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 the Oseledets theorem.[1]

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. [2]

References

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

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[1] "Cocycle - Encyclopedia of Mathematics".

[2] Hatcher, Allen (2002). Algebraic Topology (1st ed.). Cambridge: Cambridge University Press. p. 198. ISBN 9780521795401. MR 1867354.

Cocycle

Definition

Algebraic Topology

See also

References