Homology of Spheres

Algebraic Topology · Axiom Academy

Computing the homology groups of n-spheres using simplicial and singular homology Let's begin with the simplest non-trivial example: the circle S¹. We can triangulate S¹ as a 1-dimensional simplicial complex with vertices and edges forming a loop. The boundary map ∂₁ sends each edge to the difference of its endpoints. Since we traverse a complete loop, the sum of all boundaries is zero, giving us a 1-cycle that is not a boundary. For S², imagine triangulating the sphere's surface. Unlike the circle, there are no 1-dimensional holes - every loop can be continuously contracted to a point on the surface. However, the 2-sphere encloses a 3-dimensional region. This "hollow" interior manifests as a nontrivial H₂ group. The sum of all 2-simplices (triangular faces) forms a 2-cycle that generates this homology. The pattern extends beautifully to all dimensions. For the n-sphere Sⁿ, we have exactly two nontrivial homology groups: H₀(Sⁿ) ≅ ℤ: There is exactly one connected component. Hₙ(Sⁿ) ≅ ℤ: The n-dimensional "surface" of Sⁿ forms a fundamental n-cycle. Hₖ(Sⁿ) = 0 for k ≠ 0, n: All intermediate dimensions have no holes.

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