Cohomology Explorer

Cohomology assigns algebraic invariants to shapes by tracking how numeric labels — cochains — placed on vertices, edges and faces interact through the coboundary operator δ. This simulator builds a genus-g handlebody as an explicit simplicial mesh, lets you paint a random 1-cochain on its edges, and computes its coboundary on every face live in the browser. Watching which cochains vanish under δ (closed) versus which arise as someone else's coboundary (exact) makes the quotient definition of Hᵏ concrete, while the Betti numbers b₀, b₁, b₂ update immediately as you add or remove handles — directly illustrating χ = b₀ − b₁ + b₂ and b₁ = 2g for a closed orientable surface of genus g.