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.

How to Cite This Entry:
Cocycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cocycle&oldid=55165
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article