Cocycle

Short description: Closed cochain

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 the 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 $\phi$ is a cocycle, then $d\circ \phi =\phi \circ \partial =0$, which means cocycles vanish on boundaries.

See also

• Čech cohomology
• Cocycle condition