Cocycle
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 the Oseledets theorem.[1]
Definition
Algebraic Topology
Let X be a CW complex and [math]\displaystyle{ C^n(X) }[/math] be the singular cochains with coboundary map [math]\displaystyle{ d^n: C^{n-1}(X) \to C^n(X) }[/math]. Then elements of [math]\displaystyle{ \text{ker }d }[/math] are cocycles. Elements of [math]\displaystyle{ \text{im } d }[/math] are coboundaries. If [math]\displaystyle{ \varphi }[/math] is a cocycle, then [math]\displaystyle{ d \circ \varphi = \varphi \circ \partial =0 }[/math], which means cocycles vanish on boundaries. [2]
See also
- Čech cohomology
- Cocycle condition
References
- ↑ "Cocycle - Encyclopedia of Mathematics". https://encyclopediaofmath.org/wiki/Cocycle.
- ↑ Hatcher, Allen (2002) (in English). Algebraic Topology (1st ed.). Cambridge: Cambridge University Press. p. 198. ISBN 9780521795401. https://www.math.cornell.edu/~hatcher/AT/ATpage.html.
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