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.
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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