Unraveling chaos: A topological analysis of simplicial homology groups and their foldings

Definition 2.1

The chaotic space is a new space that includes several physical characters, every one of them homeomorphic to the initial one, either with a fixed point or without a fixed point, as shown in Figures 1 and 2. We denote the chaotic space by X ˆ = ⟨ X 1 , X 2 , … ⟩ .

Definition 2.2

The chaotic cycle is an original cycle that carries several alternative cycles that are homeomorphic to each other, and the chaotic cycle ∑ e ˆ i can be represented as α ˆ = ⟨ α n 0 , α n 1 , … ⟩ and the chaotic point x ˆ is denoted as x ˆ = ⟨ x 0 , x 1 , … ⟩ . Also, we have two different chaotic cycles:

1. Chaotic cycles of type ( I ): All cycles ⟨ α n 0 , α n 1 , … ⟩ are identical physical characters.

2. Chaotic cycles of type ( I I ) ⟨ α n 0 , α n 1 , … ⟩ represent different physical characters; for example, α n 0 represents the density, α n 1 represents the colors, α n 2 represents the elasticity, etc. Moreover, we can represent the chaotic elements g ˆ of chaotic group G ˆ by g ˆ = ⟨ g 1 , g 2 , … ⟩ .

Definition 2.3

Given a chaotic simplicial complex L ˆ , we define the chaotic t -chain group C ˆ t ( L ˆ ) as the chaotic free abelian group that is generated by the oriented chaotic t -simplexes of L ˆ . Also, C ˆ t ( L ˆ ) = 0 ˆ whenever, t > dim ( L ˆ ) . Moreover, the chaotic t -chain is an element of C ˆ t ( L ˆ ) .

For ( 1 ≤ j ≤ N t ) , let σ ˆ t , j be t -chaotic simplexes in L ˆ . We can write x ˆ ∈ C ˆ t ( L ˆ ) as x ˆ = ∑ j = 1 N t x ˆ j σ ˆ t , j , x ˆ j ∈ Z ˆ ; the chaotic x ˆ j are referred to as the chaotic coefficients of x ˆ . The structure of a chaotic group is provided as the sum of two chaotic t -chains, y ˆ = ∑ j y ˆ j σ ˆ t , j and z ˆ = ∑ j z ˆ j σ ˆ t , j , which is defined as y ˆ + z ˆ = ∑ j ( y ˆ j + z ˆ j ) σ ˆ t , j . The chaotic unit element is 0 ˆ = ∑ j 0 ˆ . σ t , j , while the inverse chaotic element of x ˆ is − x ˆ = ∑ j ( − x ˆ j ) σ ˆ t , j . Note that C ˆ t ( L ˆ ) is a chaotic free abelian group of rank N t , C ˆ t ( L ˆ ) = Z ˆ ⊕ Z ˆ ⊕ ⋯ ⊕ Z ˆ . ︸ N t We must introduce the chaotic boundary operator before defining the chaotic cycle group and the chaotic boundary group. We must mention that the chaotic boundary of a t -chaotic simplex σ ˆ t by ∂ ˆ t σ ˆ t should be seen as an operator on σ ˆ t to construct its chaotic boundary. Now, we will discuss the case of chaotic boundaries in the chaotic lower-dimensional simplexes. Since there is no chaotic boundary on a 0 ˆ -chaotic simplex, we set ∂ ˆ 0 q ˆ 0 = 0 ˆ . For a chaotic 1-simplex ( q ˆ 0 q ˆ 1 ) , we define ∂ ˆ 1 ( q ˆ 0 q ˆ 1 ) = q ˆ 1 − q ˆ 0 .

Definition 2.4

For t > 0 , let σ ˆ t ( q ˆ 0 … q ˆ t ) be an oriented chaotic t -simplex. The boundary ∂ ˆ t σ ˆ t of σ ˆ t is defined by

the symbol q ˆ j ^ , which means that the vertex q ˆ j is to be omitted. For example,

For t = 0 , we define ∂ ˆ 0 σ ˆ 0 = 0 ˆ . Also, ∂ ˆ t shows linearly on a chaotic element x ˆ = ∑ x ˆ j σ ˆ t , j ∈ C ˆ t ( L ˆ ) as ∂ ˆ t x ˆ = ∑ x ˆ j ∂ ˆ t σ ˆ t , j ∈ C ˆ t − 1 ( L ˆ ) , ∂ ˆ t defines the chaotic boundary operator ∂ ˆ t : C ˆ t ( L ˆ ) → C ˆ t − 1 ( L ˆ ) , and this operator is a chaotic homomorphism. For an n -dimensional chaotic simplicial complex L ˆ , there is a chaotic sequence of chaotic free abelian groups and chaotic homomorphisms,

We denote this sequence by C ˆ ( L ˆ ) , which represents the chaotic chain complex corresponding to L ˆ . However, the kernel and image of a homomorphism ∂ ˆ t are worth studying. We call x ˆ a chaotic t -cycle, if ∂ ˆ t ( x ˆ ) = 0 ˆ , with x ˆ ∈ C ˆ t ( L ˆ ) . The set of all chaotic t -cycles Z ˆ t ( L ˆ ) satisfies Z ˆ t ( L ˆ ) ≤ C ˆ t ( L ˆ ) (where ≤ denotes a subgroup). From now on, we will denote Z ˆ t ( L ˆ ) as the chaotic t -cycle group. Also, Z ˆ t ( L ˆ ) = ker ∂ ˆ t . Moreover, if t = 0 , ∂ ˆ 0 x ˆ = 0 , we obtain Z ˆ 0 ( L ˆ ) = C ˆ 0 ( L ˆ ) .

Definition 2.5

Given a chaotic t -dimensional simplicial complex L ˆ , if x ˆ ∈ C ˆ t ( L ˆ ) and there is a chaotic element z ˆ ∈ C ˆ t + 1 ( L ˆ ) for which x ˆ = ∂ ˆ t + 1 z ˆ , then x ˆ is called a chaotic t -boundary. The set of chaotic t -boundaries B ˆ t ( L ˆ ) is a chaotic subgroup of C ˆ t ( L ˆ ) and is called the chaotic t -boundary group. Note that B ˆ t ( L ˆ ) = im ∂ ˆ t + 1 and B ˆ t ( L ˆ ) are chaotic subgroups of C ˆ t ( L ˆ ) .

Theorem 2.6

If Z ˆ t ( L ˆ ) and B ˆ t ( L ˆ ) are chaotic t-cycles and chaotic t-boundary groups of C ˆ t ( L ˆ ) , then B ˆ t ( L ˆ ) ⊆ Z ˆ t ( L ˆ ) .

Proof

Let x ˆ ∈ B ˆ t ( L ˆ ) , then x ˆ = ∂ ˆ t + 1 z ˆ for some z ˆ ∈ C ˆ t + 1 ( L ˆ ) and so, we obtain ∂ ˆ t x ˆ = ∂ ˆ t ( ∂ ˆ t + 1 z ˆ ) = 0 ˆ ; thus, x ˆ ∈ Z ˆ t ( L ˆ ) . Hence, B ˆ t ( L ˆ ) ⊆ Z ˆ t ( L ˆ ) .□

Definition 2.7

Given a chaotic n -dimensional simplicial complex L ˆ , the chaotic t-homology group H ˆ t ( L ˆ ) , 0 ≤ t ≤ n is defined by H ˆ t ( L ˆ ) = Z ˆ t ( L ˆ ) ⁄ B ˆ t ( L ˆ ) . Also, H ˆ t ( L ˆ ) = 0 ˆ for t > n or t < 0 and H ˆ 0 ( L ˆ ) = Z ˆ .

Example 2.8

Let L ˆ = { a ˆ } . The chaotic 0-chain is C ˆ 0 ( L ˆ ) = { n a ˆ : n ∈ Z } = Z ˆ , where Z ˆ = { … , − 2 ˆ , − 1 ˆ , 0 ˆ , 1 ˆ , 2 ˆ , … } . Obviously, Z ˆ 0 ( L ˆ ) = C ˆ 0 ( L ˆ ) and B ˆ 0 ( L ˆ ) = 0 ˆ since ∂ ˆ 0 a ˆ = 0 ˆ and a ˆ is not a boundary of any space. This implies that H ˆ 0 ( L ˆ ) = Z ˆ 0 ( L ˆ ) ⁄ B ˆ 0 ( L ˆ ) = C ˆ 0 ( L ˆ ) = Z ˆ . Similarly, if L ˆ = { a ˆ , b ˆ } is a chaotic simplicial complex consisting of two chaotic 0-simplexes, then,

Now, we want to construct the chaotic simplicial homology group of S ˆ 1 , let L ˆ be a chaotic triangulation of S ˆ 1 , as shown in Figure 3, and L ˆ = { a ˆ , b ˆ , c ˆ , ( a ˆ b ˆ ) , ( b ˆ c ˆ ) , ( c ˆ a ˆ ) } , where ( c ˆ a ˆ ) = ( e 1 j , j = 1 , 2 , … ) , ( a ˆ b ˆ ) = ( e 2 j , j = 1 , 2 , … ) , and ( b ˆ c ˆ ) = ( e 3 j , j = 1 , 2 , … ) . Since, no (2-chaotic simplexes exist) in L ˆ , yield B ˆ 1 ( L ˆ ) = 0 ˆ and H ˆ 1 ( L ˆ ) = Z ˆ 1 ( L ˆ ) ⁄ B ˆ 1 ( L ˆ ) = Z ˆ 1 ( L ˆ ) . Let z ˆ ∈ Z ˆ 1 ( L ˆ ) , then z ˆ = n 1 ( a ˆ b ˆ ) + n 2 ( b ˆ c ˆ ) + n 3 ( c ˆ a ˆ ) , where n 1 , n 2 , n 3 ∈ Z .

Now ∂ ˆ 1 z ˆ = n 1 ( b ˆ − a ˆ ) + n 2 ( c ˆ − b ˆ ) + n 3 ( a ˆ − c ˆ ) = ( n 3 − n 2 ) a ˆ + ( n 1 − n 2 ) b ˆ + ( n 2 − n 3 ) c ˆ = 0 ˆ . This holds true whenever, n 1 = n 2 = n 3 , and so, Z ˆ 1 ( L ˆ ) = { n 1 ( ( a ˆ b ˆ ) + ( b ˆ c ˆ ) + ( c ˆ a ˆ ) ) : n 1 ∈ Z } . This proves that Z ˆ 1 ( L ˆ ) = Z ˆ and H ˆ 1 ( L ˆ ) = Z ˆ 1 ( L ˆ ) = Z ˆ .

Now, to find H ˆ 0 ( L ˆ ) , we obtain Z ˆ 0 ( L ˆ ) = C ˆ 0 ( L ˆ ) and B ˆ 0 ( L ˆ ) = { ∂ ˆ 1 [ n 1 ( a ˆ b ˆ ) + n 2 ( b ˆ c ˆ ) + n 3 ( c ˆ a ˆ ) ] : n 1 , n 2 , n 3 ∈ Z } = { ( n 3 − n 1 ) a ˆ + ( n 1 − n 2 ) b ˆ + ( n 2 − n 3 ) c ˆ : n 1 , n 2 , n 3 ∈ Z } . Define a chaotic surjective homomorphism Φ ¯ : Z ˆ 0 ( L ˆ ) → Z ˆ by Φ ¯ ( m 1 ( a ˆ ) + m 2 ( b ˆ ) + m 3 ( c ˆ ) ) = m 1 + m 2 + m 3 .

Clearly, Ker Φ ¯ = Φ ¯ − 1 { 0 ˆ } = B ˆ 0 ( L ˆ ) , and so Z ˆ 0 ( L ˆ ) ⁄ Ker Φ ¯ = Im Φ ¯ = Z ˆ or H ˆ 0 ( L ˆ ) = Z ˆ 0 ( L ˆ ) ⁄ B ˆ 0 ( L ˆ ) = Z ˆ . Hence,

Theorem 3.1

Let L ˆ 1 and L ˆ 2 be two chaotic triangulations of S ˆ 1 1 and S ˆ 2 1 , respectively. Then, there are different types of foldings F : L ˆ 1 → L ˆ 2 that induce F ˜ : H ˆ t ( L ˆ 1 ) → H ˆ t ( L ˆ 2 ) such that F ˜ ( H ˆ t ( L ˆ 1 ) ) is isomorphic to one of 0 , 0 ˆ , Z , or Z ˆ .

Proof

If F : L ˆ 1 → L ˆ 2 is a folding for which F ( L ˆ 1 ) = L ˆ 2 , that is restricted on the original cycle must yield a folding on the chaotic cycles (physical characters), as shown in Figure 4, and so we obtain the induced folding F ˜ : H ˆ t ( L ˆ 1 ) → H ˆ t ( L ˆ 2 ) for which

Also, it follows from the folding F as shown in Figure 5, for all chaotic edges into original edges, that the yield F : H ˆ t ( L ˆ 1 ) → H ˆ t ( L ˆ 2 ) in which

Moreover, if F is folding on some chaotic edge into the original, as shown in Figure 6, we obtain the induced folding F ˜ : H ˆ t ( L ˆ 1 ) → H ˆ t ( L ˆ 2 ) in which

Similarly, for the folding with a singularity of all chaotic edges (physical characters) and its original into another one, as shown in Figure 7, we obtain

Lemma 3.2

Let M ˆ 1 and M ˆ 2 be chaotic manifolds, and let L ˆ 1 and L ˆ 2 be their respective chaotic triangulations. Then, there exists a folding F : L ˆ 1 → L ˆ 2 that maps F ˜ : H ˆ t ( L ˆ 1 ) → H ˆ t ( L ˆ 2 ) such that F ˜ ( H ˆ t ( L ˆ 1 ) ) = H ˆ t ( F ( L ˆ 1 ) ) .

Proof

If F : L ˆ 1 → L ˆ 2 is a folding in which F ( L ˆ 1 ) is homeomorphic to a chaotic triangulation of S ˆ 1 , then

Theorem 3.3

Let Q ˆ 1 and Q ˆ 2 be two chaotic triangulations of two chaotic tori, T ˆ 1 and T ˆ 2 respectively. There are different foldings F : Q ˆ 1 → Q ˆ 2 that induce F ˜ : H ˆ t ( Q ˆ 1 ) → H ˆ t ( Q ˆ 2 ) such that F ˜ ( H ˆ t ( Q ˆ 1 ) ) ≈ G 1 × G 2 , where G 1 and G 2 are one of 0 , 0 ˆ , Z , or Z ˆ .

Proof

The proof of this theorem is similar to the proof of Theorem 3.1.□

Corollary 3.4

Let L ˆ 1 and L ˆ 2 be two chaotic triangulations of S 1 t and S 2 t , respectively, and F : L ˆ 1 → L ˆ 2 is a folding where dim F ( S j t ) = dim ( S j t ) , t ≥ 2 , j = 1 , 2 . Then, there are many types of induced folding of F ˜ : H ˆ t ( L ˆ 1 ) → H ˆ t ( L ˆ 2 ) such that

Proof

The proof is obvious.□

Theorem 3.5

If C ˆ = { ( R ˆ , Θ ˆ ) : 1 ˆ ≤ R ˆ ≤ 2 ˆ } , and L ˆ is a chaotic triangulation of S ˆ 1 , then there exists a sequence of chaotic unfoldings { η j : L ˆ j − 1 → L ˆ j : j = 1 , 2 , … m } , which leads to a sequence of unfoldings { η ˜ j : H ˆ t ( L ˆ j − 1 ) → H ˆ t ( L ˆ j ) : j = 1 , 2 , … m } in which lim m → ∞ ( η ˜ m ( H ˆ t ( L ˆ m − 1 ) ) ) is either a chaotic infinite cyclic group or a chaotic identity group.

Proof

Let C ˆ = { ( R ˆ , Θ ˆ ) : 1 ˆ ≤ R ˆ ≤ 2 ˆ } , S ˆ 1 = { ( R ˆ , Θ ˆ ) : R ˆ = 1 ˆ } , and η 1 : L ˆ 0 → L ˆ 1 , η 2 : L ˆ 1 → L ˆ 2 , … , η m : L ˆ m − 1 → L ˆ m be a sequence of unfoldings for which lim m → ∞ η m ( L ˆ m − 1 ) = L ˆ , where L ˆ is a chaotic triangulation of S ˆ 1 , as shown in Figure 8, which induces η ˜ 1 : H ˆ t ( L ˆ 0 ) → H ˆ t ( L ˆ 1 ) , η ˜ 2 : H ˆ t ( L ˆ 1 ) → H ˆ t ( L ˆ 2 ) , … , η ˜ m : H ˆ t ( L ˆ m − 1 ) → H ˆ t ( L ˆ m ) such that

Hence, lim m → ∞ ( η ˜ m ( H ˆ t ( L ˆ m − 1 ) ) ) is a chaotic infinite cyclic group or chaotic identity group.□

Theorem 3.6

Let L ˆ be the chaotic triangulation of a chaotic manifold M ˆ . If { η j : L ˆ j − 1 → L ˆ j , j = 1 , 2 , … , m } is a sequence of unfoldings on L ˆ j − 1 into L ˆ j , then H ˆ t ( lim m → ∞ ( η m ( M ˆ ) ) ) and H ˆ t ( η m ( M ˆ ) ) do not necessarily have to be equal.