De Rham Cohomology — Mathematics Knowledge System

Overview

De Rham cohomology computes the cohomology of a smooth manifold using differential forms rather than singular cochains. The k-th de Rham cohomology group H^k_{dR}(M) consists of closed k-forms modulo exact k-forms. The de Rham theorem establishes a canonical isomorphism between de Rham cohomology and singular cohomology with real coefficients, connecting analysis and topology.

Intuition

Integration is the bridge between analysis and topology in de Rham theory. A closed form is one whose exterior derivative is zero; an exact form is the exterior derivative of something. A closed form that is not exact detects a 'hole': integration around a loop enclosing a hole gives a nonzero result. Stokes' theorem ensures that exact forms integrate to zero on cycles, making this notion well-defined.

Formal Definition

Definition

Let M be a smooth n-manifold and Omega^k(M) the space of smooth k-forms. The exterior derivative d: Omega^k(M) -> Omega^{k+1}(M) satisfies d^2 = 0, giving the de Rham complex. The k-th de Rham cohomology group is H^k_{dR}(M) = ker(d: Omega^k -> Omega^{k+1}) / im(d: Omega^{k-1} -> Omega^k). The de Rham theorem states H^k_{dR}(M) ≅ H^k(M; R) for all k.

0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n(M) \to 0

De Rham complex

H^k_{\mathrm{dR}}(M) = \ker\!\left(d:\Omega^k\to\Omega^{k+1}\right) \big/ \operatorname{im}\!\left(d:\Omega^{k-1}\to\Omega^k\right)

k-th de Rham cohomology

H^k_{\mathrm{dR}}(M) \cong H^k(M;\mathbb{R})

De Rham's theorem

\int_{\partial\sigma} \omega = \int_\sigma d\omega

Stokes' theorem (key tool)

Theorems

Theorem 1: De Rham's Theorem

For a smooth manifold M, there is a natural isomorphism H^k_{\mathrm{dR}}(M) \cong H^k(M;\mathbb{R}) for all k, given by integration: [\omega] \mapsto (\sigma \mapsto \int_\sigma \omega).

Theorem 2: Poincaré Lemma

Every closed form on a contractible open subset U \subset \mathbb{R}^n is exact: H^k_{\mathrm{dR}}(U) = 0 for k \geq 1.

Theorem 3: Hodge Decomposition

On a compact Riemannian manifold, every cohomology class in H^k_{\mathrm{dR}}(M) has a unique harmonic representative: \Omega^k = d\Omega^{k-1} \oplus \delta\Omega^{k+1} \oplus \mathcal{H}^k, where \mathcal{H}^k = \ker(\Delta).

Worked Examples

R^n is contractible, so by the Poincaré lemma every closed form of degree k >= 1 is exact.
A 0-form (smooth function) is closed iff df = 0 iff f is constant. So H^0_{dR}(R^n) = R (constant functions).
For k >= 1: ker(d) = im(d), so H^k_{dR}(R^n) = 0.
$H^k_{\mathrm{dR}}(\mathbb{R}^n) = \begin{cases}\mathbb{R} & k=0 \\ 0 & k\ge 1.\end{cases}$

Answer: H^0_{dR}(R^n) = R, H^k_{dR}(R^n) = 0 for k >= 1.

Practice Problems

Difficulty 7/10

Compute H^*_{dR}(S^2).

Difficulty 8/10

Prove that omega = (x dy - y dx)/(x^2 + y^2) is a closed but not exact 1-form on R^2 \ {0}.

Difficulty 9/10

State the Mayer-Vietoris sequence for de Rham cohomology and use it to compute H^*_{dR}(T^2).

Common Mistakes

Common Mistake

Every closed form is exact on any manifold.

The Poincaré lemma guarantees this only on contractible sets. On a circle, dtheta is closed but not exact (its integral around S^1 is 2pi ≠ 0).

Common Mistake

De Rham cohomology captures integer topological information.

De Rham cohomology with real coefficients loses torsion information. For example, it cannot distinguish RP^2 from a point (both have H^1_{dR} = 0), while Z/2 coefficients do.

Quiz

Summary

De Rham cohomology H^k_{dR}(M) is defined as closed k-forms modulo exact k-forms on a smooth manifold.
The de Rham theorem establishes H^k_{dR}(M) ≅ H^k(M; R), with the isomorphism given by integration via Stokes' theorem.
The Poincaré lemma shows H^k_{dR}(contractible) = 0 for k >= 1, which is the local triviality of the de Rham complex.
Hodge theory on compact Riemannian manifolds provides canonical harmonic representatives for each de Rham class.
De Rham cohomology is computable and provides a bridge between differential geometry (forms, curvature) and topology (cohomology, characteristic classes).

References

BookBott, R. & Tu, L. — Differential Forms in Algebraic Topology. Springer, 1982.
WebsiteWikipedia — De Rham cohomology