In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous dynamical systems, cocycles are used to describe particular kinds of map, as in Oseledets theorem.
Definition
Algebraic Topology Let X be a CW complex and C n ( X ) {\displaystyle C^{n}(X)} be the singular cochains with coboundary map d n : C n − 1 ( X ) → C n ( X ) {\displaystyle d^{n}:C^{n-1}(X)\to C^{n}(X)} . Then elements of ker d {\displaystyle {\text{ker }}d} are cocycles. Elements of im d {\displaystyle {\text{im }}d} are coboundaries. If φ {\displaystyle \varphi } is a cocycle, then d ∘ φ = φ ∘ ∂ = 0 {\displaystyle d\circ \varphi =\varphi \circ \partial =0} , which means cocycles vanish on boundaries.
See also Čech cohomology Cocycle condition
References