Simplicial Homology — Betti Numbers & Boundary Operators

β_k = dim(ker ∂_k) − dim(im ∂_{k+1}) — counting independent k-dimensional holes

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Simplicial homology: Given a simplicial complex K, the boundary operator ∂_k maps k-chains to (k-1)-chains. ∂_k([v₀,...,v_k]) = Σᵢ (−1)ⁱ [v₀,...,v̂ᵢ,...,v_k]. Key: ∂∘∂=0 (boundaries are cycles). Betti numbers: β₀ = #components, β₁ = #independent loops, β₂ = #voids. Euler characteristic: χ = β₀ − β₁ + β₂ = V − E + F (topological invariant). Examples: Circle S¹: β=(1,1,0), χ=0. Sphere S²: β=(1,0,1), χ=2. Torus T²: β=(1,2,1), χ=0. The Smith normal form (or Gaussian elimination mod 2) computes exact Betti numbers from boundary matrices.