coboundary

In mathematics, a coboundary is an element of a cochain group that lies in the image of a coboundary operator in a cochain complex. A cochain complex consists of objects C^0, C^1, C^2, … together with homomorphisms δ^n: C^n → C^{n+1} called coboundary operators, satisfying δ^{n+1} ∘ δ^n = 0. For each n, the n-cocycles are ker δ^n, and the n-coboundaries are im δ^{n-1}. An element of im δ^{n-1} is by definition a coboundary. The cohomology group H^n is defined as ker δ^n modulo im δ^{n-1}.

In concrete contexts, coboundaries have explicit descriptions. In group cohomology of a group G with G-module A, a 1-cochain f: G → A has coboundary δf: G×G → A given by δf(g,h) = g·f(h) − f(gh) + f(g), with δ^2 ∘ δ^1 = 0. In de Rham cohomology, the coboundary operator is the exterior derivative d on differential forms; a form is a coboundary if it is the exterior derivative of a form of one degree lower (an exact form). In Čech cohomology, δ combines values on overlaps via alternating sums.

Coboundaries are distinct from cocycles, which are cochains annihilated by δ^n (ker δ^n). They are also distinguished from boundaries in homology; however, coboundaries form the images that define cohomology groups, capturing the part of local data that arises as a coboundary rather than as genuine cohomological information. Coboundaries thus help describe how local structures fail to glue together globally and are central to the construction of cohomology theories.