ncocycles

An n-cocycle is a type of cochain that satisfies a specific compatibility condition in a cohomology theory. In a typical setting, one fixes a space or a topological object X and an abelian group A (or a suitable coefficient system). The n-cochains C^n(X; A) are functions that assign coefficients in A to oriented n-simplices (or to local data in other models). The coboundary operator δ maps C^n to C^{n+1} and satisfies δ∘δ = 0, so the image of δ is contained in the kernel of δ.

An n-cocycle, then, is an element z in C^n(X; A) with δz = 0. The set of all n-cocycles is denoted Z^n(X; A). By contrast, an n-coboundary is any element that can be written as δb for some b in C^{n−1}(X; A); these form the group B^n(X; A) of coboundaries. The nth cohomology group is defined as the quotient H^n(X; A) = Z^n(X; A) / B^n(X; A). Cohomology groups measure obstructions to extending local data or to gluing pieces together in a global way.

Cocycles appear in several flavors. In singular or simplicial cohomology, they are explicit cochains on chains; in de Rham cohomology, n-cocycles are closed differential forms (dω = 0) with exact forms (dη) as coboundaries. Across contexts, n-cocycles provide a stable invariant that captures global structure from local information. The term is commonly written as n-cocycles.