Homology of the Torus

Algebraic Topology · Axiom Academy

Computing H₀(T²) ≅ ℤ, H₁(T²) ≅ ℤ², H₂(T²) ≅ ℤ using triangulation The torus can be triangulated with 1 vertex, 3 edges (a, b, c), and 2 triangular faces. Excellent! You've computed the complete homology of the torus. H₀(T²) ≅ ℤ: The torus is connected, giving one generator for H₀. H₁(T²) ≅ ℤ²: Two independent 1-cycles (meridian and longitude) that cannot be contracted, giving two ℤ generators. H₂(T²) ≅ ℤ: The torus surface itself forms a fundamental 2-cycle, captured by a single ℤ generator. Geometric Meaning: H₁ detects the two "holes" through the torus, while H₂ captures the 2-dimensional surface enclosing a solid region. The torus is a fundamental example showing how homology captures topological features: connectivity, loops, and surfaces!

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