de Rham cohomology

de Rham cohomology
Name	de Rham cohomology
Field	Élie Cartan, Hodge theory, Algebraic topology
Introduced	Georges de Rham
Keywords	Differential form, Cohomology, Manifold

de Rham cohomology. de Rham cohomology is an invariant assigning graded vector spaces to differentiable manifolds via closed and exact differential forms, introduced by Georges de Rham and developed in the wake of work by Élie Cartan, Henri Cartan, Beno Eckmann, and Samuel Eilenberg. It connects analysis on Riemannian manifolds, algebraic invariants from Algebraic topology, and structural results in Hodge theory and Morse theory, and it underpins comparisons such as the de Rham isomorphism linking to singular homology and Čech cohomology.

Introduction

De Rham cohomology arose from attempts by Georges de Rham to classify closed differential forms modulo exact forms on smooth manifolds, influenced by concepts in Poincaré conjecture-era topology and by the formalism of Élie Cartan's exterior calculus; contemporaneous advances by Henri Poincaré, Élie Cartan, Hermann Weyl, and Élie Cartan's students framed modern interpretations. The theory yields graded groups H^k_{dR}(M) capturing global features of Morse theory-relevant spaces studied by Marston Morse, Andrey Kolmogorov, and later formalized in works by Raoul Bott and John Milnor. Connections to Hodge decomposition emerged through contributions by W. V. D. Hodge, while algebraic perspectives tie to results by Alexander Grothendieck and Jean-Pierre Serre.

Differential Forms and the de Rham Complex

Differential forms form a graded algebra equipped with the exterior derivative d introduced by Élie Cartan; the de Rham complex Ω^*(M) = (Ω^0(M) → Ω^1(M) → Ω^2(M) → ⋯) yields cohomology groups H^k_{dR}(M) = ker d / im d, a construction paralleling chain complexes in Samuel Eilenberg and Norman Steenrod's categorical frameworks. The formal properties of d, including d^2 = 0, reflect homological algebra structures developed by Henri Cartan and Jean-Louis Koszul, and the algebraic duality between forms and chains evokes pairings studied by Lefschetz and Hermann Weyl. Techniques from Riemannian geometry and Ricci flow considerations introduced by Richard Hamilton and later used by Grigori Perelman inform analytical treatments of forms via Hodge theory associated to Atiyah–Singer index theorem contexts by Michael Atiyah and Isadore Singer.

Computation and Examples

Computations of de Rham cohomology for basic spaces use tools developed by Élie Cartan and computational devices popularized by John Milnor and Raoul Bott; classic examples include H^*(ℝ^n) triviality shown via contraction by Henri Poincaré (the Poincaré lemma), H^*(S^n) established using Mayer–Vietoris sequences in the spirit of Emil Artin and Heinrich Maschke, and H^*(T^n) for the torus linked to product structures exploited by Bernhard Riemann and Felix Klein. Calculations on projective complex manifolds draw on methods of Hodge and André Weil, while computations on Lie groups employ results from Élie Cartan's classification and the cohomology of homogeneous spaces used by Hermann Weyl and Harish-Chandra.

Cohomology Ring and Algebraic Structures

The wedge product endows de Rham cohomology with a graded-commutative ring structure analogous to cup products in Alexander duality contexts and elaborated in algebraic topology by Edward H. Spanier and Norman Steenrod. The ring structure on H^*(M) interacts with intersection theory on smooth projective varietys investigated by Alexander Grothendieck and Jean-Pierre Serre, and with characteristic classes introduced by Shiing-Shen Chern, Raoul Bott, and Hirzebruch. Structural theorems such as the Künneth formula and Poincaré duality in de Rham form rely on frameworks developed by Hermann Weyl and Solomon Lefschetz, and tie into notions used in the proofs of the Hirzebruch–Riemann–Roch theorem by Friedrich Hirzebruch and in index calculations by Atiyah and Singer.

de Rham Theorems and Relations to Other Cohomologies

The de Rham theorem, establishing isomorphism between de Rham cohomology and singular cohomology with real coefficients, follows methods echoing constructions by Samuel Eilenberg and Norman Steenrod and later categorical treatments by Alexander Grothendieck. Over complex manifolds the Dolbeault isomorphism connects to Hodge decomposition due to W. V. D. Hodge and further developments by Phillip Griffiths and Joseph Harris. Algebraic de Rham cohomology, developed by Alexander Grothendieck and Pierre Deligne, relates to étale cohomology as advanced by Jean-Pierre Serre and Alexander Grothendieck's school, while comparisons with Čech cohomology reflect techniques from Henri Cartan and Jean-Louis Verdier.

Applications and Consequences

De Rham cohomology provides invariants used in classification problems encountered by John Milnor and Raoul Bott, informs existence results for closed forms relevant to Sullivan's work in rational homotopy theory, and underlies the analytical foundations of Hodge theory used by W. V. D. Hodge, Pierre Deligne, and Gerald Faltings. In geometric analysis it influences the study of calibrated geometries in works by Reinhardt Bryant and Shing-Tung Yau, and in symplectic topology it supports results stemming from Alan Weinstein and Yasha Eliashberg on formality and obstructions. In mathematical physics, de Rham methods appear in path integral heuristics of Edward Witten and in gauge theory insights from Michael Atiyah and Nathan Seiberg.

Generalizations and Extensions

Generalizations include algebraic de Rham cohomology of Alexander Grothendieck and Pierre Deligne, relative de Rham cohomology used in Lefschetz pencil constructions by André Weil and Wolfgang Kuhnel, and equivariant de Rham theories developed with influences from Bertram Kostant and N. Berline and M. Vergne. Extensions to singular spaces adopt tools from Hironaka's resolution of singularities and perverse sheaf technology by Alexander Beilinson and Joseph Bernstein, while noncommutative de Rham-type theories echo frameworks proposed by Alain Connes and Maxim Kontsevich in deformation quantization and derived algebraic geometry advanced by Jacob Lurie and Bertrand Toën.

Category:Algebraic topology