The evolution of this folded-symplectic geometry happened in 1969 by Martinet. However, on a local four-orbifold the author would like to demonstrate a two form consisting a singularity of type Σ2,0 & represented in coordinate system asydy∧dz+dx∧dt to show folded-symplectic form. It can also be denoted as 4R with the help of fold map ),,,2(),,,(2wxzywxzyto validate the name fold symplectic. Cannas da Silva has startedto unveil the folded-symplectic geometry in late nineties to emphasize on the existence of spin-c structures to the existence of symplectic orbifolds. In literature researchers illustrate an unfolding concept in which a folded-symplectic orbifold can be divided into symplectic pieces. Guillemin, Cannas and Piresrevealed compact Numerical origami thus the folding hypermethods� fibrates over a compact base by a locally free circle action. It shows that Numerical origami orbifolds as well as templates of unimodular polytopes in the dual of the Lie Integral of the torus, T∗. This is nothing but the generalization of Delzant‘s theorem which illustrates one-to-one correspondence among compact symplectic orbifolds & polytopes in T∗. Furthermore, the above polytope is delzant which is simple, smooth rather rational. The labelled polytopes & symplectic Numerical manifolds, it can be observed thatpolytopes have combinatorial properties to achieve some geometric information regarding Numerical manifolds. The main aim of this work is to broad these fundamental concepts to Numerical, foldedsymplectic orbifolds as well as non-compact. Lee proposed in literature about isomorphismamong Numerical, folded symplectic 4-orbifolds.

Numerical orbifold can be denoted as (Ő2m, η) if equipped with an effective, Hamiltonian action of a torus (T) with dimension m which is half the dimension of the orbifold.

An integral system states that a symplectic orbifold Ő2m equipped either with m linearly independent Poisson functions p1, p2,p3. . . , pnor Hamiltonian vector fields.Thus, a Numerical orbifold is an integrable system for which these above functions taken in a manner that the Hamiltonian vector of the Poisson functions are unit periodic. The moment map θ(Ő) is represented as convex polyhedron or Newton polytope of Ő. All polytopes emerging from folded Numerical orbifolds fulfil the below specified:

1. For every vertex vexists there are m edges leaving it. 2. These edges having relation v + twi (i = 1,.., m) where wi∈ (Qm)∗ (= ΛWeight )). 3. These weights w1,.., wm form a basis of the weight lattice ΛWeight, for every vertex v. Remark: Ő is a folded Numerical manifold if and only if the above mentioned condition 1 and 2 are satisfied.

Numerical orbifolds (according to Delzant theorem) are arrangedin terms of moment polytopes, or we can call it as, if Ő1, Ő2 are two folded Numerical orbifolds with moment maps θ(Ő1) and θ(Ő2) and θ(Ő1) =θ(Ő2), then it is said that there exist a Tnfolded symplectic equivariant diffeomorphism amongŐ1 andŐ2. If a polytope P fullfill above mentioned 1-3 proposition one can say a folded Numerical symplectic orbifold Ő with θ(Ő)=P.

Thus, },:{1jinridaRaP for diϵ Rn, ηiϵ Rn.

If the polytope P is illustrated as an n-dimensional polyhedron in Rn, then (a) ithdimentional face (Mi)of polytope Pif and only if Mi is an i-simplex (b) IntMi is nothing but congruent to the interior of the mentioned i-simplex. (c) All point in given polytope P is in the interior of exactly single face.

A facet is defined for an m-dimensional polytope P is nothing but an (m − 1)-dimensional face. The count of desired facets in polytope P is f: the facets primitive. The exact vector spaces V sequence can be represented as: 00nEplRRr Where, kkE: and thus ηk∈ΛWeight = Hom (Rn , 2πR), this resultsinto 1)1()1(1nEplAA And it shows )(EKer is a torus. The well-known fact for the moment map for the action of A(1)p on Cp is shown below:

arrrrGpp),......,(2

1),.....,(:2211.

However, pa,.....,1. Thus, for the action of given  overCp the moment map is expressed by θ*(Ő) where

00***

rRRrpEp.

The moment polytope is demonstrated with the help of the right triangle with three vertices (0, 1), (0, 0) & (1, 0). Assumeηkcan be represented for ith face normal vector. η1 = (0, −1), η2 = (−1, 0) and η3 = 1/2 (1, 1). 23RRrE kkE:

)1,0()0,0,1( )0,1()0,1,0( )1,1()1,0,0(

 

  

χ1=χ2=0 and χ3=1/2

101

r = R(1, 1, 1) ⊂ R3 = Ker(E) and )1(A thus, C3 can be minimized w.r.t. action of & demonstrated by:

21"*

2

3212321

"*

321232221321

" * 3* 3

/)0().(

2 1 2 1

)(2 1)},,({ ),,(),,(2 1),,( )1,1,1(,)(

:

)1,1,1(:



   





 



 





Theorem 1: Let us assume (Ő, η, θ :Ő → T∗ ) be a Numerical, folded-symplectic orbifold with moment map θ : Ő → T∗, where T* is the Lie Integral of the torus T acting on Ő. Let us say the fold f⊆ Ő is co-orientable. Thus, a. Ő/T is generally represented as a orbifold with corners b. The moment map θfall downwards to a

Definition: Let us say corners C and assume unimodular map with orbifold represented by *:tC. Where t is nothing but Lie Integral of torus T. Thus, )("CO can be defined as a category having triple objects. (Ő, η, η : Ő → C) Hence, Numerical folded symplectic orbifold described by (Ő, η, θ.η). where, η is quotient map &θ.η is moment map with torus T. However, a morphism among objects (Őj, ηj, ηj : Ő → C) in case of two j=1,2. So the commutative diagram for 2"1":OO andit is also seen that θ ∗η2 = η1. Whereas, θ is elaborated as an equi-variant symplectic morphism that preserves moment maps. By the given definition, all morphism is invertible, thus it can be said )("CO is a groupoid. Theorem 2: Isomorphism family of objects ingroupoid )("CO having mapping both one-to-one with L2(Ő; IT × R), where IT = ker(exp: t → T) is nothing but the integral lattice form of the torus orbifold T whichworks on various objects of )("CO.

The main focus behind this topic to understand a Numerical orbifold from its moment polytope. After studying the fundamentalsof Morse theory, it can be determinedthe following: firstly the homology of symplectic Numerical orbifolds; Secondlyto find suitable Morse functionality given by a moment map w.r.t. a fit subgroup of circle. In this section, we emphasis on basic surgery constructions lied over symplectic reduction, which clasp in the family of symplectic Numerical orbifolds.

The below mentioned two theorems illustrate principle neighbourhoods of fixed points. The proofs of these theorems depend on the equivariant class of the Moser trick.

Assume thatorbifold pair (Ő, η) is 2m-dimensional symplectic orbifold provided witha symplectic action of a compact Lie group L, and suppose f be a fixed point. So, the Darboux chart for orbifold is a chart (D, a1, a2….an, b1, b2,…..bn) centred at f and L-equivariant w.r.t. a linear action of Lover R2m such that η





1

|

Though, there exists a Darboux chart centred at every point of a symplectic orbifold by the above Darboux theorem.

distinguished by the weights over(TfŐ). So any specified symplectic action is known to be Hamiltonian. In order to develop the mathematical calculation of the Betti numbers for symplectic Numerical orbifold with the help of moment map as defined in Morse functionality, further identify the general image of moment map around a fixed point f of a Hamiltonian torus action.

Let us suppose that ),,,(2"mmTO canbe defined as a Hamiltonian Tm-space, where f is said to be a fixed point. Then there occurs a chart (D, a1, a2….an, b1, b2,…..bn) centred at f as well as weights m,.......,1 belongs to Zm. It can be illustrated by

iiiDdbda1|

and •

)(2

1)(|22

iDbaf

This theorem promises the phenomenon of a Darboux chart centred at any fixed point f where the moment map seems to be the moment map for a linear action on R2m.

SupposeŐrepresents m-dimensional orbifold and Morsedefine a smooth function over orbifold �: Ő → R if and only if all critical points of this function are non-degenerate. However, the index of a bilinear function having maximal dimension of a subspace of R with this relation B: Rn×Rn → R for non-positive values of B.The invalidity of B is nothing but the dimension of its nullspace, such that, the subspace containing of every η belongs to Rn to demonstrate the equality B(η,η) = 0 for each η element ofRn. Thus, a critical point r of fixed point f i.e. �: Ő → R is non-degenerate when the Hessian relation satisfy following: Hr: Rn×Rn → R consist invalidity equal to zero. Although r is nothing but a non-degenerate critical point for orbifold �: Ő → R. The index of this smooth function at point r is known as the index of the hessian relation mentioned above. It is well known a1, a2….an, b1, b2,…..bn) centred at r in a manner that 221221)(....)()(......)()(|nDaaaarSS Where ξ represents the index of � at point r. So, it can be said that non-degenerate critical points are mandatory isolated. Suppose � defines a Morse basis over orbifold Ő. For x∈ R, we consider the equality shown below:



A. COz)("

B. 

COz)1()()1("

C. 01"0"1"......)(.......)()(CCCOyOyOy. Thus, after these findings it is quite easy to say that a Morse basis over orbifold is known to be a Morse function for which the above mentioned inequalities are to be equalities.

Let us suppose that ),,,("TO canbe defined as a 2m-dimensional symplectic Numerical orbifold. Now, choose a vector V whose elements do not depends on U by selecting a proper generic direction in Rm. This outcome ensures followings:

• The single-dimensional subset, TV belongs to Tm, caused by the vector V is opaque in Tm

• Vector V of the moment polytope is having sequence to the facets represented by

)(:"O

• All the vertices of ϒ have various projections along vector V.

Let us suppose that Gis known to be the tautological line bundle on Qm-1, it means:

estimate to Q represented by ([Q], c) → [Q]. The fiber of G over the point [Q] belongs to Qm-1is nothing but the complex line in Cm introduced by that point.

The symplectic orbifold blow-up of Cm at the genesis is known as the total space of the bundle G. The respectivesymplectic orbifold blow-down map is represent the map ζ : G → Cm illustrated by following: ζ([Q], c) = c. It canalso be observed that the total space of G may be putrefiedin a manner that nothing is common in two groups, }}0{\|)0],{([:mCQQU and qmcCqcQU},0{\|)],{([: for some η ϵ C*}. The above set U is known as the exceptional diffeo-morphic divisor to Qm-1 and finds mapped to the creation by ζ. Furthermore, the limitation of ζ to the complementary group ∩ is said to be a diffeomorphism over Cm\{0}. Thus, the conclusive remarks is that Gis procured from Cm by replacing with the image of Qm-1.

A symplectic m-dimensional manifold/orbifold(Ő)is illustrated with the help of a closed second form ηwhere, ηmdiminishes transversally as well as η is confined maximally non-degenerate hyper methods H.This is the way of introducing folded-symplectic form which is nothing but the conjunction of more than one symplectic manifolds.A Numerical, folded symplectic orbifold can be proved that a folded-symplectic manifold pair (Őm, η) equipped with an effective, Hamiltonian action of a torus (T) with dimension m. In this chapter classification of Numerical orbifold has been discussed in terms of Delzant theorem, symplectic reduction as well as Morse fundamentals. Furthermore, moment polytope also has been illustrated using Darboux theorem.

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Research Scholar of OPJS University, Churu, Rajasthan