Computing Simplicial Homology

Algebraic Topology · Axiom Academy

EXAMPLE Computing Simplicial Homology Step-by-step calculation for a triangulated 2-simplex (triangle) We consider a filled triangle with vertices v₀, v₁, v₂ and all faces. Excellent work! You've computed the simplicial homology of a filled triangle. Chain Complex: The computation starts by identifying chain groups C₀, C₁, C₂ and boundary maps ∂₁, ∂₂. Cycles vs Boundaries: Homology measures "holes" by computing ker(∂ₙ)/im(∂ₙ₊₁) - cycles that are not boundaries. Contractible Spaces: A filled triangle has trivial homology H₁ = H₂ = 0 because it's contractible to a point. General Pattern: For any contractible simplicial complex, only H₀ ≅ ℤ is nontrivial (reflecting connectivity). This technique extends to computing homology of any simplicial complex - from higher-dimensional simplices to complex surfaces!

This is the written version of the interactive lesson above. See the full Algebraic Topology course.