A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
The de Rham cohomology Hᵏ(M) = {closed k-forms}/{exact k-forms} measures the topological complexity of a manifold using differential forms. It detects "holes" of various dimensions: H⁰ counts connected components, H¹ detects loops that are not boundaries, and Hⁿ detects orientation. The de Rham theorem proves this analytic construction equals the purely topological singular cohomology, establishing one of the deepest bridges between analysis and topology. De Rham cohomology is computable, functorial, and fundamental to modern geometry and physics.
The exterior derivative d creates a chain complex of differential forms: 0 → Ω⁰(M) →d Ω¹(M) →d Ω²(M) →d ... →d Ωⁿ(M) → 0. The identity d² = 0 means every exact form (image of d) is closed (kernel of d), so im(d) ⊂ ker(d) at each stage. The de Rham cohomology Hᵏ(M) = ker(d : Ωᵏ → Ωᵏ⁺¹) / im(d : Ωᵏ⁻¹ → Ωᵏ) measures the "gap" between closed and exact forms. A nonzero element of Hᵏ(M) is a closed form that cannot be written as dα — it represents a topological obstruction.
The simplest examples are illuminating. For H⁰: closed 0-forms are locally constant functions, so H⁰(M) ≅ ℝᵇ⁰ where b₀ is the number of connected components. For H¹: on the circle S¹, the form dθ is closed but not exact (∫_{S¹} dθ = 2π ≠ 0, but θ is not a globally defined function). So H¹(S¹) ≅ ℝ, reflecting the hole in S¹. On the 2-torus T², H¹(T²) ≅ ℝ² (two independent loops) and H²(T²) ≅ ℝ (the area form). On ℝⁿ, the Poincaré lemma gives Hᵏ(ℝⁿ) = 0 for k ≥ 1 — no topology means no cohomology.
The de Rham theorem states that H*_dR(M) is naturally isomorphic to the singular cohomology H*_sing(M; ℝ). The isomorphism is given by integration: a closed k-form ω acts on a k-cycle σ by ω(σ) = ∫_σ ω. Stokes' theorem ensures this is well-defined on equivalence classes (adding an exact form to ω does not change the integral over a cycle, and integrating over a boundary gives zero by Stokes). The Betti numbers bₖ = dim Hᵏ(M) are topological invariants: b₀ = number of components, b₁ = number of independent loops, bₙ = 1 if M is compact and oriented, 0 otherwise. The alternating sum χ(M) = Σ(-1)ᵏ bₖ is the Euler characteristic.
De Rham cohomology has a ring structure: the wedge product of closed forms is closed, and wedging with an exact form gives an exact form, so ∧ descends to cohomology. The resulting ring (H*(M), ∧) is isomorphic to the cup product ring of singular cohomology. Cohomology rings distinguish manifolds that Betti numbers alone cannot — for instance, CP² and S² ∨ S⁴ have the same Betti numbers but different cohomology rings. The Mayer-Vietoris sequence, Künneth formula, and Poincaré duality are powerful computational tools. In physics, de Rham cohomology classifies gauge field configurations (the first Chern class), magnetic monopole charges, and topological sectors of quantum field theories.