Simplicial complexes simplicial homology p-chain vector spaces Cp(K Let o1., n be p dimensional simplices in a complex K (K)is the vector space spanned by o1,,On over Zii p dimensional boundary map dp 0n:Cp(K)→Cn-1(K an(o=sum of(p -1)boundaries of o p dimensional homology groups Hp(K) 1+10 a=0 Hp()=kernel(ap)/image(@p+1)
Simplicial complexes & simplicial homology • 𝒑-chain vector spaces 𝑪𝒑(𝑲) : Let 𝜎1,…,𝜎𝑛 be 𝑝 dimensional simplices in a complex 𝐾, 𝐶𝑝(𝐾) is the vector space spanned by 𝜎1,…,𝜎𝑛 over 𝐙𝟐; • 𝒑 dimensional boundary map 𝝏𝒑 : 𝜕𝑝: 𝐶𝑝 𝐾 → 𝐶𝑝−1 𝐾 𝜕p 𝜎 = 𝑠𝑢𝑚 𝑜𝑓 (𝑝 − 1) 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠 𝑜𝑓 𝜎 • 𝒑 dimensional homology groups 𝑯𝒑(𝑲) : 𝜕𝑝+1 ∘ 𝜕𝑝 = 0 𝐻𝑝 𝐾 = 𝑘𝑒𝑟𝑛𝑒𝑙(𝜕𝑝)/𝑖𝑚𝑎𝑔𝑒(𝜕𝑝+1)
Simplicial complexes simplicial homology p dimensional Betti number Bp Bn=dim(h,(k)) Betti numbers are topological invariants
Simplicial complexes & simplicial homology • 𝒑 dimensional Betti number 𝜷𝒑 : 𝛽𝑝 = 𝑑𝑖𝑚(𝐻𝑝(𝐾)) • Betti numbers are topological invariants
Filtratⅰons A filtration is a sequence of simplicial complexes Ki such that each K; is a subcomplex of Ki+1,i.e ∈k;Ckj+1∈…
Filtrations A filtration is a sequence of simplicial complexes {𝐾𝑖 } such that each 𝐾𝑗 is a subcomplex of 𝐾𝑗+1 , i.e. ⋯ ⊂ 𝐾𝑗 ⊂ 𝐾𝑗+1 ⊂ ⋯
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o8 88。08080