cohomologie

Cohomologie (cohomology in English) is a branch of algebraic topology that assigns to a topological space a sequence of algebraic objects, typically groups or modules, that encode global structural information. The construction uses cochains, a coboundary operator, and cohomology groups defined as the quotient of cocycles by coboundaries. For a space X and a group of coefficients G, the nth cohomology group H^n(X;G) measures the failure of local data to glue together globally. Cohomology is contravariant: a continuous map f: X → Y induces a homomorphism f^*: H^n(Y;G) → H^n(X;G). The groups are often enriched with a graded ring structure via the cup product, giving H^*(X;G) a multiplicative aspect.

Several cohomology theories exist. Singular cohomology is built from real or abelian-group valued functions on singular simplices; de Rham cohomology uses differential forms on smooth manifolds, with closed forms modulo exact forms. The de Rham theorem identifies de Rham cohomology with singular cohomology with real coefficients for smooth manifolds. Čech and sheaf cohomology are important in algebraic geometry and complex analysis, providing tools to study global sections of sheaves.

Examples illustrate the invariants. For any nonempty connected X, H^0(X;G) ≅ G. The circle S^1 has H^1(S^1;Z) ≅ Z, reflecting its 1-dimensional hole, while the 2-sphere S^2 has H^2(S^2;Z) ≅ Z. The universal coefficient theorem relates cohomology to homology with chosen coefficients, and duality theorems like Poincaré duality connect cohomology groups in complementary degrees on oriented manifolds.

Cohomology plays a central role in classifying bundles, detecting holes, and informing geometric and physical theories through its algebraic structure.