Cocycle

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.

Algebraic Topology
Let X be a CW complex and $$C^n(X)$$ be the singular cochains with coboundary map $$d^n: C^{n-1}(X) \to C^n(X)$$. Then elements of $$\text{ker }d$$ are cocycles. Elements of $$ \text{im } d $$ are coboundaries. If $$ \varphi$$ is a cocycle, then $$d \circ \varphi = \varphi \circ \partial =0 $$, which means cocycles vanish on boundaries.