coboundaries

Coboundaries are a standard concept in cohomology theory. In a cochain complex, there are abelian groups C^n (n-cochains) connected by coboundary maps δ^n: C^n → C^{n+1} with the property that δ^{n+1} ∘ δ^n = 0. For each n, the n-coboundaries are the elements of C^n that arise as δ^{n-1}(a) for some a ∈ C^{n-1}. Equivalently, B^n = im δ^{n-1} ⊆ C^n. By convention, B^0 = 0 since there is no δ^{-1} acting on C^0.

The n-cocycles are the elements Z^n = ker δ^n, i.e., cochains whose coboundary is zero. The cohomology group in degree n is defined as H^n = Z^n / B^n, i.e., cocycles modulo coboundaries. Thus coboundaries represent cochains that come from a previous degree and are regarded as "trivial" in cohomology.

Coboundaries occur in many contexts, including singular cohomology of topological spaces, where C^n(X; A) consists of n-cochains with coefficients in an abelian group A, and δ^n is the coboundary operator. Similar constructions appear in group cohomology and sheaf cohomology. The concept is dual to that of a boundary in homology, where boundaries are the images of boundary maps ∂_n: C_n → C_{n-1} and cocycles correspond to cycles in cohomology.