Cocycle
This article relies largely or entirely on a single source. (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.[1]
Definition
[edit]Algebraic Topology
[edit]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
[edit]References
[edit]- ^ "Cocycle - Encyclopedia of Mathematics".
- ^ Hatcher, Allen (2002). Algebraic Topology (1st ed.). Cambridge: Cambridge University Press. p. 198. ISBN 9780521795401. MR 1867354.