Kohomologie

Kohomologie, or cohomology, is a central construction in algebraic topology and geometry. For a topological space X and an abelian group A, it assigns a sequence of abelian groups H^n(X; A), the n-th cohomology groups. These groups arise from a cochain complex C^n(X; A) with coboundary maps δ^n: C^n → C^{n+1} satisfying δ^{n+1} ∘ δ^n = 0, and H^n(X; A) = ker δ^n / im δ^{n-1}. The theory is contravariant in X: a continuous map f: X → Y induces a homomorphism f^*: H^n(Y; A) → H^n(X; A).

Several flavors exist, including singular cohomology, de Rham cohomology on smooth manifolds (cohomology of differential forms with the exterior derivative), and Čech or sheaf cohomology used in algebraic geometry. The de Rham theorem identifies H^n_dR(M) with H^n(M; R) for smooth manifolds, linking differential forms to topological invariants.

With a cup product, Kohomologie becomes a graded ring H^*(X; A) = ⊕_n H^n(X; A), providing a multiplicative structure. The theory is compatible with product spaces via the Kunneth theorem and with changes of coefficients via the universal coefficient theorem, which relates H^n(X; G) to Hom(H_n(X), G) and Ext(H_{n-1}(X), G). Poincaré duality further connects cohomology and homology on oriented closed manifolds.

Examples: for a point, H^0 ≅ A and H^n = 0 for n > 0; for the circle S^1, H^0 ≅ A and H^1 ≅ A. Applications include classifying maps, obstructions in bundle theory, and defining characteristic classes; cohomology also provides invariants (Betti numbers) and underpins physical concepts such as conserved fluxes and gauge fields.