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

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Algebraic Topology

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Let X be a CW complex and   be the singular cochains with coboundary map  . Then elements of   are cocycles. Elements of   are coboundaries. If   is a cocycle, then  , which means cocycles vanish on boundaries. [2]

See also

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References

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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.