Cochains

Cochains are algebraic objects used to construct cohomology theories in algebraic topology. For a topological space X and a fixed abelian group G (often Z, R, or a related group), an n-cochain assigns to every singular n-simplex in X an element of G. The set of all such functions is denoted C^n(X; G). The collections {C^n(X; G)} together form a cochain complex with the coboundary map δ: C^n(X; G) → C^{n+1}(X; G) defined by δφ(σ) = φ(∂σ), where ∂σ is the boundary of the singular simplex σ. Since δ∘δ = 0, one can form the nth cohomology group H^n(X; G) = ker δ / im δ, the cocycles modulo coboundaries.

Cochains generalize functions on the space and are dual to chains. They can be used with various coefficient theories, leading to singular cohomology, de Rham cohomology via differential forms as cochains, or Čech cohomology via open covers.

Key properties include functoriality: a continuous map f: X → Y induces a pullback map f^*: C^n(Y; G) → C^n(X; G) and thus a homomorphism f^* on cohomology. The cohomology groups form a contravariant functor of spaces with coefficients in G and are invariants under homotopy equivalence.

An additional algebraic structure is the cup product, which endows H^*(X; G) with a graded ring structure. On the cochain level there is a compatible product that satisfies δ(φ ∪ ψ) = δφ ∪ ψ + (−1)^{deg φ} φ ∪ δψ.