Folding on manifolds and their fundamental group

ABSTRACT In this paper, the induced sequence of folding and unfolding on the fundamental group will be obtained from a sequence of folding and unfolding on a manifold. The limit of folding and unfolding on the fundamental group is deduced. The sequences of the commutative diagram of fundamental groups will be achieved from the sequences of the commutative diagram of manifolds. The connection between a manifold and a fundamental group is assigned.


Introduction and definitions
Classical results in algebraic topology give that groups are naturally realized as fundamental groups of spaces. For instance, free groups arise as fundamental groups of wedges of circles, any group can be realized as the fundamental group of some CW-complex, and pushouts of groups arise via the van Kampen theorem. This ability to construct geometric interpretations of discrete groups has important applications in both topology and algebra. The fundamental group was introduced by Poincare on a topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. Intuitively, it records information about the basic shape, or holes, of the topological space [1]. Let X be a topological space and x 0 ∈ X, the set π 1 (X, x 0 ) = {homotopy classes of loops in (X, x 0 )}, together with the product operation [f ][g] = [f .g] is called the fundamental group where π 1 is a functor map between categories [1]. The fundamental groups of some types of a manifold were studied in [2][3][4][5][6][7]. An n-dimensional manifold is a Hausdorff space M such that each point in M has a neighbourhood homeomorphic to R n [8]. A submanifold of a manifold M is a subset N which itself has the structure of a manifold [1,8]. Let X be a topological space and let A ⊆ X be a subspace. We say A is a retract of X if there exists a continuous map r : X → A such that r(a) = a, ∀a ∈ A [8,9]. Given spaces X and Y where x 0 ∈ X and y 0 ∈ Y, and X ∩ Y = ∅, then we define the wedge sum X ∨ Y as the quotient of X ∩ Y by identification x 0 ∼ y 0 [1]. Let M 1 and M 2 be two smooth manifolds of dimension m 1 and m 2 respectively. A map F : M 1 → M 2 is said to be an isometric folding of M 1 into M 2 if for every piecewise geodesic path ϒ : I → M 1 the induced path F • ϒ : I → M 2 is piecewise geodesic and of the same length as ϒ [10]. If F does not preserve length it is called topological folding [11]. A map φ : M 1 → M 2 is said to be unfolding of M 1 into M 2 if, for every piecewise geodesic path ϒ : I → M 1 , the induced path φ • ϒ : I → M 2 is piecewise geodesic with a length greater than ϒ [12]. For more information about the folding on manifolds and topological spaces, see [13][14][15][16][17].
is an infinite cyclic group.
(ii) Let where F 2 is a free abelian group of rank 2.

Theorem 2.5:
is an infinite cyclic group.
Theorem 2.6: Let X 0 denote the field of real numbers. Then there is a sequence of unfoldings φ i : X i−1 → X i with variation curvature which induce foldingsφ i :

) is a free group on a countable set of generators.
Proof: Consider the following sequence of unfoldings with variation curvature: φ 1 : Figure 5, which induce a foldingφ i : Therefore, lim m→∞ (φ m (π 1 (X m−1 ))) is a free group on a countable set of generators.

Theorem 2.8: Suppose that M is a manifold of dimension one, N is a submanifold of M and let
Since π 1 is a functor between manifolds and fundamental groups, we get the following sequence of the commutative diagram of fundamental groups:

Conclusion
In the present paper, we achieved the limit of folding and unfolding on the fundamental group. The relation between limits of foldings and retractions on the induced fundamental groups from viewpoint of a commutative diagram is obtained. New types of folding on the fundamental groups are deduced.