Homology of Spheres

Core Idea

The n-sphere Sⁿ has singular homology H_k(Sⁿ) = Z for k = 0 and k = n, and H_k(Sⁿ) = 0 otherwise. This computation, established via the Mayer-Vietoris sequence or the long exact sequence of the pair (Dⁿ, S^n-1), is one of the most important results in algebraic topology. It provides the foundation for degree theory, the Brouwer fixed point theorem, and the classification of maps between spheres, and it reveals that each sphere has exactly one nontrivial "hole" in its own dimension.

Explainer

The computation of the homology of spheres is a cornerstone of algebraic topology: nearly every major theorem and application refers back to H_*(Sⁿ). The result is clean and beautiful: H_k(Sⁿ) is Z when k = 0 or k = n, and zero otherwise. The 0-dimensional homology H_0(Sⁿ) = Z reflects that Sⁿ is connected (for n >= 1; S⁰ consists of two points, giving H_0(S⁰) = Z²). The n-dimensional homology H_n(Sⁿ) = Z detects the single "n-dimensional hole" — the cavity enclosed by the sphere. All intermediate homology vanishes: Sⁿ has no holes of any dimension other than 0 and n.

The Mayer-Vietoris induction is the most elegant computation method. Decompose Sⁿ into two open hemispheres U and V, each contractible (they deformation retract to a point). Their intersection U intersect V deformation retracts to the equatorial (n-1)-sphere S^n-1. The Mayer-Vietoris long exact sequence reads: ... -> H_k(U) direct sum H_k(V) -> H_k(Sⁿ) -> H_{k-1}(S^n-1) -> H_{k-1}(U) direct sum H_{k-1}(V) -> ... Since U and V are contractible, H_k(U) = H_k(V) = 0 for k > 0, and the sequence collapses to isomorphisms H_k(Sⁿ) = H_{k-1}(S^n-1) for k >= 2. Starting from S⁰ (two points with H_0 = Z², all higher homology zero), we get: H_1(S¹) = H_0(S⁰)/corrections = Z, H_2(S²) = H_1(S¹) = Z, and inductively H_n(Sⁿ) = Z.

An alternative approach uses the long exact sequence of the pair (Dⁿ, S^n-1). Since the disk Dⁿ is contractible, H_k(Dⁿ) = 0 for k > 0. The long exact sequence ... -> H_k(Dⁿ) -> H_k(Dⁿ, S^n-1) -> H_{k-1}(S^n-1) -> H_{k-1}(Dⁿ) -> ... gives isomorphisms H_k(Dⁿ, S^n-1) = H_{k-1}(S^n-1) for k >= 2. The relative homology H_k(Dⁿ, S^n-1) is isomorphic to the reduced homology of the quotient Dⁿ/S^n-1 = Sⁿ (by excision-type arguments), giving the same inductive formula. Both methods reduce to the same recursion and yield the same answer.

The fundamental class [Sⁿ], the generator of H_n(Sⁿ), is the homological incarnation of the sphere's orientation. For any triangulation of Sⁿ, the fundamental class is represented by the sum of all n-simplices with orientations consistent with the global orientation. The fact that H_n(Sⁿ) = Z means that every n-cycle on Sⁿ is a multiple of the fundamental class — it wraps around the sphere some integer number of times. This integer is the foundation of degree theory: for a continuous map f : Sⁿ -> Sⁿ, the induced map f_* : H_n(Sⁿ) -> H_n(Sⁿ) is multiplication by an integer deg(f), the degree of f. The degree is the single most important homotopy invariant of such maps and connects to winding numbers, Brouwer's theorem, and the Borsuk-Ulam theorem.

The homology of spheres also reveals a key limitation of the fundamental group and motivates the study of higher-dimensional invariants. The sphere Sⁿ for n >= 2 has trivial fundamental group (every loop can be contracted to a point), yet its homology is nontrivial in dimension n. The fundamental group is blind to these higher-dimensional holes. This is precisely why homology (and later, higher homotopy groups and cohomology) are essential: they detect topological features that no single algebraic invariant can capture alone. The spheres, as the simplest spaces with holes of each dimension, serve as the calibration targets for all these invariants.