Wednesday 24 July 2013

Distance Theory Algebraically Supplemented Note 3 Homology Group

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-1Δq    ≤  q
k < j    εj ( Pk ) = Pk
j ≤ k    εj ( Pk ) = Pk+1
Space    X
Continuous map     σΔq 
Free module generated from all the <map σ> s     q ( X ) 
Boundary operator δ q ( X )  q-1 ( X )
δσ = σεj
δσ = 0     q-dimensional cycle
All the q-dimensional cycles      Zq X )
c, c’     q-dimensional cycles
x  q+1 ( X )
c - c’ δx      
and c’ are homolog.
Quotient group of Zq X ) that are homolog each other     Hq ( )
Hq ( ) is q-dimensional homology group.

2
Space     M
Fixed base point of M     x M
Unit interval     I
Continuous map from I     γI 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 +qΩM )

[References]



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