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.

Definition

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.

See also

• Čech cohomology
• Cocycle condition