Cohomologi

Cohomologi, or cohomology in English, is a mathematical tool in algebraic topology that assigns to a topological space a sequence of abelian groups (or modules) H^n(X; A) that encode global properties. It is conceptually dual to homology and is especially useful for studying maps, geometry, and the ways spaces can be glued together.

There are several concrete constructions of cohomology theories. Singular cohomology uses cochains on all singular simplices of a space; de Rham cohomology on smooth manifolds uses differential forms and the exterior derivative; Čech cohomology uses open covers; and sheaf cohomology generalizes these ideas to broader settings and more flexible coefficients.

Definition and basic structure: For a space X and coefficient group A, one forms a cochain complex C^n(X; A) with a coboundary operator δ: C^n → C^{n+1}. The n-th cohomology group is H^n(X; A) = ker(δ^n) / im(δ^{n-1}). The construction is contravariant in X: a continuous map f: X → Y induces a pullback map f^*: H^n(Y; A) → H^n(X; A).

Key properties and theorems: Cohomology is functorial, and there are long exact sequences arising from pairs (X, A) or cofibrations. The Mayer–Vietoris sequence relates the cohomology of a space to that of a cover. The universal coefficient theorem connects cohomology with different coefficient groups, and the Kunneth formula describes cohomology of products. For oriented manifolds, Poincaré duality links cohomology to homology.

Applications: Cohomology detects holes, classifies certain geometric structures (for example line and principal bundles), and provides invariants used in geometry, topology, and mathematical physics. It also underpins characteristic classes and various obstruction theories.