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 $ K $ with coefficients in a non-Abelian group $ G $ is a function $ \sigma \rightarrow f ( \sigma ) \in G $, defined on the set $ K _ {1} $ of one-dimensional simplices of $ K $, such that $ f ( \sigma ^ {(} 0) ) f ( \sigma ^ {(} 2) ) = f ( \sigma ^ {(} 1) ) $ for any two-dimensional simplex $ \sigma \in K $. Two cocycles $ f $ and $ g $ are said to be cohomologous if there exists a function $ h: K _ {0} \rightarrow G $ such that $ f ( \tau ) h ( \tau ^ {(} 0) ) = h ( \tau ^ {(} 1) ) g ( \tau ) $ for any one-dimensional simplex $ \tau \in K $. The cohomology classes of one-dimensional cocycles form a pointed set $ H ^ {1} ( K; G) $. 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.