1 Boundary and domain is defined by the following.
Sequence that consists of Abelian group {cn} and homomorphism {∂n} is presented.
Sequence satisfies ∂no∂n+1 = 0
∂n is boundary.
The equality means that boundary of set’s boundary is null set.
{cn, , ∂n} is called chain complex.
∂ is called boundary operator.
∂Ω = 0
Ω is called closed domain.
Complementary set of closed domain in universal set becomes open domain.
2 Path, initial point and terminal point is defined by the following.
Continuous map ψ from closed interval I = [0, 1] to topological space X gives image J =ψ(I).
ω = (J, ψ) that is given from mapψ and image is called path.
Topological space in which path is given is called arcwize connected.
ψ(0) = x0 is called initial point of path ω.
ψ(1) = x1 is called terminal point of path ω. When x0= x1 is presented, path is called closed path or loop and initial point is called base point.
3 Time, homotopic and homotopy class is defined by the following.
Closed interval I is presented.
t ∈ I
t is time in I.
ψ(t) = x0
On continuous map H : I ╳ I = X, H(0, t) = x0 and H(1, t) = x1 is presented.
ω1 = (J1, ψ1), ω2= (J2, ψ2)
ω1 and ω2 are called homotopic path that is expressed by ω1 ∼ ω2.
H is called homotopy.
All of path, namely equivalence class is called homotopy class that is expressed by [ω].
4 Fundamental group (first homotopy group) is defined by the following.
Homotopy class [ω] that has base point x0 of closed path ω in topological space X is expressed by π1(X, x0).
[ω1], [ω2] ∈π1(X, x0)
When [ω1], [ω2] = [ω1ω2] is defined to product, π1(X, x0) becomes group.
Groupπ1(X, x0) is called fundamental group (first homotopy group) that has base point x0.
Fundamental group that becomes unit group in topological space X is called simply connected (1-connewcted).
Unit group is ge = eg = g against arbitrary g.
5 Deformation is defined by the following.
Topological space X and Y, continuous map f and g are presented.
f : X → Y , g : Y →X
f ○ g ∼ idY
g ○ f ∼ idx
id is identity mapping.
X and Y becomes same homotopy equivalence (homotopy tipe) that is expressed by X ≃ Y.
For example, anulus, Möbius band and solid tolus are homotopy equivalence with circle.
Topological space X and its subspace A is presented.
When A has continuous map r that is called retraction, A is called retaract of X.
When A has retraction r : X → A and homotopy H : X ╳ [0, 1] → X, A is called deformation retarct.
When X has point a∈X and a is deformation retarct from X, X is called contractible space.
For example, sphere is contractible.
6 Connect is defined by the following.
There exists theorem by KODAIRA Kunihiko that gives the following.
All of K3-curved surfce are connected by deformation.
7 Calabi-Yau manifold is K3-curved surface.
8 Torus that is Calabi-Yao manifold is connected by the KODAIRA’s theorem.
Refer to the following papers.
Torus Chain For MACLANE Saunders Tokyo June 10, 2006
Breaking Symmetry of Language For TARSKI Tokyo June 11, 2006
9 In Riemann surface, y2 = (x-α)(x-β)(x-γ) and adding point at infinity is expressed by torus.
10 In space of complex numbers, language is expressed by torus.
Tokyo May 19, 2007
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