Cohomology

Cohomology is a family of invariants used in topology, geometry, and related fields to classify and measure the structure of spaces. For a topological space X and an abelian group G, one obtains a sequence of abelian groups H^n(X; G), called the cohomology groups, which detect features such as connectedness, holes, and voids across dimensions.

The standard construction uses a cochain complex. Cochains C^n(X; G) consist of algebraic data assigned to n-dimensional building blocks of X, together with a coboundary operator δ: C^n → C^{n+1} satisfying δ∘δ = 0. The n-th cohomology group is defined as H^n(X; G) = ker δ / im δ, capturing cocycles modulo coboundaries. Different choices of cochain models yield equivalent cohomology theories.

Several flavors of cohomology are used in practice. Singular cohomology uses maps from standard simplices to X; de Rham cohomology uses differential forms on smooth manifolds with the exterior derivative; Čech cohomology and sheaf cohomology apply in more algebraic or geometric contexts. The coefficients G may be integers, real numbers, or other abelian groups. The de Rham theorem states that on smooth manifolds, real-valued de Rham cohomology is isomorphic to singular cohomology with real coefficients.

Cohomology is contravariant: continuous maps induce pullback homomorphisms H^n. It also carries a graded ring structure via the cup product, turning the direct sum of cohomology groups into a graded-commutative algebra. Key tools include the universal coefficient theorem, Poincaré duality for oriented closed manifolds, and long exact sequences such as Mayer–Vietoris.

Examples: for the n-sphere S^n, H^0 ≅ Z and H^n ≅ Z with other groups trivial; for the torus T^2, H^1 ≅ Z^2 and H^2 ≅ Z. Beyond topology, cohomology informs geometry, algebraic geometry, and mathematical physics, with variants like group cohomology and sheaf cohomology addressing broader contexts.