Cocycle

A cochain which is annihilated by the coboundary mapping, in other words, a cochain that vanishes on boundary chains. The concept of a cocycle generalizes the concept of a closed differential form on a smooth manifold with a vanishing integral over a boundary chain.

In accordance with the different versions of the concept of a cochain, there are different versions of cocycles. For example, an Aleksandrov–Čech cocycle in a topological space is a cocycle of the nerve of some open covering of the space. Only one-dimensional cocycles with non-Abelian coefficients need special discussion. A one-dimensional cocycle of a simplicial set with coefficients in a non-Abelian group is a function , defined on the set of one-dimensional simplices of , such that for any two-dimensional simplex . Two cocycles and are said to be cohomologous if there exists a function such that for any one-dimensional simplex . The cohomology classes of one-dimensional cocycles form a pointed set . Similarly one defines one-dimensional cocycles and their cohomology classes in the Aleksandrov–Čech sense, with coefficients in a sheaf of non-Abelian groups. The cohomology groups of these cocycles are related to fibre bundles with a structure group.

For references see Cochain.