# Cocycle

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{{Short description|Closed cochain}}

{{One source|date=October 2022}}
In [mathematics](/source/mathematics) a '''cocycle''' is a closed [cochain](/source/cochain_(algebraic_topology)).<!--<ref>{{cite book | last = Warner | first = Frank W. |title = Foundations of Differentiable Manifolds and Lie Groups | year=1983}} page 173</ref>-->  Cocycles are used in [algebraic topology](/source/algebraic_topology) to express obstructions (for example, to integrating a [differential equation](/source/differential_equation) on a [closed manifold](/source/closed_manifold)).  They are likewise used in [group cohomology](/source/group_cohomology).  In [autonomous](/source/autonomous_system_(mathematics)) [dynamical system](/source/dynamical_system)s, cocycles are used to describe particular kinds of map, as in [Oseledets theorem](/source/Oseledets_theorem).<ref>{{cite web | url=https://encyclopediaofmath.org/wiki/Cocycle | title=Cocycle - Encyclopedia of Mathematics }}</ref>

==Definition==
===Algebraic Topology===

Let ''X'' be a [CW complex](/source/CW_complex) and <math>C^n(X)</math> be the singular [cochains](/source/Chain_complex) with coboundary map <math>d^n: C^{n-1}(X) \to C^n(X)</math>. Then elements of <math>\text{ker }d</math> are '''cocycles'''. Elements of <math> \text{im } d </math> are '''coboundaries'''. If <math> \varphi</math> is a cocycle, then <math>d \circ \varphi = \varphi \circ \partial =0 </math>, which means cocycles vanish on boundaries. <ref>{{Cite book|first=Allen|last=Hatcher|authorlink=Allen Hatcher|url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html|title=Algebraic Topology|date=2002|publisher=[Cambridge University Press](/source/Cambridge_University_Press)|isbn=9780521795401|edition= 1st|location=Cambridge|language=English|mr=1867354|page=198}}</ref>

==See also==
* [Čech cohomology](/source/%C4%8Cech_cohomology)
* [Cocycle condition](/source/Cocycle_condition)

==References==
{{Reflist}}

Category:Algebraic topology
Category:Cohomology theories
Category:Dynamical systems

{{topology-stub}}

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Adapted from the Wikipedia article [Cocycle](https://en.wikipedia.org/wiki/Cocycle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cocycle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.