De Rham cohomology

De Rham cohomology is a cohomology theory in differential geometry and algebraic topology that uses the differential forms on a smooth manifold to study its topological properties. It was introduced by the Swiss mathematician Georges de Rham in his 1931 thesis, providing a profound bridge between the local analytic data of calculus and the global topological structure of spaces. The central result, De Rham's theorem, establishes an isomorphism between these differential form cohomology groups and the singular cohomology groups with real coefficients, revealing deep connections between analysis, geometry, and topology. This theory has become a fundamental tool in areas ranging from mathematical physics to complex geometry.

Definition and basic properties

The construction begins on a smooth manifold \(M\). One considers the complex of differential forms \(\Omega^*(M)\), equipped with the exterior derivative \(d\). This derivative satisfies \(d \circ d = 0\), making \((\Omega^*(M), d)\) a cochain complex. The \(k\)-th de Rham cohomology group is defined as the quotient of the space of closed \(k\)-forms (forms \(\omega\) with \(d\omega = 0\)) by the space of exact \(k\)-forms (forms \(\omega\) with \(\omega = d\eta\) for some \((k-1)\)-form \(\eta\)). This group, denoted \(H_{\text{dR}}^k(M)\), is a real vector space. Key properties include its invariance under diffeomorphism, making it a topological invariant of the manifold. The theory is functorial: a smooth map \(f: M \to N\) induces a pullback map \(f^*: H_{\text{dR}}^*(N) \to H_{\text{dR}}^*(M)\) on cohomology. Important operations like the cup product in cohomology correspond under this isomorphism to the wedge product of differential forms, giving the cohomology ring a rich algebraic structure.

De Rham's theorem

The seminal result, proved by Georges de Rham, states that for a smooth manifold \(M\), the de Rham cohomology groups are isomorphic to the singular cohomology groups \(H^k(M; \mathbb{R})\) with real coefficients. This isomorphism is induced by integrating closed differential forms over smooth singular chains. The theorem's proof relies on sheaf-theoretic arguments and the Poincaré lemma, which asserts the local exactness of closed forms. This equivalence shows that the analytically defined de Rham cohomology captures the same topological information as the combinatorially defined singular cohomology. The theorem was later placed in a broader sheaf-cohomology framework by mathematicians like Jean-Pierre Serre and Henri Cartan, connecting it to Čech cohomology and the theory of sheaves.

Examples and computations

For the Euclidean space \(\mathbb{R}^n\), the Poincaré lemma implies all cohomology groups vanish for \(k > 0\), so \(H_{\text{dR}}^k(\mathbb{R}^n) = 0\). For the circle \(S^1\), the first cohomology group is one-dimensional, generated by the angular form \(d\theta\). The cohomology of the \(n\)-sphere \(S^n\) is nontrivial only in dimensions \(0\) and \(n\). For the torus \(T^2 = S^1 \times S^1\), the first cohomology group is two-dimensional, reflecting its two independent non-contractible loops. Computations often use the Mayer–Vietoris sequence, a powerful tool derived from the sheaf properties of differential forms, to break a manifold into simpler pieces. For compact Kähler manifolds, such as complex projective space \(\mathbb{CP}^n\), de Rham cohomology further refines into the Hodge theory decomposition, relating harmonic forms to topological invariants.

Applications

This theory has extensive applications across mathematics and physics. In symplectic geometry, it is crucial for understanding Moser's theorem and the classification of symplectic forms up to deformation. In mathematical physics, de Rham cohomology provides the language for Maxwell's equations in terms of differential forms on spacetime, where the fields are interpreted as elements of cohomology classes. It is fundamental in gauge theory and the study of characteristic classes, such as the Chern class and Pontryagin class, which are defined via de Rham cohomology using curvature forms. In dynamical systems, it aids in the study of integrals of motion and Hamiltonian mechanics. The theory also underpins modern approaches to index theory, as seen in the work of Michael Atiyah and Isadore Singer.

Generalizations and related theories

Numerous generalizations extend the core ideas. For non-smooth spaces like algebraic varieties, Alexander Grothendieck developed algebraic de Rham cohomology, using the Zariski topology and Kähler differentials. In complex geometry, Dolbeault cohomology generalizes the de Rham complex to complex manifolds by separating the exterior derivative into \(\partial\) and \(\bar{\partial}\) operators. For manifolds with singularities or stratified spaces, theories like intersection homology and L^2-cohomology have been developed. The Hodge decomposition theorem, a cornerstone of Hodge theory, refines de Rham cohomology on compact Riemannian manifolds. Furthermore, the theory connects to étale cohomology in arithmetic geometry and to noncommutative geometry pioneered by Alain Connes, where the role of the manifold is played by a C*-algebra.

Category:Differential topology Category:Homology theory Category:Differential geometry