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The aim of this study is to establish that, the equivalent class
*m*,*n*]. The study proved from homotopical point of view that
*m*,*n*] ,

The properties of a topological space that were developed so far have been depended on the choice of topology, the collection of open sets. Taking a different tack, we introduce a different structure, algebraic in nature, associated to a space together with a choice of base point ( X , x 0 ) . The structure will allow us to bring to bear the power of algebraic arguments [

Furthermore, topologists of the early 20th century dreamed of a generalisation to higher dimensions of the non-abelian fundamental group, for applications to problems in geometry and analysis for which group theory had been successful.

For some decades now, the theorem Π 1 ( S , x o ) is a group, established to be a fundamental group with respect to “ ∘ ” in the interval [0, 1] [

This led to a search for the uses of groupoids in higher homotopy theory, and in particular for higher homotopy groupoids. The basic intuitive concept was generalising from the usual partial compositions of homotopy classes of paths to partial compositions of homotopy classes (of some form) of complexes. But a search for such constructs proved abortive for some years from 1966 [

Recently, the concept of fundamental group was used on the ageing process of human. The homotopy relates the topological shape of the infant to the topological shape of the adult. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. This complex connected cylindrical shape of the body x = S 1 × I , described by the homotopic functions f , h : x → x provided the ageing process in vertical interval I = [ 0, β ] . But this work is limited to the interval [ 0, β ] [

Therefore this research intends to establish the proof that Π 1 ( S , x o ) is a group in other domains other than [0, 1]. In this paper, we defined homotopy and presented some group properties. We then described the fundamental group and its related properties such as group homomorphisms. We also looked at useful definitions and theorems that play an important role in computing fundamental groups, and our main result was actually built upon these theorems. In the end, we established the proof that the equivalent class Π 1 ( S , x o ) is a fundamental group in the interval [ m , n ] ∀ n , m ∈ Z + .

Throughout this paper we assumed the knowledge of basic algebra and general topology.

Definition 2.1.1

Let X be a topological space. A path in X from x_{0} to x_{1} is a continuous map f : I → X such that f ( 0 ) = x 0 and f ( 1 ) = x 1 . We say that x_{0} is the initial point and x_{1} the final point [

Definition 2.2.1

Suppose X is a space and x 0 ∈ X is a choice of base point in X. The space of based loops in X denoted Ω ( X , x 0 ) , is the subspace of map ( [ 0,1 ] , X ) ,

Ω ( X , x 0 ) = { λ map ( [ 0,1 ] , X ) | λ ( 0 ) = λ ( 1 ) = x 0 } (1)

Composition of loops determines a binary operation ∗ : Ω ( X , x 0 ) × Ω ( X , x 0 ) → Ω ( X , x 0 ) . We restrict the notion of homotopy when applied to the space of based loops in X in order to stay in that space during the deformation.

Definition 2.2.2

Given two based loops λ and μ , a loop homotopy between them is a homotopy of paths H : [ 0,1 ] × [ 0,1 ] → X with H ( t , 0 ) = λ ( t ) , H ( t , 1 ) = λ ( t ) and H ( 0 , s ) = H ( 1 , s ) = x 0 . That is, for each s ∈ [ 0,1 ] , the path t ↦ H ( t , s ) is a loop at x 0 .

Proposition 2.2.3

Continuous mappings F : W → X and G : Y → Z induce well-defined functions F ∗ : [ X , Y ] → [ W , Y ] and G ∗ : [ X , Y ] → [ X , Z ] by F ∗ ( [ h ] ) = [ h ∘ F ] and G ∗ ( [ h ] ) = [ G ∘ h ] for [ h ] ∈ [ X , Y ] .

Proof. We need to show that if h ≃ h ′ , then h ∘ F ≃ h ′ ∘ F and G ∘ h ≃ G ∘ h ′ . Fixing a homotopy H : X × [ 0,1 ] → Y with H ( x ,0 ) = h ( x ) and H ( x ,1 ) = h ′ ( x ) , then the desired homotopies are H F ( w , t ) = H ( F ( w ) , t ) and H G ( x , t ) = G ( H ( x , t ) ) .

To a space X we associate a space particularly rich in structure, the mapping space of paths in X, map ( [ 0,1 ] , X ) . Recall that map ( [ 0,1 ] , X ) is the set of continuous mappings Hom ( [ 0,1 ] , X ) with the compact-open topology. The space map ( [ 0,1 ] , X ) has the following properties:

1) X embeds into map ( [ 0,1 ] , X ) by associating to each point x ∈ X to the constant path, c x ( t ) = x for all t ∈ [ 0,1 ] .

2) Given a path λ : [ 0,1 ] → X , we can reverse the path by composing with t ↦ 1 − t . Let λ − 1 ( t ) = λ ( 1 − t ) .

3) Given a pair of paths λ , μ : [ 0,1 ] → X for which λ ( 1 ) = μ ( 0 ) , we can compose paths by

λ ∗ μ ( t ) = { λ ( 2 t ) , if 0 ≤ t ≤ 1 / 2 μ ( 2 t − 1 ) , if 1 / 2 ≤ t ≤ 1 (2)

Thus, for certain pairs of paths λ and μ , we obtain a new path λ ∗ μ ∈ map ( [ 0,1 ] , X ) . Composition of paths is always defined when we restrict to a certain subspace of map ( [ 0,1 ] , X ) [

Definition 2.2.4

Let X be a topological space, and x 0 a point in X. The fundamental group of X is the set of path homotopy classes [ f ] of loops f : I → X based at x 0 , together with the operation “ ∘ ”. We denote it by Π 1 ( X , x 0 ) [

Definition 2.2.5

Given two loop classes [ f ] and [ g ] we define:

1) [ f ] ∗ [ g ] = [ f ∗ g ] .

2) The inverse of [ f ] is given by [ f − 1 ] , that is [ f ] − 1 , where f − 1 ( t ) = f ¯ ( t ) = f ( 1 − t ) .

Theorem 2.2.6

If X is a convex subset of ℜ n , and if a , b ∈ X , α t ( s ) = ( 1 − t ) α 0 ( s ) + t α 1 ( s ) . This defines a homotopy between α 0 and α 1 .

Theorem 2.2.7

The relation f ≃ g is an equivalence relation on the set, Hom ( X , Y ) , of continuous mappings from X to Y.

Proof. Let f : X → Y be a given mapping. The homotopy H ( x , t ) = f ( x ) is a continuous mapping H : X × [ 0,1 ] → Y and so f ≃ f .

If f 0 ≃ f 1 and H : X × [ 0,1 ] → Y is a homotopy between f 0 and f 1 , then the mapping H ′ : X × [ 0,1 ] → Y given by H ′ ( x , t ) = H ( x ,1 − t ) is continuous and a homotopy between f 1 and f 0 that is f 1 ≃ f 0 .

Finally, for f 0 ≃ f 1 and f 1 ≃ f 0 , suppose that H 1 : X × [ 0,1 ] → Y is a homotopy between f 0 and f 1 , and H 2 : X × [ 0,1 ] → Y is a homotopy between f 1 and f 2 . Define the homotopy H : X × [ 0,1 ] → Y by

H ( x , t ) = { H 1 ( x ,2 t ) , if 0 ≤ t ≤ 1 / 2 H 2 ( x ,2 t − 1 ) , if 1 / 2 ≤ t ≤ 1

Since H 1 ( x , 1 ) = f 1 ( x ) = H 2 ( x , 0 ) , the piecewise definition of H gives a continuous function. By definition, H ( x ,0 ) = f 0 ( x ) and H ( x ,1 ) = f 2 ( x ) and so f 0 ≃ f 2 [

Definition 2.2.8

The equivalence classes of maps from X to Y as in the Theorem 2.2.7 are called the homotopy classes ad denoted by [f] as the homotopy class of the map f. While we use the notation 〈 γ 〉 when γ is a loop in X based at x 0 ∈ X .

Definition 2.3.1

The equivalent classes [ f ] determined by homotopy modulo x 0 on the collection C ( S , x ∘ ) of all closed paths f on S based at x ∘ ∈ S are called homotopy classes of C ( S , x ∘ ) . The collection of these homotopy classes is denoted by Π 1 ( S , x ∘ ) [

Definition 3.1.1

1) If f , g ∈ Π 1 ( S , x 0 ) we define the juxtaposition f ∘ g of f and g as follows:

( f ∘ g ) ( s ) = { f ( 2 s ) if m 2 ≤ s ≤ n 2 g ( 2 s − n ) if m + n 2 ≤ s ≤ n , ∀ m , n ∈ Z +

Thus f ∘ g ∈ Π 1 ( S , x 0 ) and “ ∘ ” is a binary operation on Π 1 ( S , x 0 ) .

2) If [ f ] , [ g ] ∈ Π 1 ( S , x 0 ) , then let [ f ] ∘ [ g ] = [ f ∘ g ]

Proposition 3.1.2

Let X , Y be topological spaces and let A a subset of X. Then ∼ A is ann equivalence relation on the set Map A ( X , Y ) of maps from X to Y which agree with with a given map on A.

Proof. For notational convenience, drop the subscript A from the notation.

1) Reflexive property f ∼ f : Define H ( x , t ) = f ( x ) . This is the composition of f with the projection of X × I on X. Since it is a composition of two continuous maps, it is continuous.

2) Symmetric property f ∼ g ⇒ g ∼ f : Suppose H : X × I → Y is a homotomy (relative to A) of f tog. Let H ′ ( x , t ) = H ( x ,1 − t ) . The H ′ ( x , 0 ) = H ( x , 1 ) = g ( x ) and similarly for t = 1 . Also, if H ( a , t ) = f ( a ) = g ( a ) for a ∈ A , the same is true for H ′ . H ′ is a composition of two continuous maps. What are they?

3) Transitive property f ∼ g , g ∼ h ⇒ f ∼ h : This is somewhat harder. Let H ′ : X × I → Y be a homotopy (relative to A) from f to g, and let H ″ be such a homotopy of g to h. Define

H ( x , t ) = { H ′ ( x ,2 t ) , for 0 ≤ t ≤ 1 / 2 H ″ ( x ,2 t − 1 ) , for 1 / 2 ≤ t ≤ 1

Note that the definitions agree for t = 1 / 2 . We need to show H is continuous.

Corollary 3.1.3

π 1 ( ℝ n \ { 0 } ) ≅ π 1 ( S n − 1 ) .

Theorem 3.1.4

(Eckmann-Hilton). Let G be a group space and e ∈ G be the identity point. Then p i 1 ( G , e ) is abelian.

Proof. To show p i 1 ( G , x ) is abelian, we will show that for any two loops γ , δ : I → G we have γ ∗ δ ∼ δ ∗ γ . Indeed for this, we construct a homotopy between the two loops above.

Theorem 3.1.5

(Van Kampen’s theorem). Suppose X = ∪ i = 1 n A α where each A α contains a green basepoint x 0 ∈ X so that each A α is path connected and each A α ∩ A β is path connected. We have homomorphisms π 1 ( A α , x 0 ) → π 1 ( X 0 , x 0 ) induced by the inclusions A α → X and homomorphisms i α β : π 1 ( A α ∩ A β , x 0 ) → π 1 ( A α , x 0 ) induced by the inclusions A α ∩ A β → X .

1) The homomorphism Φ : π 1 ( A 1 , x 0 ) ∗ ⋯ ∗ π 1 ( A n , x 0 ) → π 1 ( X , x 0 ) is surjective.

2) If further each A α ∩ A β ∩ A γ is path connected, then the kernel of Φ is the minimal normal subgroup N generated by all the elements of the form i α β ( ω ) i α β ( ω ) − 1 for ω ∈ π 1 ( A α ∩ A β , x 0 ) so Φ induces an isomorphism π 1 ( X , x 0 ) ≃ π 1 ( A 1 , x 0 ) ∗ ⋯ ∗ π 1 ( A n , x 0 ) / N .

Theorem 3.1.6

Π 1 ( S , x ∘ ) is a fundamental group with respect to “ ∘ ” in the general interval [ m , n ] .

Proof

1) “ ∘ ” is associative. We need to show that ( f ∘ g ) ∘ k x ¯ ˜ ∘ f ∘ ( g ∘ k ) for f , g , k ∈ Π 1 ( S , x ∘ ) .

[ ( f ∘ g ) ∘ k ] ( s ) = { ( f ( 2 s ) g ( 2 s − n ) ) ∘ k = { f [ 2 ( 2 s ) ] g [ 2 ( 2 s ) − n ] k ( 2 s − n ) = { f ( 4 s ) if m 4 ≤ s ≤ n 4 g ( 4 s − n ) if m + n 4 ≤ s ≤ n 2 k ( 2 s − n ) if m + n 2 ≤ s ≤ n , ∀ m , n ∈ Z +

and

[ f ∘ ( g ∘ k ) ] ( s ) = { f ∘ ( g ( 2 s ) k ( 2 s − n ) ) = { f ( 2 s ) g [ 2 ( 2 s − n ) ] k [ 2 ( 2 s − n ) − n ] = { f ( 2 s ) if m 2 ≤ s ≤ n 2 g ( 4 s − 2 n ) if n + 2 m 4 ≤ s ≤ 3 n 4 k ( 4 s − 3 n ) if m + 3 n 4 ≤ s ≤ n , ∀ m , n ∈ Z +

We define a homotopy between ( f ∘ g ) ∘ k and f ∘ ( g ∘ k ) as follows:

h ( s , t ) = { f ( 4 s n + t ) if 〈 s , t 〉 ∈ I 2 and t ≥ 4 s − m n m g ( 4 s − t − n ) if 〈 s , t 〉 ∈ I 2 and m + n 4 s ≥ t ≥ 4 s − 2 n k ( 4 s − t − 2 n 2 − t ) if 〈 s , t 〉 ∈ I 2 and 4 s − 2 n ≥ t , ∀ m , n ∈ Z +

Then the following is true:

h ( s , m ) = { f ( 4 s ) if 0 ≥ 4 s − n [ i .e . m 4 ≤ s ≤ n 4 ] g ( 4 s − n ) if 4 s − n ≥ 0 ≥ 4 s − 2 n [ i .e . m + n 4 ≤ s ≤ n 2 ] k ( 2 s − n ) if 4 s − 2 ≥ 0 [ i .e . m + n 2 ≤ s ≤ n ]

h ( s , n ) = { f ( 2 s ) if n ≥ 4 s − n [ i .e . 0 ≤ s ≤ m 2 ] g ( 4 s − 2 n ) if 4 s − n ≥ n ≥ 4 s − 2 n [ i .e . m + n 4 ≤ s ≤ 3 n 4 ] k ( 4 s − 3 n ) if 4 s − 2 n ≥ 1 [ i .e . m + 3 n 4 ≤ s ≤ n ]

Thus h ( s , m ) = [ ( f ∘ g ) ∘ k ] ( s ) and h ( s , n ) = [ f ∘ ( g ∘ k ) ] ( s ) for all s ∈ I n .

Also h ( m , t ) = f ( m ) = x 0 and h ( n , t ) = k ( n ) = x 0 for all t ∈ I n .

Hence ( f ∘ g ) ∘ k x ¯ ˜ 0 f ∘ ( g ∘ k ) .

2) We show that the constant mapping c : I n → { x 0 } is such that [c] is the identity element of Π 1 ( S , x 0 ) with respect to “ ∘ ”. Thus we must show that f ∘ c x ¯ ˜ 0 f for any f ∈ Π 1 ( S , x 0 ) . Let h : I 2 → S be defined as follows:

h ( s , t ) = { f ( 2 s n + t ) if 〈 s , t 〉 ∈ I 2 and t ≥ 2 s − m n m x ∘ if 〈 s , t 〉 ∈ I 2 and 2 s − m n m ≥ t , ∀ m , n ∈ Z +

Then

( f ∘ c ) ( s ) = h ( s , m ) = { f ( 2 s ) if 0 ≥ 2 s − n [ i . e . m 2 ≤ s ≤ n 2 ] x 0 if 2 s − n ≥ 0 [ i . e . m + n 2 ≤ s ≤ n ]

and h ( s , n ) = f ( s ) if 1 ≥ 2 s − n [i.e. m ≤ s ≤ n ]. Thus h ( s , m ) = ( f ∘ c ) ( s ) and h ( s , n ) = f ( s ) for all s ∈ I n . More so, h ( m , t ) = f ( m ) = x ∘ and h ( n , t ) = f ( n ) = x 0 for all t ∈ I n .

Hence f ∘ c x ¯ ˜ 0 f .

3) Finally we want to show that each homotopy class [ f ] ∈ Π 1 ( S , x 0 ) has an inverse [ g ] ∈ Π 1 ( S , x 0 ) such that [ f ] ∘ [ g ] = [ c ] . Thus we want to show that if f ∈ Π 1 ( S , x 0 ) there exist a g ∈ Π 1 ( S , x 0 ) such that f ∘ g x ¯ ˜ 0 c . Let g ( s ) = f ( n − s ) ∀ s ∈ I n . Since g ( m ) = f ( n ) = x ∘ = f ( m ) = g ( n ) , g ∈ Π 1 ( S , x ∘ )

by definition, we have

( f ∘ g ) ( s ) = { f ( 2 s ) if m 2 ≤ s ≤ n 2 g ( 2 s − n ) = f ( 2 n − 2 s ) if m + n 2 ≤ s ≤ n , ∀ m , n ∈ Z +

We then define homotopy h between f ∘ g and c as follows:

h ( s , t ) = { x 0 if 0 ≤ s ≤ n t 2 f ( 2 s − t ) if n t 2 ≤ s ≤ n 2 g ( 2 s + t − n ) if n 2 ≤ s ≤ n − n t 2 x 0 if n − n t 2 ≤ s ≤ n , ∀ m , n ∈ Z +

Since f and g are continuous, h is continuous and we have

h ( s , m ) = { f ( 2 s ) if m 2 ≤ s ≤ n 2 g ( 2 s − n ) if m + n 2 ≤ s ≤ n , ∀ m , n ∈ Z +

and

h ( s , n ) = x 0 if m ≤ s ≤ n

Thus h ( s , m ) = ( f ∘ g ) ( s ) and h ( s , n ) = c ( s ) ∀ s ∈ I n . Also h ( m , t ) = h ( n , t ) = x ∘ for all t ∈ I n . Hence f ∘ g x ¯ ˜ ∘ c .

This paper, in full accordance with the principles of homotopy, has been able to establish the proof that Π 1 ( S , x ∘ ) is a fundamental group in the general interval [ m , n ] , ∀ m , n ∈ Z + .

In general, Π 1 ( S , x ∘ ) depends upon x ∘ . However, in the case of an arc-wise-connected space S, we can show that Π 1 ( S , x ∘ ) is independent of x ∘ .

Sincere thanks to Mrs. Belinda Zigli and Prof. William Obeg-Denteh for their immersed supports. The authors wish to acknowledge the support of their respective Universities and the anonymous referees for their helpful comments towards the improvement of the paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Delali1, Z.D., William, O.-D., Lewis, B. and Kwame, A.R. (2021) The Homotopical Proof of as a Fundamental Group in a General Interval. Advances in Pure Mathematics, 11, 377-385. https://doi.org/10.4236/apm.2021.115024