EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWOFIXED POINTS
ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONG

Abstract. The main purpose of this paper is to introduce the main result of [6] and apply
it to show a ring structure of the equivariant cohomology of a certain class of torus orbifolds
which have exactly two fixed points. We also show that a torus orbifold over the suspensionof an (n − 1)-dimensional simplex is equivariantly homeomorphic to the 2n-dimensional sphere
quotiented by a product of cyclic groups.

1. Introduction
The relationship between the equivariant cohomology of certain smooth manifolds M 2n , withhalf-dimensional torus actions Tn = S 1 ×· · ·×S 1 , and the combinatorics of their quotient spaces iswell-known. A toric manifold, defined as a compact non-singular toric variety, admits the locallystandard action of a half-dimensional torus. This implies that its quotient space is a manifold withfaces (see Definition 2.3). The equivariant cohomology is then simply given as the face-ring of thisquotient space. This can also be realized as the face-ring of the complete regular fan associatedto the toric variety. A quasitoric manifold, first defined in [9], is a topological generalization of atoric manifold whose quotient space is a simple polytope (an example of a manifold with faces)
but which is more general than a projective toric variety. Its equivariant cohomology is then givenby the face-ring associated to the simple polytope.
Torus manifolds, appearing in [14], are a wider class of manifolds M 2n , with half-dimensional
torus action, that contain both toric and quasitoric manifolds. The main example of a torus
manifold that is neither a toric nor a quasitoric manifold is the even-dimensional sphere S 2n ⊂Cn ⊕ R, where the torus acts coordinatewise on the first n coordinates. To any torus manifold we
can associate a combinatorial object, called a torus graph (see [17]), which is an n-valent graphwhose vertices correspond to the fixed points of M and whose edges are labelled by irreducible
torus representations. The underlying graph for S 2n is the one with exactly two vertices and nedges between them. We can then calculate the graph equivariant cohomology of the torus graph,
which is a ring of piecewise polynomials on the graph. When the ordinary cohomology of M istrivial in all odd degrees, then the torus action is locally standard and the equivariant cohomologyof M is isomorphic to the graph cohomology of the associated torus graph. It is also possible togive explicit generators and relations for this ring via Thom classes of the torus graph and we cansee that this is isomorphic to the face-ring of the quotient space which is, in this case, a manifoldwith faces.
When we move from manifolds to orbifolds the picture slightly changes. In the case of singulartoric varieties, including toric varieties having orbifold singularities, it was proved in [10] that its
equivariant cohomology with integer coeﬃcients is given by the piecewise polynomials on its fan if
its ordinary odd degree cohomology vanishes. For toric manifolds the ring of piecewise polynomials
on the fan is isomorphic to the face-ring of its quotient but this is not true for orbifolds in general.
In [5] it is shown that the equivariant cohomology of projective toric orbifolds and quasitoricorbifolds, under the condition of vanishing odd degree cohomology, can be realized as a subring of
the usual face-ring of the orbit space/fan that satisfies an intergrality condition.
2010 Mathematics Subject Classification. Principal: 55N32, Secondly: 57R18; 13F55.
Key words and phrases. orbifold; torus action; torus orbifold; equivariant cohomology; face ring.
The second author was supported by JSPS KAKENHI Grant Number 17K14196. The third author has been
supported by the POSCO Science Fellowship of POSCO TJ Park Foundation.
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ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONG

In this paper we give the main result from [6] where we focus on torus orbifolds. Let T n be
an n-dimensional torus, i.e., an n-dimensional commutative, compact, connected Lie group. A
torus orbifold is a 2n-dimensional, closed, oriented orbifold with an eﬀective T n -action whose
fixed point set is non-empty. This notion is a generalization of torus manifolds as introduced by
Hattori-Masuda in [14].
In Section 2 we introduce some basic definitions that we need including that of an orbifoldwith group action which our main objects, torus orbifolds, are an example of. We then give a
constructive definition of an orbifold over a combinatorial object known as a manifold with faces.
These will be examples of torus orbifolds and their quotient spaces are manifolds with faces. We
restrict out attention to these cases.
The suspension of an (n − 1)-simplex gives the simplest example of an n-dimensional manifoldwith faces – one with exactly two vertices. In Section 3 we show that every manifold with exactly
two vertices is equivalent, in a combinatorial sense, to the suspension of a simplex and then showthat a torus orbifold over it is equivariantly homeomorphic to the quotient of an even dimensional
sphere by a product of cyclic groups.
In Section 4 we give the main result from [6] which computes the equivariant cohomology ofa torus orbifold X whose odd degree ordinary cohomology is trivial. To do this we associate toeach torus orbifold over a manifold with faces a labelled graph, called a torus orbifold graph, which
generalizes the torus graphs of [17]. We can then describe the equivariant cohomology of X as thering of piecewise polynomials on the associated torus orbifold graph if H odd (X) = 0. We showthat this is isomorphic to a weighted face ring that gives us a description of HT∗ (X) in terms ofgenerators and relations. Restricting to torus orbifolds X over the suspension of a simplex wecompute these ring structures explicitly.
2. Torus orbifolds over manifolds with faces
In this section, we introduce 2n-dimensional torus orbifolds over a manifold with faces Q. We
first prepare some notation.
We set[n] := {1, . . . , n}and
Rn+ := {(x1 , . . . , xn ) ∈ Rn | xi ≥ 0, for all i = 1, . . . , n}.
A compact, connected, commutative, Lie group is called a torus, often denoted by T . We use thefollowing symbols for a torus T :• t := Lie(T ), the Lie algebra of T ;• tZ := exp−1 (e) ⊂ t is the lattice, where e ∈ T is the identity element and exp : t → T isthe exponential map;
• tQ := tZ ⊗Z Q.
We also use the following symbol as the standard n-dimensional torus:
Tn := {(z1 , . . . , zn ) ∈ Cn | |z1 | = · · · = |zn | = 1}.
This is nothing but the product of n unit circles in C. It is well-known that every n-dimensional
torus T is isomorphic to Tn .
In this paper, we often identify t∗Z := Hom(tZ , Z) with H 2 (BT ; Z) ∼= Zn . This identification is2
canonically defined as follows. It is well-known that elements in H (BT ; Z) are given by the first
Chern classes of line bundles ET ×T Cρ → BT , where Cρ is the 1-dimensional representation space
defined by ρ : T → S 1 . Moreover, the representation ρ lifts to the Lie algebra homomorphismρ̂ : t → R; therefore, ρ̂ ∈ t∗ . Since T and S 1 are abelian groups, we can take ρ̂ ∈ t∗Z . This
gives a homomorphism from H 2 (BT ; Z) to t∗Z . By comparing their dimensions, we can check thishomomorphism is isomorphism.
Moreover, in this paper, we use the symbol ∼= for the equivalence between two algebraic objects(i.e. group isomorphism or algebra isomorphism) and the symbol ≃ for the equivalence between
two geometric objects (i.e. homeomorphism or equivariant homeomorphism).

EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWO FIXED POINTS

3

2.1. Torus orbifolds. We first briefly recall the notion of a torus orbifold defined in [14, Section
12]. Let X = (X, U) be a 2n-dimensional orbifold whose underlying topological space is X andwhose (maximal) orbifold atlas is U = {(Uα , Vα , Hα , pα )}, i.e., U satisfies the following conditions:• {Uα } is an open covering of X;• Vα is an open subset in R2n ;• Hα is a finite subgroup of O(2n) acting on Vα ;• a continuous map pα : Vα → Uα which is an Hα -equivariant map that induces a homeomorphism Vα /Hα ≃ Uα , where Hα acts on Uα trivially.
We often denote X by X and call (Uα , Vα , Hα , pα ) an orbifold chart of X. A continuous map
f : X → X ′ from an orbifold X to another orbifold X ′ is called an orbifold map if for every x ∈ X
there are two orbifold charts (Uα , Vα , Hα , pα ) for X around x and (Uα′ , Vα′ , Hα′ , p′α ) for X ′ aroundf (x) and a continuous map fα : Vα → Vα′ such that the following diagram commutes:
Vα
pα

fα

/ Vα′
p′α

 f |Uα / Uα′Uα

If fα can be taken as a smooth map for every orbifold chart, then we call an orbifold map f asmooth map.
Let G be a Lie group. A G-action on an orbifold X is a smooth map φ : G × X → X which
satisfies the usual rules of being a group action. We often denote an orbifold (or any topologicalspace) X with G-action by (X, G) or (X, G, φ) if we emphasize the action φ : G × X → X. A
2n-dimensional closed oriented orbifold (X, T ) with an eﬀective n-dimensional torus T -action iscalled a torus orbifold if it has a nonempty fixed point set X T .
Remark 2.1. In this paper, an equivariant map f : (X, G, φ) → (X ′ , G′ , φ′ ) means a weak equivariant map, i.e., there is a group homomorphism h : G → G′ such that f (φ(g, x)) = φ′ (h(g), f (x))
for all (g, x) ∈ G × X.
Let (X, T ) be a torus orbifold. A special orbifold chart (Ux , Vx , Hx , px ), (also called a good local
chart, see e.g. [1]), around x ∈ X is an orbifold chart satisfying the condition that p−1
x (x) is asingle point xe ∈ Vx . The following fact is well-known (see, for instance, [11, Proposition 2.12]).
Lemma 2.2. Let x ∈ X be a fixed point in a torus orbifold (X, T ). Then, there exists a T invariant open neighborhood Ux of x and a special orbifold chart (Ux , Vx , Hx , px ) around x which
satisfies the following conditions:(1) there is a finite covering Tex of T such that Tex /Hx ≃ T and Tex acts on Vx ;(2) the continuous map px is an equivariant map between (Vx , Tex ) and (Ux , T ) which induces
an equivariant homeomorphism between (Vx /Hx , Tex /Hx ) and (Ux , T ).
A connected component of the fixed point set of a circle subgroup Si of T is a suborbifold,say Xi . This suborbifold Xi is called a characteristic suborbifold if Xi is a (2n − 2)-dimensional
orbifold and contains at least one fixed point of the T -action. In the definition of torus orbifolds in[14], we also need to choose an “invariant normal orientation” for every characteristic suborbifold
Xi . We will explain this in Section 2.3 for the case of torus orbifolds over manifolds with faces.
2.2. Manifolds with faces. In order to define a torus orbifold over Q, we next recall the definitionof an n-dimensional manifold with faces, also called a face acyclic nice manifold with corners (see[7, Chapter 10], [18, Section 5] or [16, Section 2.2] for more details). Let Qn (= Q) be an ndimensional topological manifold with boundary (we will always assume that Q is connected). Achart with corners, or simply a chart, for Qn is a pair (V, ψV ), where V is an open subset of Qn
andψV : V → Rn+

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ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONG

is a homeomorphism from V to an open subset ΩV ⊂ Rn+ . Two charts with corners (V, ψV ), (W, ψW )−1
are said to be compatible if the transition function ψV ◦ψW: ψW (V ∩W ) → ψV (V ∩W ) is a stratapreserving homeomorphism. We call a collection of compatible charts with corners {(V, ψV )} which
covers Qn an atlas. A maximal atlas is called the structure with corners of Qn . A topologicalmanifold with boundary together with a structure with corners is called a (topological) manifoldwith corners.
Let p ∈ Qn be a point of an n-dimensional manifold with corners Qn . If an element p ∈ V
is in a chart (V, ψV ), then we can assign a number, say d(p) ∈ [n] ∪ {0}, to p by the number ofzero-coordinates of ψV (p) ∈ Rn+ . This number d(p) is called the depth of p. We call the closure ofa connected component of d−1 (k) (0 ≤ k ≤ n) a codimension-k face. The codimension 1, (n − 1)and n faces are called facets, edges and vertices, respectively. The set of all edges and vertices
is called the one-skeleton (or the graph) of Qn . By restricting the structure with corners on Qn
to faces, we may regard each codimension-k face as an (n − k)-dimensional (sub)manifold withe ∗ (F ) = {0}, then we callcorners. If each face F of a manifold with corners Qn is acyclic, i.e. HnQ a face acyclic manifold with corners.
Definition 2.3 (Manifold with faces). An n-dimensional face acyclic manifold with corners Q is
said to be an n-dimensional manifold with faces (or a face acyclic nice manifold with corners) if
Q satisfies the following conditions:(1) for every k ∈ [n], there exists a codimension-k face;∩k(2) a connected component of i=1 Fi for distinct k facets is a codimension-k face whenever∩kthere are exactly k distinct facets
i=1 Fi ̸= ∅; conversely, for each codimension-k face H,∩kF1 , . . . , Fk such that H is a connected component of i=1 Fi .
Let Q1 , Q2 be manifolds with faces. If there is a homeomorphism f : Q1 → Q2 which preserves
faces for two manifolds with faces Q1 and Q2 , then we call Q1 and Q2 isomorphic (in the sense of
manifolds with faces). We can also define a weaker equivalence relation called a combinatoriallyequivalence among manifolds with faces as follows. We may regard the set of faces of a manifoldwith faces Q as a partially ordered set by the inclusions of faces, say S(Q). If there is a bijective
map between S(Q1 ) to S(Q2 ) which preserves the order, then we call two manifolds with faces Q1
and Q2 combinatorially equivalent. It is easy to check that if Q1 and Q2 are isomorphic, then Q1and Q2 are combinatorially equivalent. However, the converse is not true, see Remark 3.3.
2.3. Torus orbifolds over manifolds with faces. In this paper, we introduce a constructive
definition of a torus orbifold X over a manifold with faces Q (also see [9, 19] for details).
Let Q be an n-dimensional manifold with faces. We write the set of facets of Q as
F(Q) := {F1 , . . . , Fm }.
Let T be an n-dimensional torus. We identify the lattice tZ of the Lie algebra of T with Zn . A
function
λ : F(Q) → tZ
is called a characteristic function if it satisfies the following condition:• {λ(Fi1 ), . . . , λ(Fik )} is linearly independent whenever Fi1 ∩ · · · ∩ Fik ̸= ∅, for 1 ≤ k ≤ n.
Let us define the following quotient space:(2.1)

X(Q, λ) := (Q × T )/ ∼,where the equivalence relation ∼ is given by(2.2)(x, t) ∼ (y, s) if and only if x = y and t−1 s ∈ TF (x) ,where F (x) is the unique face of Q containing x in its relative interior and TF (x) is the k-dimensional
torus generated by λ(Fi1 ), . . . , λ(Fik ) ∈ tZ , where F (x) = (Fi1 ∩ · · · ∩ Fik )o is a codimension-k face(the symbol F o denotes a connected component of the set F ). To be more precise, TF (x) is the
k-dimensional torus subgroup of T defined by
TF (x) := (Rλ(Fi1 ) × · · · × Rλ(Fik )) / (Zλ(Fi1 ) × · · · × Zλ(Fik ))EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWO FIXED POINTS

5Note that if x ∈ Q is a vertex then TF (x) ∼= T.
It is easy to check that X := X(Q, λ) has the natural T -action induced from the T -action on
the second factor of Q × T , and its projection onto the orbit space X → X/T is induced fromthe projection onto the first factor Q × T → Q, i.e., X/T is equipped with the structure of amanifold with faces by the induced homeomorphism X/T ≃ Q. In this sense, we may identify
the orbit space of X with Q, and we call the projection π : X → Q the orbit projection of X.
Note that if F (x) (for x ∈ Q) is a codimension-k face then the inverse π −1 (x) is an (n − k)dimensional orbit which is homeomorphic to T /TF (x) . More precisely, one can define an orbifold
structure on X(Q, λ) similarly as in [19, Section 2.1] (also see [6] for a more precise account).
Moreover, one can see that {π −1 (v) | v ∈ V (Q)} is the set of fixed points. Hence, we conclude
that X(Q, λ) is a torus orbifold. Conversely, let (X, T ) be a 2n-dimensional torus orbifold. We
assume that the orbit space X/T is isomorphic to some n-dimensional manifold with faces Q andthat the projection π : X → X/T ≃ Q can be regarded as an orbit projection, i.e., π −1 (Fi ) isa characteristic suborbifold Xi for every facet Fi ∈ F(Q). Let Si be the circle subgroup of T
fixing Xi . Then, we can define the characteristic function λ : F(Q) → tZ by a choice of a nonzero
vector vi ∈ tZ such that Si = expRvi . Note that there are infinitely many choices of such nonzero
vectors, i.e. if we take a primitive vector si such that Si = expRsi , then for any r ∈ Z \ {0} theequality Si = expR(rsi ) holds. In order to determine the nonzero vector, we need to choose a“normal orientation” of Xi in the following way. Due to [14, Lemma 12.1], for every x ∈ Xi there
is a special chart (Ux , Vx , Hx , px ) around x which satisfies the following properties:
2n⊥(1) the tangent space of x
e = p−1
again, splits into Wix ⊕ Wix,
e Vx ≃ R
x (x) in Vx , say Vx := Tx⊥−1where Wix is tangent to px (Ux ∩ Xi ) and Wix may be regarded as the normal vector
space of x ∈ Xi ;⊥;(2) Hx acts on Wix(3) there is a (connected) finite cover Sei of Si and a lifting of the action of Si to the action of
Sei on Vx for any point x ∈ Xi ;⊥(4) the lifted action of Sei preserves (acts on) the normal vector space Wix(⊂ Vx ) non-trivially;⊥e(5) Si acts on Vx = Wix ⊕ Wix eﬀectively.
⊥
We choose an orientation of Wix
for every x ∈ Xi . We call this orientation a normal orientationof Xi . On the other hand, there are exactly two primitive vectors si and −si such that expRsi =expR(−si ) = Si . Note that the choice of signs determines the orientation of Si . We take theorientation of Si as the orientaion whose lifted Sei -action preserves the orientation of the given
normal orientation of Xi . Therefore, we can determine the primitive vector si ∈ tZ for Xi without
the sign ambiguity. Moreover, the continuous map px : Vx → Ux ≃ Vx /Hx is an equivariant map
with respect to the finite covering Sei → Si ≃ Sei /Hx , i.e., the ri (> 0)-times rotation map between
circles. Thus we may regard Sei = expR(ri si ) and Si = expRsi (also see (3.2)). Consequently,
once we choose a normal orientation for each of the characteristic suborbifolds Xi in the torus
orbifold X (we call a torus orbifold with normal orientations of each Xi an omnioriented torus
orbifold), the characteristic function λ : F(Q) → tZ is uniquely determined without ambiguity of
scalar multiplications, i.e., λ(Fi ) = vi := ri si . We also have the following lemma:
Lemma 2.4. Given a torus orbifold (X, T ) with X/T ≃ Q, let λ be the characteristic function determined by the appropriate choice of omniorientation as above. Then, X is equivariant
homeomorphic to X(Q, λ).
We call a(n) (omnioriented) torus orbifold X a torus orbifold over Q if X is equivariantly
homeomorphic to X(Q, λ) for some characteristic function λ.
Remark 2.5. Note that a choice of scalar multiplications of the λ(Fi )’s changes the orbifold structure on X; however, it does not change the equivariant homeomorphism type of X. Namely, ifλ(Fi ) = ri λ′ (Fi ) (Fi ∈ F(Q)) for some ri ∈ Z \ {0}, then X(Q, λ) ≃ X(Q, λ′ ) (equivariantly
homeomorphic).
Example 2.6 (Spindle). We denote the cyclic k-group by Ck and consider the natural surjection
pk : C → C/Ck . We define a spindle S 2 (m, n), for m, n ̸= 0, as follows. The underlying topological

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ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONG

space of S 2 (m, n) is homeomorphic to the 2-dimensional sphere S 2 . Denote the northpole in S 2 as
N and the southpole as S. The orbifold structure of S 2 (m, n), for m, n > 0, is the maximal orbifold
atlas U = {(Uα , Vα , Hα , pα )} which contains the following two orbifold charts: (1) (US , C, Cm , pm )for m > 0 around S whose open neighborhood US is defined by S 2 \ {N }; (2) (UN , C, Cn , pn ) forn > 0 around N whose open neighborhood UN is defined by S 2 \ {S}. The orbifold S 2 (m, m)
is also called a rugby ball and S 2 (m, 1) or S 2 (1, n) is also called a tear drop. If m is negative,then we consider the orbifold chart on US as (US , C, C−m , p−m ), where C is C with the reversedorientation, C−m acts on it by the multiplication and p−m : C → C/C−m is the natural surjection.
Similarly, we define the orbifold chart on UN when n < 0.
Note that the standard T1 -action on C induces (US , T1 /Cm ) ≃ (US , T 1 ) and (UN , T1 /Cn ) ≃(UN , T 1 ) by pm and pn respectively. Moreover, because the underlying space S 2 has an eﬀective T 1 action, S 2 (m, n) also has an eﬀective T 1 -action. Then, there are two fixed point {N, S}. Therefore,(S 2 (m, n), T 1 ) is a torus orbifold and its orbit space S 2 (m, n)/T 1 is the interval [−1, 1] such that{−1} (resp. {1}) corresponds to S (resp. N ). Then, (US , C, Cm , pm ) (resp. (US , C, C−m , p−m ))has the natural extension of the T 1 -action whose normal orientation is determined by the standard
orientation of C (resp. C) and pm : C → US ≃ C/Cm (resp. p−m : C → US ≃ C/C−m ) induces thehomomorphism Te1 → T 1 ≃ Te1 /C±m by the m-times rotation. Similarly we have the T 1 -extension
on the orbifold chart on UN . Hence, the characteristic function λm,n : {−1, 1} → Z \ {0} is definedby λm,n (−1) = m and λm,n (1) = n. This means that the pair ([−1, 1], λm,n ) defines the spindle
S 2 (m, n) (see Figure 1), i.e., S 2 (m, n) is a torus orbifold over the manifold with corners [−1, 1].
Note that for any m, n ∈ Z \ {0}, (S 2 (m, n), T 1 ) is equivariantly homeomorphic to (S 2 , T 1 ) ≃2(S (1, 1), T 1 ). We also note that (S 2 (1, 1), T 1 ), (S 2 (1, −1), T 1 ), (S 2 (−1, 1), T 1 ), (S 2 (−1, −1), T 1 )are torus manifolds with four omniorientations (they give four invariant stably complex structures
on S 2 ), also see Remark 2.5.

P

Q

Figure 1. The characteristic pair of the spindle S 2 (m, n).

3. The topology of a torus orbifold over the suspension of a simplex
Let Q be a manifold with faces with two vertices, we call such Q a manifold with two vertices
for short. It is easy to check that there exists a characteristic function λ on Q. By definition, theT -action on X(Q, λ) has exactly two fixed point. Conversely, if there are exactly two fixed pointson X(Q, λ) then its orbit space has exactly two vertices.
Definition 3.1. A torus orbifold X is said to be a torus orbifold with two fixed points if there isa manifold with two vertices Q such that X is a torus orbifold over Q.
In this section, we introduce some topological properties of the special class of torus orbifolds
with two fixed points. By the definition of a manifold with faces, an n-dimensional manifold Q
with two vertices, for n ≥ 2, has the following properties:(1) there exist exactly n facets, say F(Q) = {F1 , . . . , Fn };∩k(2) if 0 < k < n, the intersection j=1 Fij is connected, i.e., a codimensioin-k face of Q;(3) conversely, for every codimensional k face H (k ̸= 0, n), there exist exactly k distinct∩kfacets Fi1 , . . . , Fik such that H = j=1 Fij ;∩n(4) j=1 Fj = {p, q} (the set of all vertices of Q).
If n = 1, Q is nothing but the interval [−1, 1], i.e., there are only two facets {−1} and {1}.
The typical example of an n-dimensional manifold with two vertices is the suspension Σ∆n−1of the (n − 1)-dimensional simplex ∆n−1 . Here, Σ∆n−1 is defined by∆n−1 × [−1, 1]/ ∼,EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWO FIXED POINTS

7

where [−1, 1] is the interval and the equivalence relation ∼ is defined by collapsing ∆n−1 × {−1}(resp. ∆n−1 × {1}) to the vertex p (resp. q). Note that a codimension-k face H of Σ∆n−1 , fork = 0, . . . , n − 1, is determined by the suspension ΣF of some codimension-k face F in ∆n−1 andcodimension-n faces are the two vertices p, q in Σ∆n−1 . It is easy to check the following property:
Proposition 3.2. Let Q be an n-dimensional manifold with two vertices. Then Q is combinatorially equivalent to Σ∆n−1 .
n−1Remark 3.3. We defineby the induced topology from Rn+ , i.e., ∆n−1 :=∑nthe topology on ∆nn−1{(x1 , . . . , xn ) ∈ R+ |,i=1 xi = 1}. By the connected sum with a homology sphere and Σ∆
we can construct an n-dimensional manifold with two vertices which is not isomorphic to Σ∆n−1(also see [8]).

Remark 3.4. We also note that Σ∆n−1 can also be thought of as taking ∆n and collapsing a facet.
If n ≥ 2, the characteristic function λ : F(Σ∆n−1 ) → tZ ∼= Zn is often illustrated by thefollowing (n × n)-square matrix in GL(n; Q) ∩ Mn (Z):
λ11 · · · λ1n

.. 
..
(3.1)
Λ = (λ(F1 ) | · · · | λ(Fn )) =  ...
.
. 
λn1

···

λnn

where λ(Fi ), i = 1, . . . , n, is a nonzero vector in Z . Therefore, X(Σ∆n−1 , λ) (i.e. if we fix the
topology of the manifold with two vertices) is completely determined by the above matrix Λ. We
denote the torus orbifold over Σ∆n−1 (for n ≥ 2) determined by the matrix Λ as X(Λ). The goalof this section is to describe the equivariant topological type of a torus orbifold X(Λ).
n

3.1. The equivariant topological type of X(Λ). Let π : X(Λ) → Σ∆n−1 be the orbit projection of the T n -action on X(Λ) and Xi be the (2n − 2)-dimensional torus suborbifold defined by
π −1 (Fi ), for i = 1, . . . , n, i.e. the characteristic suborbifold associated to Fi . Since the characteristic function λi := λ(Fi ) is a vector in tZ , one can define the circle subgroup expRλi ⊂ T n , say
Si . Then Xi = X(Λ)Si is the fixed point set of the restricted Si -action.
Note that the integer square matrix Λ defined as in (3.1) induces the isomorphism Λ : Rn → Rn ;therefore, this induces the isomorphism Rn /Zn → Rn /Λ(Zn ), which we also call Λ. Hence, we candefine the following surjective homomorphism:∏nΛι / / nTn = Rn /Zn ∼ / Rn /Λ(Zn ) ∼(3.2)
T = t/tZ= t/Λ(tZ ) = i=1 Si=

where Tn is the standard n-dimensional torus in Cn and the surjective homomorphism ι is induced
from the product of the injective homomorphisms ιi : Si → T n . Put
Ten :=

n
∏

Si ,

i=1

then via the isomorphism Λ we may regard Ten as the standard torus Tn . By this identification,
the standard Ten -action on the unit 2n-dimensional sphere S 2n (⊕ni=1 Cλi ⊕ R) may be regarded as
the standard Tn -action on S2n := S 2n (Cn ⊕ R), where the symbol S 2n (V (ρ) ⊕ R) for a complex ndimensional T n -representation space V (ρ) represents the unit sphere (with respect to the standard
metric) in Cn × R ≃ V (ρ) ⊕ R with the torus action induced from the representation ρ : T n → T n .
This is known as one of the torus manifolds with two fixed points, and its characteristic submanifold
Mi ⊂ S2n , i = 1, . . . , n, is defined as(3.3)

Mi = {(z1 , . . . , zn , r) ∈ S2n ⊂ Cn ⊕ R | zi = 0}.

Therefore, its normal orientaion can be canonically determined by the orientation of the ith
complex space. Denote
G(Λ) := ker ι ◦ Λ ⊂ Tn .

8ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONG

We also denote the complex one-dimensional representation ιi ◦ Λ by
µi := ιi ◦ pi ◦ Λ : Tn → Ten → Si ,→ T n .
It is well-known that the following group is a product of cyclic groups, i.e., a finite abelian group(see Remark 3.11):n∩
ker µi ∼G(Λ) == tZ /Λ(tZ ).
i=1

Then, we can consider the following action induced from the standard action (S2n , Tn ):(S2n /G(Λ), Tn /G(Λ)) ≃ (S2n /G(Λ), T n ).
It is easy to check that there are exactly two fixed points in this action and that the characteristic
suborbifolds Xi ⊂ S2n /G(Λ), i = 1, . . . , n, are given by
Xi = Mi /G(Λ),
where the Mi ’s are defined in (3.3). Since only the ith factor of Tn fixes Mi , Xi is fixed byµi (Tn ) = Si = expRλi ⊂ T n . Here, we define the normal orientation of Xi as the normal
orientation of Mi . This shows that the characteristic function of (S2n /G(Λ), T n ) coincides withthat of (X(Λ), T n ). Hence by using Lemma 2.4, we have the following theorem:
Theorem 3.5. Assume n ≥ 2. Let X(Λ) be a torus orbifold over Q = Σ∆n−1 with a characteristic
function Λ. Then, (X(Λ), T n ) is equivariantly homeomorphic to (S2n /G(Λ), Tn /G(Λ)).
Remark 3.6. If G(Λ) is the identity group, then (S2n /G(Λ), Tn /G(Λ))) is the torus manifold(S2n , Tn ).
If n = 1, then G(Λ) is the cyclic group Cm . Therefore, (S2 /G(Λ), T1 /G(Λ))) is the rugby ballwith T 1 -action (S2 (m, m), T 1 ). It is well-known that the spindle S 2 (m, n) for |m| ̸= |n| is a bad
orbifold, i.e., it can not be obtained by the global quotient of S 2 . This shows that Theorem 3.5
does not hold for n = 1.
Theorem 3.5 leads us to the following corollary which can be obtained by applying Lemma 2.2
to the case of a torus orbifold over Q:
Corollary 3.7. Let Ux = X(Rn+ , λ|Rn+ ) be the open invariant neighborhood around a fixed point
x of a torus orbifold X(Q, λ), i.e., the λ restricts to a facet around x, say λ|Rn+ . Let Λ be the(n × n)-matrix as in (3.1) which defines the characteristic function λ|Rn+ . Then, the following
holds:(Ux , T n ) ≃ (Cn /G(Λ), Tn /G(Λ)),
where ≃ represents an equivariant homemorphism. Furthermore, there is the following special
orbifold chart around x:(Ux , Vx , Hx , px ) = (Ux , Cn , G(Λ), px : Cn → Cn /G(Λ) ≃ Ux ).
Note that G(Λ) acts on S 2n−1 = {(z1 , . . . , zn , 0) ∈ S 2n (Cn ⊕ R)}. Denote its orbit space byL(Λ) := S 2n−1 /G(Λ),
which is called an orbifold lens space in [5]. We note that this orbifold L(Λ) has a natural T n -action.
Remark 3.8. When G(Λ) is isomorphic to a cyclic group Cp and acts freely on S 2n−1 , then L(Λ)
is a lens space. Kawasaki [15] considers the case when a cyclic group Cp acts almost freely on
S 2n−1 and calls the quotient space S 2n−1 /Cp the twisted lens space which is an orbifold in general(in [2, 4], a twisted lens space is also called a weighted lens space).
We also have the following:
Corollary 3.9. The torus orbifold (X(Λ), T n ) is equivariantly homeomorphic to (ΣL(Λ), T n ),where the T n -action on the suspension ΣL(Λ) is the natural extension of the T n -action on L(Λ).
Moreover, we have the following lemma:

EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWO FIXED POINTS

9

Lemma 3.10. Assume n ≥ 2. Let N be the smallest subgroup of G(Λ) which contains all thoseelements of G(Λ) that fix points in S 2n−1 . Then H 3 (X(Λ)) ∼= G(Λ)/N .
Proof. Because n ≥ 2, S 2n−1 is simply connected. Moreover, G(Λ) acts on S 2n−1 eﬀectively.
Therefore, by [3], we have that π1 (S 2n−1 /G(Λ)) ∼= G(Λ)/N ∼= H1 (L(Λ)). This shows that
H1 (L(Λ)) is torsion if G(Λ)/N is not the identity group; namely, G(Λ) ̸= N . By using theuniversal coeﬃcient theorem, we have H1 (L(Λ)) ∼= H 2 (L(Λ)). Therefore, it follows from the2∼
Mayer-Vietoris exact sequence that H (L(Λ)) = H 3 (ΣL(Λ)) = H 3 (X(Λ)).
□
In particular, if (S 2n−1 )G(Λ) ̸= ∅, then H 3 (X(Λ)) = 0. Moreover, H odd (X(Λ)) = 0 if G(Λ) ={e}. Namely, if det Λ = ±1, then H odd (X(Λ)) = 0.
Remark 3.11. By using the Smith normal form, there are P, Q ∈ GL(n; Z) such that the sequence(3.2) can be written as follows:
(3.4)

Λ
∼
=

Tn = Rn /Zn
′

Λ
∼
=

∼
=


/ Rn /(r1 Z ⊕ · · · ⊕ rn Z)

where



Λ′ = QΛP −1 = / / T n = t/tZ

ι

Q

P


Rn /P (Zn )/ Rn /Λ(Zn ) ∼
= Ten

ι


/ / Tn /(Cr1 × · · · × Crn )


r1
..




.
rn

for some positive integers r1 , . . . , rn such r1 |r2 | · · · |rn , and Cr ∼= Z/rZ is the cyclic subgroup in
T1 . Here, we can compute ri , i = 1, . . . , n, asri =mi (Λ),
mi−1 (Λ)

where m0 (Λ) := 1 and mi (Λ) is the ith determinant divisor, i.e., the greatest common divisor of
all i × i minors of Λ.
Remark 3.12. As X(Λ) is simply connected, H 1 (X(Λ)) = 0.
Suppose that the product of cyclic groups G = Cr1 × · · · × Crn acts on S 2n−1 ⊂ Cn coordinatewise. Let N be the smallest subgroup in G which contains all those elements of G which have
fixed points. Then, N = G. Therefore, by Lemma 3.10, H 3 (S 2n /G) = 0.
4. The equivariant cohomology of X(Λ) with H odd (X(Λ)) = 0
In this section, we compute the equivariant cohomology of X(Λ) with H odd (X(Λ)) = 0 by using
the formula in [6].
4.1. Orbifold torus graph of (Q, λ) and its equivariant graph cohomology. Let (Q, λ) bea pair of a manifold with faces and its characteristic function. We shall define an orbifold torus
graph ((n, n)-type GKM-graph with rational axial function) Γ(Q, λ) := (Γ, α) of (Q, λ) as follows.
Let Γ = (V (Γ), E(Γ)) be the one-skeleton of Q. By the definition of a manifold with faces, thisbecomes an n-valent graph. Moreover, we see that each edge e is a connected component of theintersection of exactly (n − 1) facets, say Fk1 , . . . , Fkn−1 ; and we also denote one of the normal
facets of e by Fkn , i.e., Fkn ∩ e contains the initial vertex i(e) with the appropriate orientation.
Now, we define a function
α : E(Γ) → t∗Qby the following system of equations:{⟨⟩⟨α(e), λ(Fk1 )⟩ = · · · = α(e), λ(Fkn−1 ) = 0;
(4.1)
⟨α(e), λ(Fkn )⟩ = 1,

10

ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONG

where ⟨ , ⟩ denotes the natural paring between a vector space tQ and its dual space t∗Q . Note that
λ(Fi ) ∈ tZ ⊂ tQ . We call such a labeled graph (Γ, α) an orbifold torus graph of (Q, λ) and often
denote it by Γ(Q, λ).
Remark 4.1. In [6], we define an abstract orbifold torus graph like [17] without (Q, λ).
Example 4.2. The spindle S 2 (m, n) is defined by ([−1, 1], λm,n ) in Example 2.6. Set the generatorof t∗Z by x, i.e., t∗Z = Zx and t∗Q = Qx. In this case, [−1, 1] itself, say e, is the edge in this manifoldwith corners and the two vertices i(e) = {−1} and t(e) = {1} are the only facets. It follows fromthe definition of Γ([−1, 1], λm,n ) that we have the following axial function (see Figure 2):α(e) =

1x,mα(e) =

P [

1
x.
n

Q [





Figure 2. The orbifold torus graph of ([−1, 1], λm,n ).
Example 4.3. Assume n ≥ 2. Suppose that Q = Σ∆n−1 and consider the torus orbifold X(Λ)
e be the transpose of the cofactor matrix of Λ, i.e., ΛΛ
e = (det Λ)In . We may put
over Q. Let Λ µ1 .. 
e(4.2)
Λ =  . ,
µnwhere µi ∈ t∗Z is the one-dimensional representation of Tn defined by ιi ◦ Λ as in Section 3.1. Then,by equation (4.1), the vector µi ∈ t∗Z satisfies the following equation:
1
µi ,
det Λ
where ei is the edge of Σ∆n−1 which is not contained in the facet Fi (see Figure 3).
α(ei ) =

(4.3)


[\[\[\[\

Figure 3. The case when n = 2. The left figure is (Σ∆1 , λ) and the right one is
its orbifold torus graph, where µ1 = 5x − y and µ2 = −3x + y for some generators
x, y in t∗Z .
For an orbifold torus graph (Γ, α), we define the following rings.
Definition 4.4 (Graph equivariant cohomology). The following ring is said to be the (integral)graph equivariant cohomology ring:(4.4)

HT∗ (Γ, α) = {f : V (Γ) → Sym(t∗Z ) | f (i(e)) ≡ f (t(e)) mod re α(e)} ,where re is the minimal positive integer such that re α(e) ∈ Sym(t∗Z ) ∼= H ∗ (BT ; Z) and i(e)(resp. t(e)) is the initial (resp. terminal) vertex of a directed edge e.

EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWO FIXED POINTS

11⊕
Here, HT∗ (Γ, α) may be regarded as an H ∗ (BT ; Z)-subalgebra of v∈V (Γ) H ∗ (BT ; Z) by identifying Sym(t∗Z ) with H ∗ (BT ; Z) ∼= Z[x1 , . . . , xn ]. Then, re α(e) ∈ H 2 (BT ; Z) and there is a natural∗
grading in HT (Γ, α) induced by the grading of H ∗ (BT ; Z).
Denote by e the reversed oriented edge of e, i.e., i(e) = t(e) and t(e) = i(e). Note that alledges satisfy re α(e) = ±re α(e) (see [6]). We may define the cohomology of (Γ, α) over rationalcoeﬃcients as follows:
{
}
(4.5)
HT∗ (Γ, α; Q) = f : V (Γ) → Sym(t∗Q ) | f (i(e)) ≡ f (t(e)) mod α(e) .
∼ H ∗ (BT ; Q)-algerbra structure. This coincides with the
Similarly, this has the natural Sym(t∗ ) =Qdefinition of the cohomology ring of a GKM graph (of a symplectic orbifold) in [13, Section 1.7].
One can see that HT∗ (Γ, α) is a subring of HT∗ (Γ, α; Q). We call HT∗ (Γ, α; Q) the rational graphequivariant cohomology.
The next theorem is a consequence of applying the main result of [6] restricted to the case of
torus orbifolds with two fixed points.
Theorem 4.5. Assume X := X(Q, λ) satisfies H odd (X) = 0. Then, there is an isomorphism∼ H ∗ (Γ, α),H ∗ (X) =T

T

where (Γ, α) is the orbifold torus graph of (Q, λ) and HT∗ (X) := H ∗ (ET ×T X; Z) is the equivariant
cohomology of X.
Remark 4.6. The above theorem also holds for more general GKM orbifolds with vanishing odd
degree cohomology (see [6]).
4.2. Weighted face ring. Given an n-valent orbifold torus graph (Γ, α) of (Q, λ) and an (n − k)dimensional face F in Q, there is an (n − k)-valent subgraph ΓF which is the one-skeleton of F .
We also call this subgraph an (n − k)-dimensional face of (Γ, α). Each face F defines a rational
Thom class τF ∈ HT2k (Γ, α; Q) as follows:∏ i(e)=v α(e) if v ∈ V (ΓF );
e∈E(Γ
/
(4.6)
τF (v) :=F)0otherwise.
Note that deg τF = 2k for the codimension-k face F .
We formally defineτ∅ = 0,τΓ = 1.

Note that deg τ∅ = deg τΓ = 0.
Example 4.7. The following two figures (Figure 4 and Figure 5) are the examples of rational
Thom classes of the orbifold torus graph in Figure 3.
)S[\T
[\Figure 4. The rational Thom classes of the facet F in Figure 3, i.e., τF (p) =τF (q) = − 32 x + 12 y.
Let Fall (Q) be the set of all faces in Q and and ZQ := Z[τF | F ∈ Fall (Q)] be the integer(graded) polynomial ring generated by the rational Thom classes. Set the graded subring of ZQ
as{}
ZQ,λ := f ∈ ZQ ∀v ∈ V (Γ), f (v) ∈ H ∗ (BT n ; Z) .

12

ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONGS

T[\ [\

Figure 5. The rational Thom classes of the vertex p in Figure 3, i.e., τp (p) =( 52 x − 12 y)(− 32 x + 12 y) and τp (q) = 0.
Then, it is easy to see that the elements in ZQ of the form∑
(4.7)
τE τF − τE∨F
τG
G∈E∩F

are 0 for all vertices v ∈ V (Q), where the summation runs through all connected components inE ∩ F . Here the symbol E ∨ F represents the minimal face which contains both E and F ; notethat if E ∩ F ̸= ∅ then the face E ∨ F can be uniquely determined (also see [17]). Therefore, we
can define the ideal I of ZQ,λ generated by all elements defined by (4.7). Set
Z[Q, λ] := ZQ,λ /I.
We call this ring the weighted face ring of (Q, λ).
The following theorem is one of the main theorems in [6]:Theorem 4.8. Let (Γ, α) be the orbifold torus graph induced from (Q, λ). Then the following
graded rings are isomorphic:H ∗ (Γ, α) ∼= Z[Q, λ].
T

Therefore, together with Theorem 4.5, we have the following corollary:
Corollary 4.9. If the torus orbifold X(Q, λ) satisfies H odd (X(Q, λ)) = 0, then its equivariant
cohomology satisfies∼ Z[Q, λ].
H ∗ (X(Q, λ)) =T

4.3. The equivariant cohomology of X(Λ) when H odd (X(Λ)) = 0. Now we may computethe equivariant cohomology of X(Λ) when H odd (X(Λ)) = 0. Due to Section 4.1, the graphΓ = (V (Γ), E(Γ)) of Σ∆n−1 is given by
V (Γ) = {p, q},E(Γ) = {e1 , . . . , en , e1 , . . . , en }where i(ei ) = p and t(ei ) = q for i = 1, . . . , n, and the axial function α : E(Γ) → H 2 (BT ; Q) is
given by the equation (4.3), i.e.,α(ei ) = α(ei ) =

1
µi .
det Λ

In this subsection, we assume that
G(Λ) = {e}.
Then, because X(Λ) is a genuine manifold, i.e., S2n , we have H odd (X(Λ)) = 0. Since |G(Λ)| =| det Λ| = 1, we have thatα(ei ) = α(ei ) = ±µie = ±1, where Λ
e is defined in (4.2). Hence,for each i = 1, . . . , n. As det Λ = ±1, we also have det Λµi is a (primitive) vector in t∗Z . Therefore, the rational Thom class τi := τFi corresponding to a
facet Fi , i = 1, . . . , n, is an element in ZQ,λ . Indeed,τi (p) = τi (q) = ±µi ∈ H 2 (BT ; Z).

EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWO FIXED POINTS

13

Set τp and τq as the Thom classes of the vertices, i.e.,τp (p) =τq (q) =

n
∏
i=1
n
∏

(±µi ) ∈ H 2n (BT ; Z),τp (q) = 0;(±µi ) ∈ H 2n (BT ; Z),τq (p) = 0.

i=1

We first prove the following theorem:
Theorem 4.10. Let Q(∼= Σ∆n−1 ) be a manifold with two vertices and λ a characteristic function
on Q such that | det Λ| = 1. Then,Z[Q, λ] ∼= Z[τ1 , . . . , τn , τp , τq ]/⟨τ1 · · · τn − (τp + τq ), τp τq ⟩.
Proof. Due to the combinatorial structure of a manifold with two vertices Q (see Section 3),
for every∩ codimension-k face H (0 < k < n), there is a subset I ⊂ [n] such that |I| = k andH = i∈I Fi . In this case, we write H = FI and denote its rational Thom class by τI . In
particular, we denote the rational Thom class of a facet Fi by τi , for convenience.
Note that in Z[Q, λ] the following relation holds for i ̸= j:
τi τj = τΓ τ{i,j} = τ{i,j} .
Moreover, for h ∈ [n] \ {i, j}, we haveτh τ{i,j} = τ{i,j,h} = τi τj τh .
Similarly we have that
τI =

∏

τi ,

i∈I

for all non-empty proper subsets I ⊊ [n]. This shows that we can reduce the generators in Z[Q, λ]to τ1 , . . . , τn and τp , τq . Namely, the ring homomorphism
φ : Z[τ1 , . . . , τn , τp , τq ] → Z[Q, λ] = ZQ,λ /I,induced from the injection ρ : Z[τ1 , . . . , τn , τp , τq ] → ZQ,λ , is a surjective ring homomorphism. In
this case, one can easily see that ker φ ∼= Imρ ∩ I = ⟨τ1 , · · · , τn − (τp + τq )⟩ which establishes thestatement.
□
Remark 4.11. Because Z[Q, λ] does not depend on the topology of Q, this theorem also holds forany manifold with two vertices Q (which may not be Σ∆n−1 ).
Consequently, we have the following corollary:
Corollary 4.12. If H odd (X(Λ)) = 0 thenHT (X(Λ)) ∼= Z[τ1 , · · · , τn , τp , τq ]/⟨τ1 · · · τn − (τp + τq ), τp τq ⟩where deg τi = 2, i = 1, . . . , n, and deg τp = deg τq = 2n.
Remark 4.13. If G(Λ) = {e} then X(Λ) = S2n , i.e., the torus manifold. Therefore, we can also
obtain the above fact by using the main theorem of [17].
4.4. The equivariant cohomology of S 2 (m, n). In this final subsection, we apply Corollary 4.9
to the case when | det Λ| > 1.
Recall the spindle S 2 (m, n). This is homeomorphic to S 2 , therefore, H odd (S 2 (m, n)) = 0. The
orbifold torus graph of S 2 (m, n) is the one defined in Figure 2.
In this case, the rational Thom classes for the two vertices p and q are defined as follows:11
τp (p) = x, τp (q) = 0 and τq (p) = 0, τq (q) = x, respectively.
mnTherefore, we have that
Zm,n := Z([−1,1],λ ) ∼= Z[mτp , nτq , f (τp , τq )τp τq | f (τp , τq ) ∈ Z[τp , τq ]].
m,n

14

ALASTAIR DARBY, SHINTARO KUROKI, AND JONGBAEK SONG

(Note that the generators are duplicated.) Hence, with the method similar to the one demonstrated
in Section 4.1, it is easy to show the following theorem:
Theorem 4.14. The T 1 -equivariant cohomology of the spindle S 2 (m, n) is isomorphic to thefollowing ring:HT∗ (S 2 (m, n)) ∼= Zm,n /I ∼= Z[mτp , nτq ]/⟨mnτp τq ⟩,where deg τp = deg τq = 2.
It is well-known that the T 1 -action on S 2 (m, n) is equivariantly homeomorphic to the standard
T -action on S2 . Therefore, for any m, n(̸= 0), their equivariant cohomologies are isomorphic. We
finally remark the following proposition; the proposition may be regarded as the generalization ofthis fact for S 2 (m, m) = S2 /Cm .
1

Proposition 4.15. If Λ is the diagonal matrix (see Λ′ in Remark 3.11) then the torus orbifold
X(Λ) is equivariantly homeomorphic to the torus manifold obtained by Λ = In , i.e., the standard
2n-dimensional sphere S2n with Tn -action.
Proof. If Λ is the diagonal matrix Λ′ in Remark 3.11 then X(Λ) is the orbifold S 2n /Cr1 ×· · ·×Crn ,where Cr1 × · · · × Crn acts on the complex coordinates in S 2n ⊂ Cn ⊕ R standardly. Then, because
the scalar products on the characteristic functions does not change the topological type of theunderlying topological space, this is equivariantly homeomorphic to the torus manifold obtainedby Λ = In . This establishes the statement.
□

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EQUIVARIANT COHOMOLOGY OF TORUS ORBIFOLDS WITH TWO FIXED POINTS

15

Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou,
215123, Jiangsu Province, China.
E-mail address: Alastair.Darby@xjtlu.edu.cn
Department of Applied Mathematics, Okayama Uniervsity of Science, 1-1 Ridai-cho Kita-ku Okayamashi Okayama 700-0005, Japan
E-mail address: kuroki@xmath.ous.ac.jp
Department of Mathematical Sciences, KAIST, Daehak-ro 291, Daejeon 34141, Republic of Korea
E-mail address: jongbaek.song@gmail.com