cocycle
{{Short description|Closed cochain}}
{{One source|date=October 2022}}
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.{{cite web | url=https://encyclopediaofmath.org/wiki/Cocycle | title=Cocycle - Encyclopedia of Mathematics }}
Definition
=Algebraic Topology=
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. {{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}}
See also
References
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