Cohomology Explorer | mysimulator.uk

Genus-g Handlebody

Handles (genus g) 0

Rotation speed 0.15

Live invariants

b₀ (components)

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b₁ (rank H¹)

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b₂ (voids)

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χ = V−E+F

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Cochain state

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‖δf‖ (closedness)

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How it works

A simplicial cochain complex assigns numbers to vertices (0-cochains), edges (1-cochains) and faces (2-cochains). The coboundary operator δ turns a k-cochain into a (k+1)-cochain by summing values around each higher simplex's boundary, oriented consistently:

(δf)(edge u→v) = f(v) − f(u)
(δg)(face) = Σ ± g(edge) around the face boundary
δ∘δ = 0  always

A cochain is closed if δf = 0, and exact if f = δh for some lower cochain. Because δ∘δ = 0, every exact cochain is automatically closed — the interesting question is whether the reverse holds. The quotient closed ⁄ exact is the cohomology group Hᵏ, and its rank is the Betti number bₖ. Here we paint a random 1-cochain on the mesh's edges (arrows + heat colour = value), then press Apply δ to compute its 2-coboundary on every face and watch the panel report whether it vanished (closed) or not.

Genus g (handles) — each added handle attaches a tube with two new independent loops that no coboundary can kill, raising b₁ by 2 per handle: b₁ = 2g for a closed orientable surface.
1-cochain — re-randomizes the values living on each edge; the arrow colour encodes sign and magnitude.
Apply δ (coboundary) — computes the 2-cochain δf on every face from the edge values around its boundary and reports its norm: a genus-g surface has closed loops around each handle that no δ of a 0-cochain can reach, which is exactly why b₁ = 2g > 0 there.
Euler characteristic χ = V − E + F = 2 − 2g for this family of surfaces, and χ = b₀ − b₁ + b₂ ties the combinatorics directly to the cohomology ranks shown above.

Real-world relevance: this same machinery — chain complexes, coboundary maps, Betti numbers — underlies persistent homology in data analysis, mesh processing in computer graphics, and de Rham cohomology's role in electromagnetism and general relativity.

Vertex (0-simplex) Edge, 1-cochain value Face, δf result

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