Distance Theory Algebraically Supplemented 3 Point Preparatory consideration / October 12, 2007

Distance Theory Algebraically Supplemented

3

Point Preparatory consideration

TANAKA Akio

1

<Homeomorphism>

Continuous map     f : X → Y

Inverse continuous map      f ^-1 : Y → X

f ( or f ^-1 ) is homeomorphism.

X and Y are homeomorphic.

2

<Homotopy>

Two topological space     X, Y

Continuous map    f_i : X → Y   ( i = 0,1)

Family of continuous maps      f_t : X → Y    ( t∈[0,1] )

Existence of maps is homotopic.

Expression is f₀ ≅ f₁

f_t ( t∈[ 0,1] ) is homotopy.

3

<Homotopy equivalent>

Two topological space     X, Y

Continuous map    f : X → Y  g : X → Y

Composition     g ○f : X → X   f ○g : Y → Y

Homotopy     g ○f ≅ Id _X   f ○g ≅ Id_Y

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<Topological pair>

Topological space     X

Topological subspace     A

Topological pair     ( X, A )

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<Attaching space>

Two topological space     X, Y

X⋂Y = Ø

X⊃A, Y⊃B

Homeomorphism (and attaching map) h : B→A

Attaching space    X∪_hY

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<Cell complex>

n-dimensional disk     Dⁿ   (n≧1)

n-dimensional sphere     S^n-1  (n≧1)

∂Dⁿ = S^n-1

i-dimensional disk     Dⁱ

Inside of Dⁱ     Dⁱ - ∂Dⁱ = Di – S^n-1

Homeomorphism at inside of Dⁱ    i cell    (0≦i≦1)  Expression is e^i.

Closed cell of eⁱ     ēⁱ    

ēⁱ - ∂ēⁱ = eⁱ

Topological space     X

Sum of cells     X⁰ = ē⁰₁∪…∪ē⁰_m

Attaching map     h₁ : ∂X⁽¹⁾ → X⁰    

Attaching space     X¹:= X⁰∪_h1 X⁽¹⁾

Xⁿ = X^n-1∪_hn X⁽ⁿ⁾

X = Xⁿ

X is n-dimensional cell complex (or cell complex).

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<Homotopy>

Topological space     X　

Subspace A that has only a point     x₀     ( x₀ is called base point.)

Topological space pair     ( X, x₀ )

Homotopy set     [ (Iⁿ, ∂Iⁿ), ( X, x₀ ) ]     n; natural number

The set is group.

n-dimensional sphere     Sⁿ

A base point on Sⁿ     x₀

Topological space pair     (S^n, x₀)

Isomorphism      [ (Iⁿ, ∂Iⁿ), ( X, x₀ ) ] ≅ [ (S^n, x₀), ( X, x₀ ) ]

Arbitrary two points in X     x₀, x₁

Natural number     n

Isomorphism      [ (Iⁿ, ∂Iⁿ), ( X, x₀ ) ] ≅ [ (Iⁿ, ∂Iⁿ), ( X, x₁) ]

[ (Iⁿ, ∂Iⁿ), ( X, x₀ ) ] that is entered group structure is called 1-dimensional homotopy group     π₁ ( X, x₀ )

π₁ ( X, x₀ ) is called fundamental group.

π₁ ( X, x₀ ) ≅ { 1 } is called simply connected.

n-dimensional homotopy group is commutative group.

Two topological space pairs ( X, x₀ ), ( Y, y₀ ) are homotopy equivalent.

Isomorphism    π_n ( X, x₀ ) ≅π_n ( Y, y₀ )

Arbitrary two points in X     x₀, x₁

Isomorphism     h : ( X, x₀ ) → ( X, x₁ )

Homotopy equivalent

Isomorphism π_n( X, x₀ ) ≅ ( X, x₁ )

8

<Convex set>

Vector space over real number field     X

K⊂X

Arbitrary points in K     x, y

K is convex set when what x and y make line segment is contained in K.

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<Simplex>

(n+1) points in space Rⁿ     p₀, …, p_n

Minimal vertex set     σⁿ = ( p₀, …, p_n ) is called n-simplex. n is dimension ofσⁿ.

Face ofσⁿ     j-simplexσ^j = ( p₀₁, …, p_nj )

Boundary ofσⁿ     all the simplexes less or equal (n-1) dimensions     Expression is ∂σⁿ.

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<Simplicial complex>

Simplex in Rⁿ     σⁿ

Set of simplexes     S = {σⁿ }

S satisfied by next conditions is Simplicial complex.

(1) σⁿ∈S → all the faces ofσⁿ∈S

(2) σ^m₁,σⁿ₂∈S →σ^m₁⋂σⁿ₂ is face ofσ^m₁ and σⁿ₂.

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<Polyhedron>

Topological space that consists of sum of all the simplexes

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<Triangulation>

Topological space     X

Simplicial complex     K

Polyhedron of K     | K |

Making K that is homeomorphic between | K | and X is triangulation.

Topological space is regarded as simplicial complex.

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<Simplicial complex homology group>

Simplicial complexes makes simplicial complex homology group by definition of equivalent relation.

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<Subdivision>

From Simplicial complex, composed simplexes are divided to smaller simplexes.

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<Simplicial map>

By subdivision, continuous map from simplicial complexes to another one can be approximated by simplicial approximation theorem.

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<Simplicial approximation theorem>

Topological space     X

Simplicial complex     K

Polyhedron of K    | K |

Homeomorphic map     t : | K | → X

Triangulation     T = ( K, t )

Topological space     X₁, X₂

Triangulation      T ₁= ( K₁, t₁ )     T ₂= ( K₂, t₂ )

Simplicial map     f : X₁ → X₂

Simplicial map on T₁, T₂     φ: X₁ → X₂

Point     x∈X₁

φ (x)∈simplex of T₂

φ is smplicial approximation of f.

Existence of φ is called simplicial approximation theorem.

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<Dimension axiom of homology group>

Topological space pair      ( X, A )

Commutative group     h_p ( X, A )  (p = 0,1,2,…)

Topological space being consisted of a point pt  when p≧1   h_p (pt) = 0

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<CW complex>

Cell complex     X

When X satisfies next two conditions, X is CW complex.

(1) Closed cell ē of X’s cell is contained in sum-set of finite cells. The closed cell is called <closure finite>.

(2) Subset of X    U

Toward cell of X, when U ⋂ ē is open set of e, U is only open set at the time. This situation is called <weak topology>.

CW complex is locally contractible in paracompact normal space.

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<Locally contractible>

Paracompact is that locally finite is given by adequate <refinement> toward arbitrary open covering in topological space.

Normal space is satisfies T₂ (Fréchet separation axiom) and T₄ ( Tietze separation axiom) in topological space.

Topological space     X

A point of X     p

Arbitrary neighborhood of p     U

Open neighborhood of p     V

V⊂U

Toward U, there exists that inclusion map i : V → U and Constant map c_p are homotopic.

From Hausdorff space ( topological T₂ space ) to complex ( especially CW complex ), bridge is algebraically built by approach between space and a point.

Tokyo October 12, 2007

Sekinan Research Field of Language