Note 3
Homology Group
1
Points in Euclid space P0, …, Pq
Convex hull Δ q = [P0, …, Pq ]
Order jth face of convex hull εj : Δq-1→Δq 0 ≤ j ≤ q
k < j εj ( Pk ) = Pk
j ≤ k εj ( Pk ) = Pk+1
Space X
Continuous map σ: Δq →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εj
σ〇εj
δσ = 0 q-dimensional cycle
All the q-dimensional cycles Zq ( X )
c, c’ q-dimensional cycles
x ∈ C q+1 ( X )
c - c’ = δx
c and c’ are homolog.
Quotient group of Zq ( X ) that are homolog each other Hq ( X )
Hq ( 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.
Hp ( ΩM ) ╳ Hq ( ΩM ) → Hp +q( ΩM )
[References]
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