Soumen SARKAR DongYoup SUH

Abstract Torus orbifolds are topological generalizations of symplectic toric orbifolds.The authours give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using a toric topological method. As a result, they show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. They also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.

Keywords Manifold with corners, Torus action, Torus orbifolds, Equivariant cobordism

1 Introduction

Cobordism, a fundamental concept in topology, was first introduced by Pontryagin in his pioneering work on a classifiction of manifolds (see [23]). There are two known definitions of bordism; one is geometric and another is homotopy theoretic. Thom constructions were used in [31] to show that cobordism groups could be computed through homotopy theory in the early 1950’s. Now the oriented, non-oriented and complex cobordism rings of manifolds are completely known in their respective category. Conner and Floyed generalized this definition to the equivariant category to study transformation groups in the beginning of sixties. After a few years, tom Dieck introduced homotopy theoretic cobordism in equivariant category. But,in the equivariant category, these two cobordism theories are not equivalent, see [28]. Even though there have been many developments such as [14, 17—18, 29—30, 32], the equivariant cobordism rings are not determined for any nontrivial groups. The main reason is that the Thom transversality theorem may not hold in equivariant category, and hence the equivariant cobordism cannot be reduced to(equivariant)homotopy theory. Here,we consider the geometric definition of equivariant cobordism.

Orbifold, which is a natural generalization of manifold, was introduced by Satake in [27],where it was calledV-manifold. The study of oriented cobordism of orbifold first appeared in [5] where the author introduced a complete set of invariants which determined the oriented cobordism classes up to torsion. Torus orbifold is a generalization of symplectic toric orbifold(see [15]) and it was introduced in [10], where some topological invariants of these spaces were studied. Some examples of torus orbifolds are the class of quasitoric orbifolds whose topological invariants are studied in [21]. Briefly, a torus orbifold is a closed orientable 2n-dimensional orbifold having an effective action of a realn-dimensional compact torus with nonempty fixed point set. In addition, if the torus action is locally standard and the orbit space is a simple polytope,then it is called a quasitoric orbifold. When the orbifold singularities of torus orbifolds are trivial, they are called torus manifolds, see [10] for topological properties and connections between torus manifolds and multifans.

In this paper we modify the basic construction of[4, 1.5]to produce effective orbifolds with torus actions whose boundaries consist of disjoint copies of locally standard torus orbifolds.Using this construction,we can show in Theorem 5.1 that any locally standard torus orbifold is equivariantly cobordant to a disjoint union of some orbifold complex projective spaces. We also have some more equivariant cobordism results on locally standard torus orbifolds and manifolds,see Theorems 5.2—5.3.

The article is organized as follows. Following [1], we recall the definitions and some facts concerning effective orbifolds in Section 2. In Section 3, we define locally standard torus orbifoldXover a nice manifold with cornersQ, and fromXwe extract two data, one is a rational characteristic functionλ,and the other is a smooth principalTn-bundleτoverQ,which we call combinatorial and topological data ofX. We also show that from a combinatorial and topological dataλandτoverQ,we can actually construct a locally standard torus orbifoldX(Q,λ,τ).We then show thatXandX(Q,λ,τ) are equivariantly homeomorphic in Theorem 3.1. All these arguments are appropriate modifications of the basic construction in [4]. We discuss the generalization of weighted projective spaces in Subsection 3.5. These spaces are called orbifold complex projective spaces. We also define orbifold Hirzebruch surface in Subsection 3.6, which is an orbifold modification of Hirzebruch surface. In Section 4, we give explicit construction of(2n+1)-dimensional smooth orbifolds withTn-action whose boundaries are disjoint union of locally standard torus orbifolds. In Section 5, we show that any torus orbifold with a locally standard torus action is equivariantly cobordant to some copies of orbifold complex projective spaces in Theorem 5.1. We also give several further equivariant cobordism results of torus manifolds and orbifolds including orbifold Hirzebruch surfaces.

2 Classical Effective Orbifolds

An orbifold is a singular space that locally looks like the quotient of an open subset of a Euclidean space by an action of a finite group. We recall the definitions and some facts concerning effective orbifolds from[1]. The reader may also consult[19] for an excellent exposition of the foundations of the theory of differentiable orbifolds.

Definition 2.1Let X be a Hausdorfftopological space. An n-dimensional effective orbifold chart on an open subset U⊆X is given by a triple(,G,φ)where

2.G is a finite subgroup of the self-diffeomorphisms of, and has an effective G-action on,

3.φ is a map fromto X such that φ is a G-invariant map inducing a homeomorphism from/G onto U.

Definition 2.2An embedding ξ: (,H,ζ) →(,G,φ)between two orbifold charts is a smooth embedding ξ:→of manifolds such that φ°ξ=ζ.

We remark that the above embedding induces an injective homomorphismH→G, see the important points (3) and (4) after [1, Definition 1.2].

Definition 2.3Two orbifold charts(,H,ζ)on V=ζ() ⊆X and(,G,φ)on U=φ() ⊆X with V∩U/= ∅are locally compatible if for any x∈V∩U, there exists an open neighborhood W⊆V∩U of x and an orbifold chart(K,µ)on W such that there exist smooth embeddings(,K,µ)→(,H,ζ)and(K,µ)→(,G,φ).

Definition 2.4An effective orbifold atlas on X is a familyU =of locally compatible effective orbifold charts such thatis an open cover of X.

An atlas V is a refinement of an atlas U if for any chart (,H,ζ) ∈V,there exists an embeddingξ: (,H,ζ)→(,G,φ) into some chart (,G,φ)∈U.

Two orbifold atlases are said to be equivalent if they have a common refinement. We denote the equivalence class of an atlas U by [U].

Definition 2.5Let X be a para-compact Hausdorffspace equipped with an equivalence class[U]of n-dimensional effective orbifold atlases. The pair(X,U), denoted byX, is called an effective orbifold of dimension n. The space|X|:=X is called the underlying space ofX.

For simplicity of notation, we may not distinguish X andXwhen the orbifold atlas onXis clear. Throughout this paper, we assume that all orbifolds are effective.

Definition 2.6LetX = (X,U)be an orbifold and x∈X. Let(,G,φ)be an orbifold chart so that x=φ()∈φ()⊂X. The local group at x is defined to be the group Gx={g∈

The groupGxis uniquely determined up to an isomorphism. We use the notion of local group to define the singular set of the orbifold X as follows. A pointx∈Xis called a nonsingular point (or smooth point) if the groupGxis trivial, and otherwisexis called a singular point.The set of singular points of an orbifold X =(X,U)is called the(orbifold)singular set,denoted byΣX. That is,

Example 2.1LetGbe a finite subgroup ofGLn(C)and letX=Cn/G. This is an orbifold complex manifold called a quotient singularity. OrbifoldXhas the structure of an algebraic variety, arising from the algebra ofG-invariant polynomials on Cn.

Example 2.2ConsiderS2n+1=,and the action of the circle groupS1is defined by

forα∈S1, where integersai’s are relatively prime. The quotient space

has an orbifold structure,denoted by WP(a0,···,an). This orbifold is called a weighted projective space with weights {a0,···,an}. In particular, the orbifold WP(1,a) is called a teardrop.

Similarly to the definition of a manifold with boundary, we can talk about an orbifold with boundary. We write R≥0={x∈R :x≥0}.

Definition 2.7Orbifold charts of an orbifold with boundary W are given by the compatible triples{(,G,φ)}where⊂Rn−1×R≥0for some n, G is a finite group acting effectively on, and{φ()}is an open cover of W.

The orbifold boundary ofW, denoted by∂W, is the set of pointsw∈Wsuch thatw∈{φ(∂)} for some chart (,G,φ) ofW. We remark that the boundary of an orbifold depends on the orbifold charts on it. For example, [0,1) with the trivial chart {0} is the boundary, but with the chart ((−1,1),Z2,φ) where Z2-acts by reflection [0,1) has empty boundary.

Definition 2.8LetX = (X,U)andY = (Y,V)be two orbifolds. A map f:X→Y is called an orbifold map(respectively orbifold smooth map)if for any point x∈X there are charts(,G,φ)containing φ−1(x)and(,H,ζ)containing ζ−1(f(x)), such that f maps U=φ()into V=ζ(and f can be lifted to a continuous(resp. smooth)map~f:→with ζ°=f°φ.

Using this, we can define the notion of homeomorphism(resp. diffeomorphism)of orbifolds.We note that homeomorphism as topological spaces may not induce a homeomorphism as orbifolds. For example, a triangle is homeomorphic to a square as a topological space but not as an orbifold.

Definition 2.9Two orbifoldsXandYare homeomorphic(resp. diffeomorphic)if there are orbifold maps(resp. orbifold smooth maps)f:X→Y and g:Y→X such that g°f=1Xand f°g=1Y.

Definition 2.10An orbifoldX = (X,U)is orientable if for each chart(,G,φ) ∈Uthe open subset⊂Rncan be given with an orientation which is invariant under the action of G,and each embedding ξ: (~W,K,µ)→(,G,φ)of charts inUis orientation-preserving.

Note that the underlying spaceXof an orbifold X =(X,U)can be obtained by gluing local charts using transition functions in the following way. For two charts (,G,φ), (,H,ζ) ∈U withx∈U∩Vthere is a chart (~W,K,µ) with smooth embeddings

such thatx∈W⊂U∩V. Thenλ2λ−11:λ1()→λ2(~W) is aK-equivariant diffeomorphism.Thus we can glue/Gand/Hby identifyingφ()～ζ() ifλ2λ−11()=. Then we have a homeomorphism

induced from the collection {φ:→X}.

We now define the tangent bundle and the frame bundle of an effective orbifold by gluing local charts using appropriate transition function. We remark that tangent bundles and frame bundles are two of many notions originally defined for manifolds which can be extended analogously to orbifolds. For an orbifold chart (,G,φ),we consider the tangent bundleT.Then it has the induced smooth action ofG, and the natural projectionp:T/G→Uis induced fromφ. It is known that for eachx=φ() ∈U,the fiberp−1(x) is diffeomorphic toTx/Gx. Thuspis a bundle-like map whose fiber is of the form Rn/G0for some finite subgroupG0⊂GLn(R). We thus have 2n-dimensional orbifold chart (T,G,π) for each (,G,φ)∈U,whereπ:T→T/Gis the orbit map. Let

for ~y∈λ1(~W) and ~z=λ2λ−11() to produce the identification space

Definition 2.11The tangent bundle of an n-dimensional orbifoldX = (X,U)is the2ndimensional orbifold TX =(T X,TU)with the natural projection map p:T X→X, which is a smooth map of orbifolds, with fiber p−1(x)=Tx/Gxfor each x∈X.

with theG-action defined by

SinceG-action onis effective,theG-action on Fr()is free,and the orbit space Fr()/Gis a smooth manifold. On the other hand, there is a rightO(n)-action on Fr()/Gwhich is induced from the group multiplication ofO(n), and it can be seen that for each point[(~x,A)]∈Fr()/Gits isotropy group is isomorphic toGx. Moreover by taking the quotient by theO(n)-action we have the induced natural projection Fr()/G→U.

Definition 2.12The frame bundle of an orbifoldX =(X,U)is the space obtained by gluing the local chartsFr()/G→U using the O(n)-transition functions obtained from the tangent bundle of X.

One of the important properties of the frame bundle is the following theorem.

Theorem 2.1(see [1, Theorem 1.23])For a given effective n-dimensional orbifoldX, its frame bundleFr(X)is a smooth manifold with a smooth effective almost-free O(n)-action. The original orbifoldXis naturally diffeomorphic to the resulting quotient orbifoldFr(X)/O(n).

3 Torus Orbifolds

The definition and basic properties of torus orbifolds are extensively discussed in [10]. In this section,we give the definition and the basic construction of locally standard torus orbifold.All these are appropriate modifications of the quasitoric theory developed in [4], which also generalize the arguments of quasitoric orbifolds in [21].

We begin by recalling the definition of manifold with corners from [3, Section 6]. Various properties of manifold with corners and maps between them are studied in [12].

LetX⊆RnandY⊆Rmsuch thatα:X→Yis continuous. Suppose that there exists an extension:U→Vofαfor some open neighborhoodUofXandVofY. Ifαis smooth thenαis called smooth.

Definition 3.1(a)A Hausdorfftopological space Q⊂Rmis called an n-dimensional manifold with corners for n≤m, if any point q∈Q has a neighborhood U and a homeomorphism φ from U to an open subset of the positive coneRn≥0={(x1,···,xn)∈Rn|x1≥0,···,xn≥0}.The pair(U,φ)is called a local chart on Q.

Moreover, Q is called smooth manifold with corners if all the transition maps are smooth and preserve the face structure.

(b)Let P and Q be m and n-dimensional smooth manifold with corners respectively. A map f:P→Q is called smooth if the map

is smooth for any two charts(U,φ)and(V,ψ)on P and Q respectively.

(c)The map f:P→Q is called diffeomorphism if it is smooth and it has smooth inverse.Two manifolds with corners are called diffeomorphic if there is a diffeomorphism between them.

(d)A k-dimensional face of Q is defined in the natural way. Codimension-1faces of Q are called facets, and faces of dimension0are called vertices. We writeF(Q) (resp.V(Q))for the set of facets(resp. vertices)of Q.

(e)A manifold with corners is said to be nice if every codimension-2face is a connected component of the intersection of a unique collection of two facets.

From the above definitions,it is easy to see that any codimension-kface of ann-dimensional nice manifold with cornersQis a connected component of the intersection of a unique set ofkmany facets ofQfor any 0 ≤k≤n. Ann-dimensional simple polytope is a convex polytope,each of whose vertices is the intersection of exactlynfacets. So simple polytopes are nice manifold with corners. In this article, we assume that every manifold with cornersQis nice,orientable and smooth unless specifically mentioned otherwise.

For aGspaceXand anHspaceY, a map is said to beθ-equivariant for a homomorphismθ:G→Hiff(gx)=θ(g)f(x) for anyg∈Gandx∈X. When we do not need to specify the homomorphismθ,fis said to be weakly equivariant or weak-equivariant.

3.1 Notations

LetMbe a free Z-module of rankn, i.e.,M～=Zn. Let

For a submoduleKofMof rankk, let

ThenKis submodule of ~Kwith finite index, and ~Kis a rankksubmodule ofM. Moreover,the inclusionιK:K→~Kinduces a surjective (covering) homomorphism

with finite abelian kernel ker(ζK) ～= ~K/K,whereTK=KR/K～=TkandT~K= ~KR/~K～=Tk.On the other hand, the inclusionι~K: ~K→Minduces an injective homomorphism

IfKis of rankn, in particular, thenζK:TK→T~K=TMis surjective with the finite abelian kernel.

3.2 Definition of locally standard torus orbifold and its properties

LetMbe a free Z-module of rankn, i.e.,M～= Znas above, or one may assumeM= Zn.Recall that the standardTn-action on Cnis defined to be

for (t1,···,tn)∈Tnand (z1,···,zn)∈Cn.

Definition 3.2A2n-dimensional connected and closed effective orbifoldXis called a locally standard torus orbifold, if the underlying topological space X ofXhas an effective TM-action such that for every point x∈X there exist

(P1)a TM-invariant neighborhood V⊂X,

(P2)a submodule N of M of rank n with the inclusion ι:N→M and the induced surjective(covering)homomorphism ζN:TN→TM, and

(P3)an orbifold chart(,G,φ)over V with G= kerζN, whereis δ-equivariantly diffeomorphic to an open set inCnfor some isomorphism δ:TN→Tn, and φ:→V is a ζN-equivariant map which induces a Tn-equivariant homeomorphism between/G and V.

If the groupGin Definition 3.2 is trivial for eachx∈X, then the orbifoldXis called a locally standard torus manifold.

Remark 3.1(1) From (P3), each groupG= kerζN⊂TN, hence the singular set ΣX of the orbifold is contained in the singular part Sing(X,Tn) of the action where Sing(X,Tn) is the union all singularTn-orbits inX.(2) Also from (P3), the orbit spaceQ=X/Tnis ann-dimensional nice smooth manifold with corners, and X is called a locally standard torus orbifold overQ.

We also remark that even though the definition of torus manifolds in [10] assumesXto have fixed points, we do not assume this here. In the case whenQis a simple polytope and each groupGis trivial,Xis called a quasitoric manifold, which was introduced by Davis and Januszkiewicz [4] by the name of toric manifold. Also ifQis a simple polytope butG’s are not necessarily trivial, thenXis defined to be a quasitoric orbifold in [21], and there the first author and Poddar studied several geometric and topological properties of them. An orbifold is called aTn-orbifold if there is an effectiveTn-action on the underlying space.

Example 3.1Consider the unit sphere

with the followingTn-action onS2n:

The points (0,···,0,−1),(0,···,0,1) ∈S2nare the fixed points of this action. LetU1=S2n−{(0,···,0,−1)}andU2=S2n−{(0,···,0,1)}. One can show thatUiisδi-equivariantly diffeomorphic to Cnwith the standard action ofTnfor someδi∈Aut(Tn) fori=1,2. SoS2nis a torus manifold. In particular,S2nis a locally standard torus orbifold with the orbit map

where the orbit space is given by

From the defining equations, we get thatQnsis a nice manifold with corners.

LetXbe a 2n-dimensional locally standard torus orbifold overQ=X/Tnwith the orbit mapπ:X→Q. Let F(Q) = {F1,···,Fm} denote the set of facets ofQ, and let°Fidenote the relative interior ofFi. By the local characterization of orbifold charts, the isotropy group of any pointx∈π−1(°Fi) ⊂Xis a locally constant circle subgroup ofTM. It is the image of a circle subgroup ofTNunderζN. Thus it determines a locally constant vectorλi∈Mup to sign, which is not necessarily a primitive vector ofM. Sinceπ−1(°Fi) is connected, this vectorλiis uniquely determined up to sign for each facetFiofQ. This vectorλiis called the rational characteristic vector ofFi. Thus we have the following function:

We recall that an element (a1,···,an) in the ranknfree Z-moduleMis called primitive if gcd{a1,···,an}=1. In general, for ann-dimensional nice manifold with cornersP,we define the following.

Definition 3.3A rational characteristic function(or simply an r-characteristic function)on an n-dimensional manifold with corners P is a map ξ: F(P) →Znsuch that whenever Fi1∩···∩Fik/=∅, the vectors ξ(Fi1),···,ξ(Fik)are linearly independent.

By (P3) of Definition 3.2, the functionλ: F(Q) →Znin (3.1) is a rational characteristic function onQ.

On the other hand, since every locally standard torus orbifoldXis compact, its orbit spaceQis a compact nice manifold with corners. So every facet ofQhas a collar neighborhood inQ,and hence the boundary∂Qhas a collar neighborhood inQ. That is,the complementQcof the union of suitable collar neighborhoods of all facets ofQis diffeomorphic toQpreserving the face structures. Note thatXc=π−1(Qc) is the total space of a principalTn-bundle, denoted byτc:EXc→Qc. SinceQcis diffeomorphic toQas a manifold with corners,we may pull backτctoQto get a topological principalTnbundle

overQ, whereEXis a nice manifold with corners andτpreserves the face structure. Thus,from a 2n-dimensional locally standard torus orbifoldXwith the orbit mapπ:X→Q, we have obtained two data, a rational characteristic functionλ: F(Q) →Zn, and a principalTnbundleτ:EX→Q. We write these data by {(Q,λ),(EX,Q,τ)} and call it the combinatorial and topological data of a locally standard torus orbifoldX.

We remark that the definition of rational characteristic function is a slight generalization of the following well-known notion of characteristic function.

Definition 3.4The function ξ in Definition3.3is called a characteristic function if the vectors ξ(Fi1),···,ξ(Fik)are a part of a basis ofZnwhenever Fi1∩···∩Fik/=∅.

IfXis a locally standard torus manifold(instead of orbifold)overQ,then the r-characteristic functionλin (3.1) is indeed a characteristic function. We finish this subsection with some examples of rational characteristic functions.

Example 3.2The manifold with corners in Figure 1(a) is obtained from the rectangleV0V1V2V3by deleting the interiors of the circleCand the triangleV4V5V6. The manifold with corners in Figure 1(b)is obtained from the disk bounded by the circleCby deleting the interior of the pentagonV0V1V2V3V4. Some r-characteristic functions of these 2-dimensional manifolds with corners are defined in the corresponding figure.

Figure 1 Some r-characteristic functions.

Example 3.3Figure 1(c) is an eye-shapeP2with vertices {V0,V1} and edges {E0,E1}.So an eye-shape is a nice manifold with corners. Defineξ: {E0,E1}→Z2by

Figure 2 A characteristic function on a closed disc.

Thenξis an r-characteristic function if and only if {(a,b),(c,d)} is a linearly independent set in Z2.

3.3 The basic construction of locally standard torus orbifold

LetPbe ann-dimensional nice manifold with corners equipped with a rational characteristic functionξ: F(P)→Zn. Moreover,let a smooth principalTn-bundleµ:E→PoverPbe given whereEis a nice manifold with corners andµpreserves the face structure. Let{(P,ξ),(E,P,µ)}denote these data,which we call a combinatorial and topological data overP. In this subsection,we construct a 2n-dimensional locally standard torus orbifoldXwith the orbit spacePsuch that the corresponding combinatorial and topological data ofXis the given data {(P,ξ),(E,P,µ)}.Each pointx∈Plies in the relative interior°Fof a unique codimension-kfaceFofP. Ifk=0,thenF=P, and otherwiseFis a connected component of the intersectionFi1∩···∩Fikof a unique collection {Fi1,···,Fik} ⊂F(P) becausePis nice. LetK(P) = 0, and letK(F)be the rankksubmodule ofM= Zngenerated by the vectorsξ(Fi1),···,ξ(Fik). Then as we have seen in Subsection 3.1, there is a surjective homomorphismζK(F):TK(F)→T~K(F)and an injective homomorphismζ~K(F):T~K(F)→TM. Let

ThenGFis a finite abelian group isomorphic to ker(ζK(F)), andTFis a rankktorus subgroup ofTM～=Tn. We will adopt the convention thatTP=1.

From the data {(P,ξ),(E,P,µ)},we construct the spaceX(P,ξ,µ) as follow. Define an equivalence relation ～on the total spaceEof the smooth principal bundleµby

whereFis the face containingµ(x)=µ(y) in its relative interior. The quotient space

has a naturalTn-action induced by the naturalTn-action onE. The orbit space of theTnactiononX(P,ξ,µ) is diffeomorphic toPas manifold with corners, and the map

can be regarded as the orbit map, where []denotes the equivalence class ofx. In the case whenµis a trivial bundle, we denoteX(P,ξ,µ) byX(P,ξ).

Lemma 3.1The space X(P,ξ,µ)is a locally standard torus orbifold over P with the orbit map π in(3.6).

ProofLet []∈X(P,ξ,µ). Henceµ(x) ∈P. We show that []has a neighborhood which isTn-equivariantly homeomorphic toVx/Gx,whereVxis aTn-invariant open subset of Cnwith the standardTnaction, andGxis a finite subgroup of Diff(Vx).

First we assume thatµ(x) belongs to the interiorThen there is a neighborhoodUxofµ(x) inPwhich is diffeomorphic to ann-dimensional open ball in Rn>0. Clearly,is aTn-invariant neighborhood of []. LetVx=Tn×Ux,Gx={1} andρx:Vx→X(P,ξ,µ)be the inclusion. Then (Vx,Gx,ρx) is an orbifold chart which satisfies the condition (P3) of Definition 3.2.

Letµ(x) belong to the relative interior°Fof a codimension-kfaceFofPwithk >0.Then there is a neighborhoodUxofµ(x) inPsuch thatUxis diffeomorphic to Rk≥0×Rn−k>0as manifold with corners andµ−1(Ux) isTn-equivariantly homeomorphic toTn×Uxas manifold with corners. Let

LetFbe a component of the intersectionFi1∩···∩Fikof a unique collection ofkmany facets ofP. By Definition 3.3, the set {ξ(Fi1),···,ξ(Fik)} is a linearly independent set of vectors of Zn. LetK(F) be the submodule generated by {ξ(Fi1),···,ξ(Fik)}.

LetK(F)⊥:=Zn/~K(F)where ~K(F)=(K(F)⊗ZR)∩Zn. ThusK(F)⊥is a free Z-module of rankn−k. Then Znis isomorphic to ~K(F)⊕K(F)⊥. Fixing an isomorphism of these Z-modules, we getTn=T~K(F)×TK(F)⊥. On the other hand, we also have an isomorphismTn～=TK(F)×TK(F)⊥asTK(F)andTK(F)⊥arekandn−kdimensional torus respectively.

For a faceF′containingF, letK(F′) be the submodule ofK(F) generated by the rational characteristic vectors corresponding to the faceF′, and letTK(F′)be the corresponding torus group defined as before. We now define an equivalence relation ～FonTK(F)×TK(F)⊥×Uxby

whereF′is the unique face whose relative interior containsq1=q2.

Let

Using the aforementioned homeomorphism, we can get thatVxis equivariantly homeomorphic to Ck×(C∗)n−k. Thus we have a commutative diagram ofTn-equivariant maps:

where the vertical arrows are the quotient maps. Note that the mapζxis the orbit map ofGF-action onVx, whereGFis as defined in (3.2). Now the triple (Vx,Gx,ζx) withGx=GFis an orbifold chart onXx, which satisfies the condition (P3) of Definition 3.2.

Next we show that the orbifold charts(Vx,Gx,ζx)’s are compatible(see Definition 2.3). The arguments are essentially similar to [7, Subsection 2.3], but few modifications are needed. First we give coordinate structure on eachVxin the following. Let

be the diffiomorphism as manifold with corners. If we write

thenfx,j(u) ≥0,fx,j(u) = 0 if and only ifu∈Ux∩Fijforj= 1,···,k, andfx,j(u)>0 forj=k+1,···,n(k≥0). Let {ξk+1,···,ξn} be a Z-basis forK(F)⊥⊂Zn. WhenF=Q, we may considerK(F)=0 ∈Zn. Then the kernel of the map determined by the matrix

isGF. Let a=(α1,···,αn) be the standard angular coordinates, and ax=(αx,1,···,αx,n)be the angular coordinates ofTnwith respect to the basisThen we get the following transformation between these angular coordinates

Letux,j:=fx,j(u)cos(2παx,j) andwx,j:=fx,j(u)sin(2παx,j) forj= 1,···,n. We defineφx:Vx→R2nby

From the identification ～Fand condition onfxand using a property of quotient map, we get thatφxis homeomorphic onto its image which is an open subset of R2n. Then the coordinate structure onVxcan be given by (ux1,wx,1,···,ux,n,wx,n).

The group action ofGFonVxcan be given by similar relation as in[7,(2.10)]. Compatibility of charts can be explained following the arguments in [7, Subsection 2.3]. ThereforeX(P,ξ,µ)is a locally standard torus orbifold.

Remark 3.2(1) IfPand the smooth principalTnbundleµin Lemma 3.1 are orientable,then so isX(P,ξ,µ).

(2) Observe that ifξsatisfies the condition in Definition 3.4, then all local groups in the above orbifold charts are trivial. So in this case,X(P,ξ,µ)is a locally standard torus manifold.

Proposition 3.1Let X and Y be locally standard torus orbifolds over P and Q respectively such that the following diagram commutes

where f is equvariantly homeomorphic, πX,πYare orbit maps, and g is a diffiemorphic as manifold with corners. Then f is equvariantly diffeomorphic.

ProofUsing the coordinate description on each orbifold chart (Vx,Gx,ζx) in the proof of Lemma 3.1 and modifying the arguments of the proof of [7, Lemma 2.3], one can complete the proof.

Example 3.4LetP2be an eye-shape andξbe an r-characteristic function onP2as in Example 3.3, and letµbe the trivialT2-bundle overP2. SoX(P2,ξ) is a 4-dimensional orientable locally standard torus orbifold. Now we show thatX(P2,ξ) is the orbit space of a finite group action onS4. LetQ2sbe the manifold with corners as in Example 3.1. LetFi= {(x1,x2,x) ∈Q2s|xi= 0} fori= 1,2. Note thatπ−1s(Q2s) =S4⊂C2×R and the isotropy group of the points inis

Therefore the corresponding r-characteristic function is given by

up to choices of sign. Then one can show that (T2×Q2s)/～=S4=π−1s(Q2s),where ～is the equivalence relation defined in (3.4). Letf:Q2s→P2be a diffeomorphism as manifold with corners such thatf(Fi)=Eifori=1,2. Consider the map Z2→Z2determined by

This induces a surjective Lie group homomorphismφ:T2↠T2, that is,φis a finite covering homomorphism. From the definition of the equivalence relation ～, it is clear that the map

induces a surjective map

defined byfφ([t,x]～) = [φ(t),f(x)]～on the equivalence classes. The finite group kerφacts naturally onS4. Sinceφis a covering homomorphism with the finite covering group kerφ, the spaceX(P2,ξ) is diffeomorphic to the quotient spaceS4/kerφ. In particular, ifξis a characteristic function (see Definition 3.4), thenX(P2,ξ)isT2-weakly equivariantly diffeomorphic toS4.

Example 3.5ConsiderS3= {(z1,z2) ∈C2: |z1|2+|z2|2= 1} with anS1-action onS3defined by

Then the orbit spacePis a closed 2-disc. Letξ: {∂P} →Z2be the map defined byξ(∂P) =(a,b), where (a,b) is a primitive vector in Z2. Thenξis a characteristic map onP. So by Lemma 3.1 and Remark 3.2,X(P,ξ) is an orientable locally standard torus manifold. Since(a,b)is a primitive vector,there is(c,d)∈Z2such thatad−bc=1. Applying an automorphism ofT2, we may assume (c,d)=(1,0) and (a,b)=(0,1). Therefore we have the following

hereT2acts onS1×S3as follows: The firstS1factor ofT2acts onS1by left multiplication,and the secondS1factor acts onS3as in (3.9). ThusX(P,ξ)is weakly equivariantly diffeomorphic toS1×S3.

The following theorem shows that any locally standard torus orbifolds can be constructed from the basic construction explained at the beginning of this subsection.

Theorem 3.1Let X be a2n-dimensional locally standard torus orbifold over Q with the associated combinatorial and topological data{(Q,λ),(EX,Q,τ)}as in Subsection3.2. Let X(Q,λ,τ)be the locally standard torus orbifold obtained by the basic construction from the data as in Subsection3.3. Then there is a Tn-equivariant orbifold diffeomorphism from X(Q,λ,τ)to X covering the identity on Q.

ProofThe proof is similar to the quasitoric orbifold case in [21]. The basic idea of the proof is similar to that of[4,Proposition 1.8],however there are two complications which require further arguments. The first one is that the orbit spaceQis not necessarily contractible,and the second one is that we are dealing with orbifolds instead of manifolds. The former complication can be fixed by considering principalTn-bundleτ:EX→Qinstead of the trivialTn-bundleTn×Q→Q. So we need to take care of the latter complication. The main point is how to extend [4, Lemma 1.4] to the locally standard torus orbifold case, i.e., we need to construct a continuous map

which mapsτ−1(q) surjectively ontoπ−1(q) for eachq∈Q, whereπ:X→Qis the orbit map.

Let Fr(X) be the frame bundle of the effective orbifoldX. Then by Theorem 2.1 the frame bundle Fr(X) is a smooth manifold with smooth effective almost-freeO(2n)-action, and the quotient orbifold Fr(X)/O(2n) is diffeomorphic toX. On the other hand, sinceXis a locally standard torus orbifold,Xhas an effectiveTn-action which induces an effectiveTn-action on Fr(X) commuting with the above mentionedO(2n)-action.

We now apply the procedure for“blowing up the singular strata”of theTn-action on Fr(X)as explained in [4, Lemma 1.4], to obtain an effectiveTn-manifold ~Fr(X) with corners with only principal orbits. Indeed, ~Fr(X)is obtained from Fr(X)by replacing each singular stratum by its normal sphere bundle in the order from the minimal stratum to the higher ones. Then there is a natural map

which collapses each sphere bundle to the base points. TheO(2n)-action on Fr(X)also induces a freeO(2n)-action on ~Fr(X) because by Remark 3.1 the singular set ΣXof the orbifoldXis contained in the singular part Sing(X,Tn) of theTn-action onX. Furthermore,Tnacts freely on ~Fr(X) and commutes with the action ofO(2n). Let:= ~Fr(X)/O(2n). Thenis a freeTn-space with the orbit space equal toQ. Indeed,isTn-equivariantly homeomorphic to the total spaceEXof a principal bundleτ:EX→Q. The natural surjective map ~finduces a continuousTn-equivariant map

SinceO(2n)-action commutes withTn-action on ~Fr(X) and Fr(X), ify∈f−1(x) ⊂EXfor somex∈X,then isotropy ofyis the same asTF, whereFis the smallest face containingτ(y).Therefore the mapffactor through the continuous mapEX/～→X. This is aTn-equivariant homeomorphismX(Q,λ,τ) →Xcovering the identity onQ. Then Proposition 3.1 completes the proof.

We remark that blowing up of singular stratum may not be unique,but the above procedure suffices our requirement.

Definition 3.5Let{(P,ξ),(E,P,µ)}and{(P′,ξ′),(E′,P′,µ′)}be two combinatorial and topological data. They are called equivalent if there is a diffeomorphism ψ:P→P′(as manifold with corners)and a δ∈Aut(Zn)such that ξ′(ψ(F)) = ±δ(ξ(F))for each F∈F(P)and µ is isomorphic to the pull back bundle ψ∗(µ′).

Theorem 3.2Two locally standard torus orbifolds X(P,ξ,µ)and X(P′,ξ′,µ′)are Tnweakly equivariantly diffeomorphic if and only if the corresponding characteristic and topological data{(P,ξ),(E,P,µ)}and{(P′,ξ′),(E′,P′,µ′)}are equivalent.

ProofLet Ψ:X(P,ξ,µ)→X(P′,ξ′,µ′)be aTn-weakly equivariant diffeomorphism. Then it induces a diffeomorphismψ:P→P′(as manifold with corners). This follows from the facts thatTn-action on each is locally standard and the orbit map is smooth. Thus we have the following commutative diagrams,

whereπ,π′are orbit maps,Pc(resp.P′c) is the complement of a suitable collor neighborhood of∂P(resp.∂P′) inP(resp.P′),E=π−1(Pc) andE′=(π′)−1(P′c). Note thatPc(resp.P′c)is diffeomorphic toP(resp.P′) as manifold with corners. Thereforeξ′(ψ(F)) =±δ(ξ(F)) for a fixedδ∈Aut(Zn) and for allF∈F(P), andµ～=ψ∗(µ′).

Conversely,assume two data{(P,ξ),(E,P,µ)} and{(P′,ξ′),(E′,P′,µ′)}are equivalent. Soξ′(ψ(F)) = ±δ(ξ(F)) for a fixedδ∈Aut(Zn) and for allF∈F(P), andµ～=ψ∗(µ′) for some diffeomorphismψ:P→P′as manifold with corners. So there is a bundle isomorphism:E→E′such that

Since ~ψisTn-equivariant andξ′(ψ(F))=±δ(ξ(F)) for allF∈F(P), the map ~ψdescends to aTn-weakly equivariant map Ψ:X(P,ξ,µ)→X(P,ξ′,µ′). Using the construction ofX(P,ξ,µ)andX(P,ξ′,µ′), one can show that Ψ is a homeomorphism. So by Proposition 3.1, Ψ is a diffeomorphism.

We remark that the above two theorems are proved for the category of quasitoric manifolds in [4] and for the category of quasitoric orbifolds in [21]. Also similar result are discussed for the category of 2-torus manifolds in [16]. One can replace diffeomorphism/diffeomorphic by homeomorphism/homeomorphic in Definition 3.5 and Theorem 3.2 and retain the conclusion of Theorem 3.2 in homeomorphic category.

3.4 Equivariant connected sum

Here,we discuss connected sum of orientable locally standard torus manifolds along an orbit as in [8]. LetA(resp.B) be an orbit of a 2n-dimensional locally standard torus manifoldM(resp.N). Assume that the isotropy group ofAis isomorphic to that ofB. SoA(resp.B)is a subset of a connected component ofMi1∩···∩Miℓ(resp.Ni1∩···∩Niℓ) for a unique collection of characteristic submanifolds {Mi1,···,Miℓ} ofM(resp. {Ni1,···,Niℓ} ofN),where a characteristic submanifold of a locally standard torus manifold is the inverse image of a facet by the orbit map. SinceTn-action is locally standard,there areTn-invariant small enough neighborhoodsUA(resp.UB) ofA(resp.B) such thatUAandUBare weak-equivariantly diffeomorphic to Cℓ×(C∗)n−ℓ. By changing the action ofTnonNby an automorphism ofTnif necessary, we may assume thatTn-actions onUAandUBare equivalent. That is, we may assume that the isotropy group of°Mijis the same as that of°Nijforj= 1,···,ℓ. By identifying the boundary ofM−UAandN−UBvia an orientation reversing equivariant diffeomorphism, we get a manifold, denoted byM#A,BN, with a natural locally standardTnaction. SoM#A,BNis an oriented locally standard torus manifolds. For simplicitys we denote the equivariant connected sum byM#Nwhen the orbitsAandBare clear. IfA,Bare orbits of dimensionn, then we say that they are principal orbits and the spaceM#Nis a connected sum along principal orbits.

Note that we can perform the equivariant connected sum construction for locally standard torus orbifolds along the orbits which belong to the smooth part of the orbifolds.

LetQbe a nice 2-dimensional manifold with corners. So every component of∂Qis either boundary of a polygon, a circle or an eye-shape in Figure 1(c). Note thatQcan be obtained from a closed surfaceSQ, by removing the interior of finitely many non-intersecting polygons,eye-shapes or discs. See Figure 1(a) for an example.

In the following lemma, letS4be theT2-sphere in Example 3.1, andS1×S3be theT2-manifold in Example 3.5.

Lemma 3.2Let M be an orientable locally standard torus manifold(resp. orbifold)over a2-dimensional nice manifold with corners Q such that ∂Q/=∅. Then M is T2-weakly equivariantly diffeomorphic to a connected sum of several copies of4-dimensional quasitoric manifolds(resp. orbifolds), T2×SQ, S4(resp. S4/G as in Example3.4), and S1×S3.

ProofWe only prove the orbifold case. The manifold case is similar with thatG’s are trivial groups. Letλbe the r-characteristic function associated toM. SinceQis 2-dimensional manifold with corners with∂Q/= ∅, we haveH2(Q) = 0. Hence the principalT2-bundleτ:EM→Qassociated toMis trivial.

SupposeQis obtained fromSQby removing the interiors of copies of non-intersecting polygonsQ11,···,Q1r, eye-shapesQ21,···,Q2sand discsQ31,···,Q3t. For simplicity, we assume that there is only one copy ofQi=Qi1for eachi= 1,2,3. Note that the facets ofQis F(Q) =Now define r-characteristic functionsλi: F(Qi) →Z2onQito be the restrictionλ|F(Qi)for eachi=1,2,3.

Let

be the connected sum of the following 4-dimensional locally standard torus orbifolds

where all connected sums are performed along some principal orbits. SoM′is a locally standard torus manifold. Observe that the orbit spaceQ′ofM′is the connected sum ofQ1,Q2,Q3andSQat their respective interior points. SoQ′is diffeomorphic as manifold with corners toQ. The r-characteristic functionλ′associated toM′is induced byλ1,λ2andλ3. The r-characteristic functions onQandQ′, and the trivial principal bundles on them satisfy the conditions of Definition 3.5. Therefore by Theorem 3.2,M～=X(Q,ξ) isT2-weakly equivariantly diffeomorphic toX(Q′,λ′) ～=M′. By Examples 3.4—3.5, the manifoldsX(Q2,λ2) (resp.X(Q3,λ3)) areT2-weakly equivariantly diffeomorphic toS4/G(resp.S1×S3). Also the manifoldX(Q1,λ1) is a 4-dimensional quasitoric orbifold by [21]. This proves the lemma.

3.5 Orbifold complex projective space

A toric varietyXΣassociated to a simplicial fan Σ is called a toric orbifold. The spaceXΣis a compact 2n-dimensional toric variety if and only if Σ is a complete fan in Rn. It is well-known that if one considers real torusTn⊂(C∗)naction then it is a torus orbifold. More studies on toric varieties can be found in [2, 6].

Definition 3.6LetΣbe a complete simplicial fan inRnwith n+1many1-dimensional cones. The associated toric orbifold XΣis called an orbifold complex projective space of real dimension2n.

Lemma 3.3(see [25, Lemma 3.9])Let X be a quasitoric orbifold over an n-dimensional simplex. Then X is equivariantly diffeomorphic to an orbifold complex projective space of real dimension2n.

We remark that a fake weighted projective space is a holomorphic generalization of weighted projective space, see [13]. A fake weighted projective space of real dimension 2nis determined by a complete simplicial fan generated by (n+1) many primitive vectors in Zn. So a fake weighted projective space is an orbifold complex projective space. Since the primitive vectors in Z are −1 and 1, the teardropWP(1,a) is not a fake weighted projective space ifa >1 but an orbifold complex projective space.

3.6 Orbifold Hirzebruch surface

A Hirzebruch surface is a nonsingular toric variety corresponding to a complete fan given by Figure 3(A) whereb∈Z. Note that a Hirzebruch surface is a manifold. For more details on Hirzebruch surface,see[11,20]. An orbifold Hirzebruch surfaceXis defined to be a toric variety corresponding to a complete simplicial fan given by Figure 3(B), where {(ai,bi),(ai+1,bi+1)}are linearly independent vectors in Z2fori= 1,···,4 and (a1,b1) = (a5,b5). One can show thatXwith the restricted action of the compact torusT2⊂(C∗)2satisfies the condition of Definition 3.2. Therefore orbifold Hirzebruch surfaces are quasitoric orbifolds.

Figure 3 Fans for Hirzebruch surface and Hirzebruch orbifold.

Consider a rectangleP2with verticesV0,···,V3and edgesV0V1,V1V2,V2V3,V0V3. Define a map

byξ(ViVi+1) =εi(ai+1,bi+1) wherei= 0,···,3,V4=V0andεi= ±1. Then by [21, Lemma 2.2] and Theorem 3.1, the orbifold Hirzebruch surfaceXisT2-equivariantly diffeomorphic to

X(P2,ξ).

4 Construction of Orbifolds with Boundary

From now on, we assume that all locally standard torus orbifolds and manifolds are orientable. In this section, we construct (2n+1)-dimensional orientable effective orbifolds withTn-actions whose boundaries are locally standard torus orbifolds. To do this, we need the notions of face-simple manifold with marked facets and rational super characteristic function defined on facets of it.

Definition 4.1A face-simple manifold with marked facets is an(n+1)-dimensional oriented compact manifold with corners Y together with a subset{P1,···,Pm}⊂F(Y)of disjoint facets of Y, called the marked facets, such that

1.Y is nice, i.e., any codimension-k face of Y is a connected component of the intersection of a unique set of k many facets of Y for any1 ≤k≤n+1, and

2.the vertex setV(Y)is equal to

Face-simple manifold Y with marked facets P1,···,Pmis denoted by Y[P1,···,Pm]. The facetsF(Y){P1,···,Pm}is called the remaining facets of Y[P1,···Pm].

Recall that the vertex-cut VC(P)of a polytopePis the polytope obtained fromPby cutting offdisjoint cone-shape neighborhoodsUvof all verticesvofP. For each vertexvofP, letFvdenote the facet of VC(P) corresponding to the intersection

An (n+1)-dimensional polytopePis said to be edge-simple if each one dimensional face ofPis the intersection of exactlynfacets ofP, see [24]. Clearly, the vertex cut VC(P) of an edge-simple polytopePis a face-simple manifold with marked facets {Fv|v∈V(P)}.

Definition 4.2Let Y[P1,···,Pm]be an(n+ 1)-dimensional face-simple manifold with marked facets, and let{F1,···,Fm′}be the remaining facets. A function

is called a rational super characteristic function(simply an rs-characteristic function)on Y[P1,···,Pm]if the set of vectors η(Fi1),···, η(Fik)are linearly independent inZnwhenever Fi1∩···∩Fik/= ∅. The vector η(Fi)is called an rs-characteristic vector assigned to the facet Fifor i= 1,···,m′. When the vectors η(Fi1),···, η(Fik)are part of a basis ofZnwhenever Fi1∩···∩Fik/= ∅, then η is called a super characteristic function(simply an s-characteristic function).

Note that for r-characteristic function in Definition 3.3, the dimension of manifold with corners and the rank of the target module are same, while for rs-characteristic function the dimension of manifold with corners is larger than the rank of the target module by 1. Definition 4.2 is a generalization of the isotropy function of an edge-simple polytope given in [24]. In [26],the authors considered hyper-characteristic functions, and in this case the dimension of the manifold is less than the rank of the target module by 1.

Letη: {F1,···,Fm′}→Znbe an rs-characteristic function on an (n+1)-dimensional facesimple manifold with marked facetsY[P1,···,Pm]. For each marked facetPj, let F(Pj) :={Gj1,···,Gjhj} be its facets. Then for each facetGjiofPj, there exists a unique facetFjiamong the remaining facets ofY[P1,···,Pm]such thatGji=Pj∩Fji. We now define functions

forj= 1,···,m. Then the following lemma is obvious from the definition of rs-characteristic function.

Lemma 4.1The function ξjdefined in(4.1)is an r-characteristic function on Pjfor j=1,···,m.

Letµ:E→Ybe an orientable smooth principalTn-bundle overY, and letηbe an rscharacteristic function onY[P1,···,Pm]. Then the collection {(Y[P1,···,Pm],η),(E,Y,µ)} is called a combinatorial and topological data onY[P1,···,Pm]. From this data, we construct a (2n+1)-dimensionalTn-orbifold with boundary being a disjoint union of some locally standard 2n-dimensional orbifolds. The construction is quite similar to the basic construction in Subsection 3.3.

LetFbe a codimension-kface ofY[P1,···,Pm]for 0

where 〈αi:i= 1,···,s〉 denotes the submodule of Zngenerated by the vectorsαifori=1,···,s. We define an equivalence relation ～bon the total spaceEof the smooth principalTn-bundleµ:E→Yas follows: Forx,y∈E,

whereFis the unique face ofYcontainingµ(x) =µ(y) in its relative interior andTFis the subgroup ofTnas defined in (3.3). Let

be the equivalence classes. ThenW(Y,η,µ) has aTn-action induced from the action ofTnonE. Let [x]～bdenote the equivalence class ofxinW(Y,η,µ), and letπ:W(Y,η,µ)→Ybe the projection map defined byπ([x]～b)=µ(x).

Theorem 4.1Let Y[P1,···,Pm]be an(n+ 1)-dimensional face-simple manifold with marked facets, let η be an rs-characteristic function on Y[P1,···,Pm],and letµbe an orientable smooth principal Tn-bundle over Y. Then W(Y,η,µ)is a(2n+1)-dimensional orientable effective Tn-orbifold with the boundary consisting of m many disjoint orientable locally standard torus orbifolds.

ProofFor any[x]～b∈W(Y,η,µ),we show that there exists a neighborhood of[x]～bwhich isTn-equivariantly homeomorphic toV/G,whereVis aTn-invariant open subset of Cn×R≥0on whichTn-action on Cnis standard and on R≥0is trivial, andGis a finite subgroup of Diff(V). There are three cases:

1. Whenµ(x)∈°Y,

2. whenµ(x)∈°F, whereFis a face ofYwhich is not contained in any of the marked facetsPi, and

3. whenµ(x)∈Pjfor somej∈{1,···,m}.

The proofs of the first two cases are almost identical to that of Lemma 3.1. The only difference is that in Lemma 3.1 we found aTn-invariant open subsetVin Cninstead of Cn×R≥0,but this difference does not cause any difficulty here.

So we prove the third case. Now assumeµ(x) ∈Pjfor somej= 1,···,m. SinceYis a compact manifold with corners, there exists a collar neighborhoodUxofPjinY. So there is a diffeomorphismgx:Ux→Pj×[0,1) as manifold with corners and aTn-equivariant homeomorphism ~gx:µ−1(Ux)→µ−1(Pj)×[0,1) such that the following diagram commutes:

Observe that the equivalence relation ～bin (4.3) does not affect the second component ofµ−1(Pj)×[0,1) and ～bis the same as the relation ～in (3.4) onµ−1(Pj). Let {Gj1,···,Gjhj}be the facets ofPj. ThenGji=Pj∩Fjifor a unique remaining facetFjiofY[P1,···,Pm].Define

Thenξiis an r-characteristic function onPjby Lemma 4.1. LetEj=µ−1(Pj)andµj:Ej→Pjbe the restriction ofµ. Thenµj:Ej→Pjis a smooth principalTn-bundle. So we have the followingTn-equivariant homeomorphisms

whereX(Pj,ξj,µj) is the orientable locally standard torus orbifold associated to the characteristic and topological data {(Pj,ξj),(Ej,Pj,µj)}. Lety′∈X(Pj,ξj,µj)×[0,1) be the image of [x]～bunder the homeomorphism in (4.5). Sinceµ(x) ∈Pj, we havey′= (y,0) for somey∈X(Pj,ξj,µj). So there is an orbifold chart (V,G,ψ) on a neighborhood ofyinX(Pj,ξj,µj). Thus (V×[0,1),G,ψ×id) is an orbifold chart on a neighborhood of [x]～binW(Y,η,µ). One can check the compatibility of these charts using similar arguments as in the proof of Lemma 3.1.

Hence the spaceW(Y,η,µ)is an orientable compact effective orbifold whose boundary is the disjoint union of orientable locally standard torus orbifolds {X(Pj,ξj,µj) forj= 1,···,m}.Recall that by our assumptionYis orientable. So choosing an orientation ofYandTn, we get an orientation onW(Y,η,µ) and the boundary orbifolds.

When the principalTn-bundleµis trivial, we writeW(Y,η) instead ofW(Y,η,µ).

Example 4.1Two rs-characteristic functionsηiof the face-simple manifolds with marked facetsYifori= 1,2 are given in Figure 4. This figure is the same as [24, Figure 4], but the functions on them are different. Here principalT2-bundlesµionYiare trivial, sinceYiis contractible fori=1,2. In(a)all marked facetsP1,P2,P3,P4are triangles,and henceW(Y1,η1)is an orientable orbifold whose boundary consists of locally standard orbifoldsX(Pj,ξj) which are orbifold complex projective spaces defined in Subsection 3.5 forj∈{1,2,3,4}.

Figure 4 Some rs-characteristic functions on face-simple manifolds Y1 and Y2 with marked facets.

In (b), the marked facetsP1,P2,P3,P4are triangles andP5is a rectangle. ThusW(Y2,η2)has the boundary consisting of orbifold complex projective spaces forj∈{1,2,3,4} and an orbifold Hirzebruch surfaceX(P5,ξ25) defined in Subsection 3.6.

Ifηsatisfies that the set of vectors {η(Fi1),···, η(Fik)} is a part of a basis of Znwhenever the intersection of the facets {Fi1,···, Fik} is nonempty, then all the local groups in the proof of Lemma 4.1 are trivial. So, in this case, we have the following corollary.

Corollary 4.1Under the same assumption as in the last paragraph, the space W(Y,η,µ)is a(2n+1)-dimensional smooth orientable bounded Tn-manifold whose boundary is a disjoint union of orientable locally standard torus manifolds.

We remark that the above corollary is a generalization of [25, Lemma 4.4].

5 Equivariant Cobordism of Torus Orbifolds

In this section, we exhibit several explicit cobordisms among orientable locally standard torus orbifolds. First, we recall the definition of equivariant cobordism of orientable locally standard torus orbifolds.

Definition 5.1Two2n-dimensional orientable locally standard torus orbifolds X1and X2are said to be equivariantly cobordant(or torus cobordant)if there exists a(2n+1)-dimensional orientable effective Tn-orbifold W with boundary ∂W such that ∂W is Tn-equivariantly diffeomorphic to X1⊔(−X2)under an orientation preserving diffeomorphism. Here−X2denotes X2with the opposite orientation.

The above cobordism relation has transitive property. It follows from the fact that torus orbifolds considered here are compact, so one can construct a tubular neighborhood of a boundary component. Let OCnbe the group of equivariant cobordism classes of 2n-dimensional orientable locally standard torus orbifolds, where the group structure is given by the disjoint union.

Example 5.1Any torus orbifold over an eye-shape in Example 3.3 isT2-equivariantly cobordant to zero, i.e.,T2-equivariantly a boundary. Indeed, letρ: kerφ×S4→S4be the action discussed in Example 3.4. Then we have the following commutative diagram

Since (S4×I)/(S4×{0}) ～=D5, the spaceD5/kerφis an orientable orbifold with boundaryS4/kerφ. That is, the torus orbifoldX(P2,ξ) =S4/kerφover an eye-shapeP2isT2-equivariantly a boundary. So the cobordism class [X(P2,ξ)]=0 in the group OC2.

Before we give equivariant cobordism results on orientable locally standard torus orbifolds,let us give a simple result on locally standard torus manifolds over polytopes with simple holes.Polytopes with simple holes are defined in [22] as follows. LetQ0be ann-dimensional simple polytope in Rn. LetQ1,Q2,···,Qℓbe a collection of disjointn-dimensional simple polytopes in the interior ofQ0. LetQ=Q0−ThisQis called ann-dimensional polytope with simple holesQ1,···,Qℓ. Since the polytopesQ0,···,Qkare simple of same dimension in Rn,Q0induces an orientation onQ. Note thatQis a nice manifold with corners.

LetMbe an orientable locally standard torus manifold over the polytopeQwithℓsimple holesQ1,···,Qℓ. Letλbe the characteristic function associated toM, and letλibe the restriction ofλtoQifori=0,1,···,ℓ.

Lemma 5.1Let M be as above. If H2(Q) = 0, then M is equivariantly cobordant to the disjoint union M(Q0,λ0)⊔···⊔M(Qℓ,λℓ)by a(2n+1)-dimensional Tn-manifold.

ProofSinceH2(Q) = 0, any principalTn-bundle overQis trivial. Hence by Theorem 3.1,Mis equivariantly diffeomorphic toM(Q,λ). Following the proof of [22, Lemma 2.1],one can show thatM(Q,ξ)isTn-equivariantly diffeomorphic to the connected sumM(Q0,λ0)♯M(Q1,λ1)♯···♯M(Qℓ,λℓ),where the connected sum occurs along a principal orbit of quasitoric manifoldsM(Qi,λi) fori=0,···,ℓ.

On the other hand, it is well-known that equivariant connected sum of two manifolds are equivariantly cobordant to the disjoint union of them. Therefore we have the lemma.

For a positive integerk, let[k]denote the set{1,···,k}. For an integer 1 ≤j≤k, let Pj[k]denote the collection of subsets of [k] withjelements.

Lemma 5.2Let j and k be positive integers such that j≤k. Let a subsetA ⊂Pj[k]and a set{ξ1,···,ξk}of vectors inZjbe given such that for any{i1,···,ij} ∈Athe vectors ξi1,···,ξijare linearly independent. Then there exists a primitive vector ξ0∈Zjsuch that theis a linearly independent set of vectors inZjfor any{i1,···,ij}∈Aand ℓ=1,···,j. Hererepresents the omission of the corresponding entry.

ProofThej=1 case follows because |Z| = ∞. Letj >1. For eachI= {i1,···,ij} ∈A and for eachiℓ∈I, letKI,iℓbe the rank (j−1) submodule of Zjgenerated by {ξi1,···,ξiℓ−1,Thenis a finite union of rank (j−1) submodules of Zj.Therefore there exists a nonzero vector

If we chooseξ0to be a primitive vector, then it clearly satisfies the conditions in the lemma.

Now we prove one of the main theorems of this article.

Theorem 5.1Let X be an orientable2n-dimensional locally standard torus orbifold with k fixed points. Then X is equivariantly cobordant to a disjoint union of k many orbifold complex projective spaces.

ProofLetXbe a 2n-dimensional orientable locally standard torus orbifold withkfixed points. So the orbit spaceQ:=X/Tnis an orientable smooth compact manifold with corners withkvertices. Let {(Q,λ′),(EX,Q,τ′)} be the combinatorial and topological data ofXas obtained in Subsection 3.2, whereλ′is an r-characteristic function onQandτ′:EX→Qis an orientable smooth principalTn-bundle. Then by Theorem 3.1,Xis equivariantly diffeomorphic toX(Q,λ′,τ′). SupposeQis a subset of Rℓ−1for someℓ≥n+1. Let F(Q):={F1,···,Fm}be the facets ofQ, and let V(Q):={V1,···,Vk} be the vertices ofQ. LetY:=Q×△1⊂Rℓ,where △1=[0,1]is the 1-simplex. ThenYis an(n+1)-dimensional nice manifold with corners,whose facets are

where {0} and {1} are the vertices of △1. LetV0i=Vi×{0} fori= 1,···,k. We perform vertex-cut ofYaround the verticesV0ifori=1,···,kas follows. Consider an open ballBℓiin Rℓaround each vertexV0iofY⊂Rℓsuch that the followings are satisfied.

1.is diffeomorphic as manifold with corners to the (n+1)-simplex in Rn+1for eachi=1,···,k, wheredenotes the closure ofBℓiin Rℓ+1.

2.=∅wheni/=j.

3.×{1}=∅for alli=1,···,k.

ThenYQ:=Y−is the desired vertex-cut, which is an orientable smooth nice (n+1)-dimensional manifold with corners. Letfori=1,···,k. ThenQi:=Sℓi∩Yis the facet ofYQcorresponding to the vertexV0ifor eachi=1,···,k. Since each vertex ofQiis an interior point of an edge ofY, andYis an (n+1)-dimensional nice manifold with corners,Qiis diffeomorphic as manifold with corners to then-simplex for eachi=1,2,···,k. Let

So the facets and the vertices ofYQare as follows:

whereQi∩Qj∩YQis empty wheni/=j. ThenYQ[Q1,···,Qk+1] is a face-simple (n+1)-dimensional orientable manifold with marked facetsQ1,···,Qk+1.

We now define an rs-characteristic functionλonYQ[Q1,···,Qk+1]. Define

whereλ0is the vector obtained from Lemma 5.2 in the following way. For each vertexV∈V(Qi) ⊂V(YQ),there exists a unique collection of facetsF0i1,···,F0in∈F(YQ) such thatV=F0i1∩···∩F0in∩Qi. LetIV:={i1,···,in}∈Pn[m] and let A:={IV|V∈V(YQ)}. Then from Lemma 5.2 there exists a primitive vectorλ0∈Znsuch that the set ofnvectors

is linearly independent for allj=1,···,nandV∈V(YQ). One can check thatλis indeed an rs-characteristic function onYQ[Q1,···,Qk+1].

On the other hand,τ′induces an orientable smooth principalTn-bundle

So we have the pullback bundleτ:EYQ= (τ′×Id)−1(YQ) →YQvia the inclusionYQ→Y.Note thatτ:EYQ→YQis an orientable smooth principalTn-bundle. So is the pullback bundleτi:Ei→Qiofτvia the inclusionQi→YQfori=1,···,k+1. So by Theorem 4.1,W(YQ,λ,τ)is an orientable effectiveTn-orbifold whose boundary is the disjoint unionwhere～bis as defined in (4.3). The spaceEk+1/～bis equivariantly diffeomorphic toX(Q,λ′,τ′),whereτ′=τk+1:EX=Ek+1→Qk+1=Q.

Let F(Qi) = {Hi1,···,Hin+1}. ThenHij=F0ij∩Qifor a unique remaining facetF0ijofYQ[Q1,···,Qk+1] for eachj=1,···,n+1. Define a functionλi: F(Qi)→Znby

Thenλiis an r-characteristic function onQifor alli= 1,···,k. So {(Qi,λi)(Ei,Qi,τi)} is a characteristic and topological data fori=1,···,k. Note that the boundary componentEi/～bis the same asX(Qi,λi,τi), which is a quasitoric orbifold over ann-simplexQifori=1,···,k.Therefore by Lemma 3.3, eachEi/～b=X(Qi,λi,τi) is an orbifold complex projective space,which proves the theorem.

We remark that in Theorem 5.1, ifXis a locally standard torus manifold, then it is equivariantly cobordant to a disjoint union of some orbifold projective spaces. Still this theorem may be useful as there are examples of locally standard torus manifolds where it is quite complicated to compute cobordism invariants, whereas there are several studies on topological invariants of orbifold projective spaces.

Example 5.2Let △2=V0V1V2be a triangle in R2, and letCbe a circle in the interior of △2. Delete the open disk bounded byCfrom △2, and letQbe the remaining subspace of △2. A rational characteristic functionλ′onQis given as on the left of Figure 5. LetY:=Q×[0,1]. Then the rs-characteristic functionλon the face-simple manifold with marked facetsYQ[Q1,···,Q4] is given on the right of Figure 5, whereQ1, Q2andQ3are the facets ofYQcorresponding to the verticesV1×{0},V2×{0}andV3×{0},respectively,andQ4=Q×{1}.SinceH2(YQ) = 0, any principalT2-bundleτoverYQis trivial. By Theorem 4.1, the spaceW(YQ,λ) is an orientable effectiveT2-orbifold with boundary. Let F(Qi) = {Hi1,Hi2,Hi3}fori= 1,2,3. ThenHij=F0ij∩Qifor a unique remaining facetF0ijofYQ[Q1,···,Q4] fori=1,2,3. Define a functionλi: F(Qi)→Z2by

Figure 5 An r- and rs-characteristic function on a 2- and 3-dimensional nice manifold with corners respectively.

Thenλiis an r-characteristic function onQifori=1,2,3. The r-characteristic function onQ4is the same as onQ.Therefore, the boundary ofW(YQ,λ) is

whereX(Qi,λi) is an orbifold complex projective space of real dimension 4 fori=1,2,3.

The following corollary is a special case of Theorem 5.1 whenk=0.

Corollary 5.1Let X be an orientable2n-dimensional locally standard torus orbifold without a fixed point. Then X is the boundary of a(2n+1)-dimensional orientable effective orbifold with Tn-action.

Corollary 5.2Let X be a4-dimensional orientable locally standard torus manifold without a fixed point. Then X is the boundary of an orientable5-dimensional manifold with T2-action.

ProofWe are using the same notation as in the proof of Theorem 5.1. LetQ=X/T2be the orbit space and {(Q,λ′),(EX,τ′)} be the combinatorial and topological data ofX. ThenXisT2-equivariantly diffeomorphic toX(Q,λ′,τ′) by Theorem 3.1. SinceXis 4-dimensional,the orbit spaceQis a 2-dimensional nice manifold with corners,and sinceXhas no fixed point,Qhas no vertices. ThereforeQis a bounded surface, andYQ=Y=Q×△1. LetC1,···,Ckbe the boundary components ofQ. By Corollary 5.1,W(Y,λ,τ) is an orientable effectiveT2-orbifold with the boundaryX(Q,λ′,τ′)～=X. Letπ:W(Y,λ,τ)→Ybe the orbit map. Notice that the orbifold singularity ofW(Y,λ,τ) may occur only inLetUibe a neighborhood ofCi×{0} inYfori=1,···,ksuch that:

1.Uiis diffeomorphic as manifold with corners toCi×R2≥0,

2.is diffeomorphic as manifold with corners toCi×△2,

3.=∅fori,j=1,···,kandi/=j.Thenis diffeomorphic toCi× △1as manifold with corners. LetW=which is an orientableT2-manifold with boundary. Then

Note that((T2×Ci×△1)/～b)isT2-equivariantly diffeomorphic toCi×((T2×△1)/～b). The facets ofCi×△1are(Ci×△1)∩Fijfor unique facetsFi1andFj2ofY. The restriction ofλon the facets of △1=ci×△1⊂Ci×△1is given by Figure 6(a). Note that vectors defined byλare all primitive. So we may assume(a,b)=(1,0)and(c,d)=(−qi,pi)after an automorphism ofT2. Then we have the hyper characteristic function (see [26, Definition 2.1]) on △1given by Figure 6(b). So following [26, Section 2], one can get that (T2×△1)/～bis weak-equivariantly diffeomorphic to the lens spaceL(pi,qi) with the usualT2action fori=1,···,k.

Figure 6

It is shown in[9]and[26]that any lens spaceL(pi,qi)isT2-equivariantly a boundary. ThusCi×L(pi,qi)is aT2-equivariantly boundary for eachi∈{1,···,k},which proves the corollary.

We now give a couple of equivariant cobordism results on orientable 4-dimensional locally standard torus orbifolds.

Theorem 5.2Let M be a locally standard torus manifold over2-dimensional nice manifold with corners Q with ∂Q/= ∅. Then M is equivariantly cobordant(by a5-dimensional T2-manifold)to some copies ofCP2.

ProofWe stick to the notations of Lemma 3.2. SinceS4,S1×S3andT2×SQareT2-equivariantly boundaries, by Lemma 3.2,MisT2-equivariantly cobordant to the connected sumM1♯···♯Mkof some 4-dimensional quasitoric manifoldsM1,···,Mkby a 5-dimensionalT2-manifold. Since equivariant connected sum of two manifolds are equivariantly cobordant to the disjoint union of them,M1♯···♯Mkis equivariantly cobordant to the disjoint unionM1⊔···⊔Mk. By [24, Theorem 6.6], we see that eachMiis equivariantly cobordant to some copies of CP2by a 5-dimensionalT2-manifold. This proves the theorem.

In the following theorem we stick to the notations in Subsection 3.6.

Theorem 5.3Let X be an orbifold Hirzebruch surface whose fan is given in Figure3(B).If either(a1,b1) = ±(a3,b3)or(a2,b2) = ±(a4,b4), then X is equivariantly a boundary, i.e.,X is the boundary of a5-dimensional orientable T2-orbifold.

ProofSuppose (a1,b1)=±(a3,b3). LetPbe the face-simple manifold with marked facetsQ1,Q2andQ3as in Figure 7(B). Note thatQ2andQ3are eye-shapes, andPis orientable,smooth and contractible. Then we can define an rs-characteristic functionηonP[Q1,Q2,Q3],as in Figure 7(B). This induces an r-characteristic functionηionQifori=1,2,3.

Figure 7 An rs-characteristic functions on a face-simple 3-dimensional manifold.

By Theorem 4.1,theT2-spaceW(P,η)is a compact orientable effective orbifold with boundary given by

whereX(Q1,η1) isT2-equivariantly diffeomorphic to the orbifold Hirzebruch surfaceXandX(Qi,ηi) is a torus orbifold over an eye-shape fori= 2,3. From Example 5.1, we can see thatX(Qi,ηi) isT2-equivariantly the boundary of a 5-dimensional orientable orbifold withT2action fori= 2,3. This proves the theorem when (a1,b1) = ±(a3,b3). The case when(a2,b2)=±(a4,b4) is similar.

Corollary 5.3If X is a Hirzebruch surface, then X bounds a5-dimensional orientable T2-manifold.

ProofWithout loss of generality, we may assume that the complete rational fan ofXis as in Figure 3(A). Then the corresponding super characteristic functionηis given as in Figure 7(A). This induces characteristic functionsηionQifori= 1,2,3. Then by Corollary 4.1, we get an orientable 5-dimensionalT2-manifoldW(P,η) with boundary given by

whereX(Q1,η1) isT2-equivariantly diffeomorphic to the Hirzebruch surfaceXandX(Qi,ηi)is a locally standard torus manifold over an eye-shape fori=2,3. So by Example 3.4,X(Qi,ξi)isT2-weakly equivariantly diffeomorphic toS4fori= 2,3. Therefore any Hirzebruch surface bounds a 5-dimensional orientableT2-manifold.

Remark 5.1Corollary 5.3 is proved in [24, Lemma 6.1], but we give a much shorter proof here.

We conclude this section by presenting some explicit cobordism relations among orbifold complex projective spaces. By definition, an orbifold complex projective space is a quasitoric orbifold over a simplex. LetYbe an (n+1)-dimensional orientable smooth manifold with corners in some Euclidean space. Let {F1,···,Fm} be the facets and {V1,···,Vk} be the vertices ofY. Let VC(Y) be the vertex cutting ofY,F′j=Fj∩YVandQibe the facet ofV C(Y) corresponding to the vertexVifori= 1,···,k. ThenQiis diffeomorphic to ann-simplex. Applying Lemma 5.2, we can define an rs-characteristic function

on VC(Y)[Q1,···,Qk]. Letµ:E→VC(Y) be orientable principalTn-bundle. Note thatµ−1(Qi) is equivariantly diffeomorphic toTn×Qifori= 1,···,k. Then by Lemma 4.1,W(VC(Y),η,µ)is an orientable effectiveTn-orbifold with boundary being is the disjoint unionSince restriction of the rs-characteristic functionλto the facets ofQiis an r-characteristic function, (Tn×Qi)/～bis an orbifold complex projective spaceOPifori=1,···,k. So,

in the group OCn. At the end, it is natural to ask the following.

Problem 5.1(see[25, Question 5.9])What are the other torus cobordism relations among the orbifold complex projective spaces?

AcknowledgementThe first author thanks Indian Institute of Sciences, Pacific Institute for Mathematical Sciences and University of Regina for support. He also thanks Indian Statistical Institute, Kolkata and Institute of Mathematical Sciences for supporting his visiting fellowship. The authors would like to thank anonymous referee, Mainak Poddar and Nigel Ray for many helpful comments and suggestions.