Distance Theory Algebraically Supplemented Note 3 Homology Group

Distance Theory Algebraically Supplemented

Note 3

Homology Group

TANAKA Akio

1

Points in Euclid space     P₀, …, P_q

Convex hull     Δ _q = [P₀, …, P_q ]

Order j^th face of convex hull     ε_j : Δ_q-1→Δq    0 ≤ j ≤ q

k < j    ε_j ( P_k ) = P_k

j ≤ k    ε_j ( P_k ) = P_k+1

Space    X

Continuous map     σ: Δ_q→Ｘ

Free module generated from all the <map σ> s     C _q ( X )　

Boundary operator δ : C _q ( X ) → C _q-1 ( X )

δσ = σ〇ε_j

δσ = 0     q-dimensional cycle

All the q-dimensional cycles      Z_q ( X )

c, c’     q-dimensional cycles

x ∈ C _q+1 ( X )

c - c’ = δx      

c and c’ are homolog.

Quotient group of Z_q ( X ) that are homolog each other     H_q ( X )

H_q ( X ) is q-dimensional homology group.

2

Space     M

Fixed base point of M     x₀ ∈ M

Unit interval     I

Continuous map from I     γ: I →M

All that satisfy γ ( 0 ) = γ ( 1 ) = γ ( x₀ ) is called loop space.    ΩM

ΩM has definition of product in homology.

H_p ( ΩM ) ╳ H_q ( ΩM ) → H_p +q( ΩM )

[References]

<Symmetry Flow Language>

4 Homology on Language / Tokyo May 15, 2007

Tokyo November 9, 2007

Sekinan Research Field of Language