A closed orientable surface of genus g is a sphere with g handles glued on. Its topology — independent of how it is bent or stretched — is captured entirely by algebraic invariants computed from its homology groups.
χ(Σ_g) = 2 − 2g (Euler characteristic)
H₀ = ℤ, H₁ = ℤ^(2g), H₂ = ℤ (homology groups)-b₀ = 1, b₁ = 2g, b₂ = 1 (Betti numbers, rank Hₙ)
χ = b₀ − b₁ + b₂ = 2 − 2g (Euler–Poincaré formula)
- Genus g — number of handles attached to the base sphere; each handle contributes one independent 1-dimensional "hole" pair (an a-cycle and a b-cycle) to π₁ and H₁.
- Handle thickness — visual radius of each handle's tube; does not change the topology, only the embedding.
- Rotation speed — auto-rotates the surface so the whole handle structure can be traced.
- Homology loops — toggles the 2g generator loops of H₁(Σ_g): red = meridian (a-cycle, bounds a disk inside the handle), cyan = longitude (b-cycle, goes through the hole and cannot be contracted).
- Simplicial mesh — overlays the actual triangulation used to render the surface, the discrete data from which V, E, F and χ = V − E + F would be computed for a simplicial complex.
Real-world relevance: these invariants are why a coffee mug and a donut are "the same shape" to a topologist (both genus 1), and why classifying manifolds by homology/cohomology underlies applications from robot motion planning to data analysis (persistent homology / TDA) and condensed-matter topological phases.