{{Short description|Closed cochain}}
{{One source|date=October 2022}} In mathematics a '''cocycle''' is a closed cochain.<!--<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 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.<ref>{{cite web | url=https://encyclopediaofmath.org/wiki/Cocycle | title=Cocycle - Encyclopedia of Mathematics }}</ref>
==Definition== ===Algebraic Topology===
Let ''X'' be a CW complex and <math>C^n(X)</math> be the singular cochains 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|isbn=9780521795401|edition= 1st|location=Cambridge|language=English|mr=1867354|page=198}}</ref>
==See also== * Čech cohomology * Cocycle condition
==References== {{Reflist}}
Category:Algebraic topology Category:Cohomology theories Category:Dynamical systems
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