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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
a cocycle is a closed
cochain In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
. Cocycles are used in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
to express obstructions (for example, to integrating a differential equation on a
closed manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...
). They are likewise used in
group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...
. In
autonomous In developmental psychology and moral, political, and bioethical philosophy, autonomy is the capacity to make an informed, uncoerced decision. Autonomous organizations or institutions are independent or self-governing. Autonomy can also be defi ...
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s, cocycles are used to describe particular kinds of map, as in
Oseledets theorem In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in ...
.


Definition


Algebraic Topology

Let ''X'' be a
CW complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
and C^n(X) be the singular cochains with coboundary map d^n: C^(X) \to C^n(X). Then elements of \textd are cocycles. Elements of \text d are coboundaries. If \varphi is a cocycle, then d \circ \varphi = \varphi \circ \partial =0 , which means cocycles vanish on boundaries.


See also

*
ÄŒech cohomology In mathematics, specifically algebraic topology, ÄŒech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard ÄŒech. Moti ...
*
Cocycle condition 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 autonom ...


References

Algebraic topology Cohomology theories Dynamical systems {{topology-stub