Cocycle


In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions. 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 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.