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.[1]
Definition
Algebraic Topology
Let X be a CW complex and be the singular cochains with coboundary map . Then elements of are cocycles. Elements of are coboundaries. If is a cocycle, then , which means cocycles vanish on boundaries. [2]