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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 ${\displaystyle C^{n}(X)}$ 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.

See also

• Čech cohomology
• Cocycle condition

References

1. "Cocycle - Encyclopedia of Mathematics".
2. Hatcher, Allen (2002). Algebraic Topology (1st ed.). Cambridge: Cambridge University Press. p. 198. ISBN 9780521795401. MR 1867354.

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