A NEW GENERALIZATION OF ORBIFOLDS USING OF GENERALIZED GROUPS
∗ Hassan Maleki
Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
and MohammadReza Molaei
Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
Abstract. Our ultimate goal in this paper is to introduce a special type of topological spaces including manifolds and also, orbifolds. Because of using of generalized groups, we call them GG-spaces. We will study their properties, and then we will introduce a specialGG-space that is not manifold and orbifold. Finally we obtain conditions that cause aGG-space to become manifold.
Keywords: Generalized group,T-Space, Quotient space, Orbifold.
AMS Mathematical Subject Classification [2010]: 22A20, 22A99, 16W22..
1. Introduction
One of the interesting problems in geometry is to extend our definitions in order to add more objects to a certain category. We know geometric objects like torus and spheres are manifolds, but cones aren’t. Extending the notion of manifolds one can define a new structure called orbifold to include cones and some other objects as well. Intuitively, a manifold is a topological space locally modeled on Euclidean space Rn. Manifolds have origins in Carl Friedrich Gauss’s works and Bernhard Riemann’s lecture in Gottingen in 1854 laid the foundations of higher-dimensional differential geometry. As an extension of manifolds, an orbifold is a topological space locally modeled on a quotient ofRn by the action of a finite group. The simplest examples of orbifolds are cones, lens spaces andZp-teardrops. Orbifolds lie at the intersection of many different areas of mathematics, including algebraic and differential geometry, topology, algebra and string theory [10]. GG-spacesare a fascinating extension of orbifolds and manifolds. We can be roughly described aGG-space as a topological space that is locally modeled on a quotient ofRn by thegeneralized actionof atopological generalized group. GG-spaces will yield a geometrical and algebraic device useful for showing the existence of structures that are not a manifold or an orbifold such as Example (3.5).
Let us recall the definition of orbifolds. They were first introduced into topology and differential geometry by Satake [9], who called themV-manifolds. Satake described them as topological spaces generalizing smooth manifolds and generalized concepts such as de Rham cohomology and the Gauss-Bonnet theorem to orbifolds. The late 1970s, orbifolds were used by Thurston in his work on three-manifolds [10]. The nameV-manifoldwas replaced by the wordorbifoldby Thurston. An orbifoldO, consists of a paracompact, Hausdorff topological spaceXO called theunderlying space, such that for eachx∈XOand neighborhoodU ofx, there exists a neighborhoodUx⊆U, an open setU˜x∼=Rn, a finite groupGx acting continuously and effectively onU˜x which fixes0∈U˜x, and a homeomorphismϕx: ˜Ux
/
Gx →Uxwith ϕx(0) =x[2].
2. Preliminaries
Generalized groups or completely simple semi-groups [1] are an extension of groups. This notion has been studied first in 1999 [4,5, 7]. Topological generalized groups have been applied in geometry, dynamical systems and also genetic [6]. The notion of generalized action [4] is an extension of the notion of group actions. Furthermore, the notion ofT-spaces have been introduced
∗speaker
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H. Maleki and M. R. Molaei
and studies as an extentionof the notion ofG-spaces using of topological generalized groups [3].
We refer to [7,8,3] for more details. We start by recalling the notions of topological generalized groups and their generalized action on a topological space.
Definition 2.1. [5] Atopological generalized group is a Hausdorff topological spaceT which is endowed with a semigroup structure such that the following conditions hold:
• For eacht∈T, there is a unique e(t)∈T such thatt·e(t) =e(t)·t=t,
• For eacht∈T, there is s∈T such thats·t=t·s=e(t),
• For eachs, t∈T,e(s·t) =e(s)·e(t),
• The generalized group operationsm1:T →Tdefined bym1(t) =t−1andm2:T×T →T defined bym2((s, t)) =s·tare continuous maps, wheret−1∈T witht·t−1=t−1·t=e(t).
Example2.2. LetT be the topological spaceR\{0}. We can see thatT with the multiplication x·y=x|y|is a topological generalized group. The identity set e(T)is{−1,1}.
Example 2.3. IfT is the topological space
R2− {(0,0)}={reiθ| r >0 and 0⩽θ <2π} with the Euclidean metric, thenT with the multiplication
(r1eiθ1)·(r2eiθ2) =r1r2eiθ2 (1)
is a topological generalized group. We havee(reiθ) =eiθ and(reiθ)−1 = 1reiθ. So we can see the identity sete(T)is the unit circle S1. However,T is not a topological group.
Definition 2.4. LetX be a topological space and let T be a topological generalized group.
A generalized action of T on X is a continuous map λ : T ×X −→ X such that the following conditions hold:
• λ(s, λ(t, x)) =λ(s·t, x), fors, t∈T andx∈X;
• Ifx∈X, then ise(t)∈T such thatλ(e(t), x) =x.
3. Main results Definition 3.1. For each x∈T, we define
Tx={t∈T| tx=x}
called thestabilizer ofxin T. A generalized actionλof T onX is calledperfectife(T)⊆Txfor eachx∈X. Moreover,λis calledsuper perfect if for eachx∈X, e(T) =Tx.
Now we are ready to define GG-spaces. A GG-space is a topological space that is locally homeomorphic to a quotient of Rn by the generalized action of a topological generalized group.
First, we need to define charts.
Definition 3.2. Let X be a topological space. Then a chart for X is a (U,U , φ, T)e where U is an open subset of X, Ue is an open subset of Rn, T is a topological generalized group that acts continuously onUe by a generalized actionλandφ:Ue −→U is a continuous map inducing a homeomorphism betweenUe/T andU.
Definition 3.3. The collection {(Ui,Uei, φi, Ti) :i∈I} of charts ofX is said to be an atlas forX if the following properties are satisfied:
• {Ui:i∈I}is a cover ofX that closed under finite intersection;
• wheneverUi⊂Uj, there is an injective generalized group homomorphism fij:Ti,→Tj
and an embedding
ψij:Uei,→Uej
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Figure 1. TheGG-space which is not an orbifold.
such that fort∈Ti,
(2) ψij(tx) =fij(t)ψij(x)
and also
(3) φj◦ψij=φi.
Definition 3.4. AnGG-space is a pair (X,A)where X is a topological space and A is an atlas forX.
In the following example, the distinction between the geometrical structure of GG-spaces and classical geometrical structures such as Manifolds and orbifolds is well illustrated. In the Manifold theory, no center is considered for the unit circle, but in the concept of GG-spaces we are able to consider the unit circle with its center as a connected geometric structure.
Example3.5. LetY =R2andT be the generalized group of example2.3which acts onY by (r1eiθ1).(r2eiθ2) =r1r2eiθ2.
We can see thatTx=e(T), for eachx∈Y, so the action ofT is super perfect. Forx=r1eiθ1 and y=r2eiθ2,[x] = [y] if and only ifθ1 =θ2. Now supposeX :=Y/T. We can see that (X, Y, π, T) is a chart for X where π : Y → X is the projection map. Moreover, X is homeomorphic to S1∪
{(0,0)}(See Figure 1). Note thatX is a connected space with the quotient topology.
Theorem 3.6. TheGG-space(X,A)is an orbifold if every topological generalized groupTi is a finite group. Moreover,(X,A)is a manifold if every topological generalized groupTi is trivial.
Proof. Using the definition of an orbifold [10] and a manifold , we can proof this theorem. □ Note. There areGG-spaces that are not a orbifold. (See Example3.5).
Theorem 3.7. For any open connected T-space (X, T, λ) that X ⊆ Rn, the quotient space X/T is aGG-space.
Theorem 3.8. Let (X,A) be a GG-space. If every topological generalized group Ti is finite and its generalized action is super perfect, thenX is a manifold.
Proof. We know that for each x ∈ X there is a chart (U,U , φ, Te ) such that x ∈ U and Ue ⊆ Rn and a continuous mapφ : Ue → U induces a homeomorphic between Ue/T and U. We claim thatUe/T is locally euclidean ,i.e. U is locally euclidean and thenX is a manifold.
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H. Maleki and M. R. Molaei
Since the generalized action of T onUe is super perfect,tz̸=z for eacht /∈e(T)and for each z∈U. Moreover,e T is finite, so we can say that for eachz∈Ue there is a neighborhoodVe ⊆Ue of zsuch that
(4) tVe∩Ve =∅,
wheret /∈e(T).
Now we consider the projection mapπ:Ue →Ue/T. We will show thatπ(Ve)is an open subset ofUe/T that is homeomorphic to the open subsetVe ofRn. This implies thatUe/T and alsoU are locally euclidean.
We can see thatπ−1(π(Ve)) =∪
tVe, wheret∈T. Since the action ofT onUeis perfect, so every λt:X →X defined byλt(x) =tx, is a homeomorphism and so is an open map. So tVe =λt(eV) is an open subset ofUe. Soπ−1(π(Ve))is open in U. According to the quotient topology,e π(Ve)is open inU/e T. Moreover, we knew thatπ|e
V :Ve →π(Ve)is an open surjective continuous map. Also using4, it is injective. Soπ(eV)is homeomorphic toVe andU/e T is locally euclidean. Therefore U
is locally euclidean. □
References
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