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.¹

Let X be a CW complex and ${\displaystyle C^{n}(X)}$ be the singular cochains with coboundary map ${\displaystyle d^{n}:C^{n-1}(X)\to C^{n}(X)}$. Then elements of ${\displaystyle {\text{ker }}d}$ are cocycles. Elements of ${\displaystyle {\text{im }}d}$ are coboundaries. If ${\displaystyle \varphi }$ is a cocycle, then ${\displaystyle d\circ \varphi =\varphi \circ \partial =0}$, which means cocycles vanish on boundaries. ²