{}\forall~{}u\in H^{2}(BT^{n})\rangle}$

for the ordinary cohomology ring $H^{*}_{{ord}}(\widetilde{\mathcal{G}})$ as a
$\mathbb{Z}$-algebra.

(ii) The monomials $x_{\mu_{i}}$, $1\leq i\leq d$ form a $\mathbb{Z}$-basis
for $H^{*}_{{ord}}(\widetilde{\mathcal{G}})$.

###### Proof.

(i) The $H^{*}(BT^{n})$-algebra structure on $H^{*}(\widetilde{\mathcal{G}})$
is given by Lemma 7.5 and $\mathbb{Z}$ has $H^{*}({BT})$-algebra structure
given by augmentation which sends each $u\in H^{2}(BT^{n})$ to $0$. Since
$H^{*}(\widetilde{\mathcal{G}})$ is free, the corollary now follows from Lemma
7.5 due to the $H^{*}({BT})$-algebra isomorphism $\Psi^{\prime}$ of
$\mathbb{Z}[\widetilde{\mathcal{G}}]$ with $H^{*}(\widetilde{\mathcal{G}})$.

(ii) This follows by Theorem 7.6 (iii) and by the isomorphism

$H_{ord}^{*}(\widetilde{\mathcal{G}})\simeq\mathbb{Z}[\widetilde{\mathcal{G}}]\otimes_{H^{*}(BT^{n})}\mathbb{Z}.$

∎