Cocycle Explained

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]

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 ker d are cocycles. Elements of im d are coboundaries. If \varphi is a cocycle, then d \circ \varphi = \varphi \circ \partial =0 , which means cocycles vanish on boundaries. ^[2]

See also

• Čech cohomology
• Cocycle condition

Notes and References

1. Web site: Cocycle - Encyclopedia of Mathematics .
2. Book: Hatcher, Allen. Allen Hatcher

   . Allen Hatcher. Algebraic Topology. 2002. Cambridge University Press. 9780521795401. 1st. Cambridge. English. 1867354. 198.