Effect of Some Geometric Transfers on Homology Groups

In this work, we introduce the results of some geometric transformation of the manifold on the homology group. Some types of folding and unfolding on a wedge sum of manifolds which are determined by their homology group are obtained. Also, the homology group of the limit of folding and unfolding on a wedge sum of 2manifolds is deduced.


Introduction
The notion of folding on manifolds has been introduced by (Robertson, 1977).The conditional folding of manifold and a graph folding have been defined by (El-Kholy, 1981-2005).Also, the unfolding of a manifold has been defined and discussed by (M.El-Ghoul, 1988).Many authors have studied the folding of many types of manifolds.The homology groups of some types of a manifold are discussed by (M.El-Ghoul, 1990;M.Abu-Saleem, 2010).(Abu-Saleem, 2007) introduced the results of some geometric transformation of the manifold on the fundamental group.In this paper, we introduce the folding and unfolding of some types of manifolds which are determined by their homology group and we study and discuss the homology group of the limit of folding and unfolding on a wedge sum of 2-manifolds.

Preliminaries
In this section, we introduce some necessary definitions which are needed especially in this paper.
Definition 2.1 The n -dimension manifold is a Hausdorff space such that each point has an open neighborhood homeomorphic to the open n-dimensional disc , where n is positive integer (W. S. Massey,   1976).
where V is a finite set whose elements are called the vertices of X and S is a set of non-empty subsets of V .Each element , s is called an n -simplex. (Thus an abstract simplex is merely the set of its vertices).The simplexes of X satisfy the following conditions; (1) The dimension of S is n and the dimension of X is the largest of the dimensions of its simplexes (P.J. Giblin, 1977)., called the n -th homology group of S (P.J. Giblin, 1977).

Notation:
) (S H n is measure the number of independent n -dimensions of holes in S , where Definition 2.5 Let M and N be two manifolds of dimensions m and n respectively.A map is piecewise geodesic and of the same length as  .If f does not preserve length, it is called a topological folding (E.El-Kholy, 1981;S.A.Robertson, 1977).Definition 2.6 Let M and N be two manifolds of the same dimensions.A map is said to be unfolding of M into N if, for every piecewise geodesic path geodesic with length greater than  ( M. El-Ghoul, 1988).
Definition 2.7 Let M and N be two manifolds of the same dimensions and Definition 2.8 Let X and Y be spaces, and choose points obtained by identifying 0 x and 0 y to a single point Hatcher, 2001, http://www.math.coronell.edu/hatcher).

The Main Results
In this section, we will introduce the following: of any folding of 2 S is either isomorphic to  or identity group.
Proof.First, for folding with singularity of 2 S as in Figure 1(a) then . Also, folding without singularity of 2 S is a manifold homeomorphic to 2 S as in Figure 1(b) and so , )) ( ( Proof.First, for folding with singularity of T as in figure 2 . Also, folding without singularity of T is a manifold homeomorphic to T as in figure 2(b) and so , )) ( ( Therefore any folding of T is either isomorphic to  ,    or identity group.is isomorphic to  or identity group.
or identity group, for all n .

Proof.
If 2 , 1 , : or identity group.: , then we have the following: are two types of foldings such that is isomorphic to  or identity group, for all n .

Proof. Since
is isomorphic to  or identity group, for all n .: or identity group, for all , then we have the following: as in Figure 7(a), then or identity group, for all 0  n .
Theorem 3.10 If 2 , 1 , : , we have the following: as in Figure 8(a).Then, or identity group, for all 0  n .

Conclusion.
Folding and unfolding problems have been implicit for long time, but have not been studied extensively in the mathematical literature until recently .Over the past few years; there has been a surge of interest in these problems in discrete and computational geometry.This paper gives the folding and unfolding of some types of manifolds, which are determined by their homology group and we discussed the homology group of the limit of folding and unfolding on a wedge sum of 2-manifolds.
The main results can be similarly extended to some other some geometric shapes such as polyhedra .The problems lies: Can all convex polyhedra be edge-unfolded, and can all polyhedra be generally unfolded?

Figure 1
Figure 1 Corollary 3.2 The homology group of the limit of folding and unfolding of a manifold which is homeomorphic to 2 , 2  n S is the identity group.Theorem 3.3 The homology group ) (T H n of any folding of T is either isomorphic to  ,    or identity

Figure 2 Corollary 3 . 4
Figure 2 Corollary 3.4 The homology group of the limit of folding and unfolding of a manifold which is homeomorphic to 2 ,  n T is the identity group.Theorem 3.5 If 2 , 1 , : 2 2 Figure 4 Figure 6

Figure 9 D
Figure 9 the sequence,