- Source: 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 Oseledets theorem.
Definition
= Algebraic Topology
=Let X be a CW complex and
C
n
(
X
)
{\displaystyle C^{n}(X)}
be the singular cochains with coboundary map
d
n
:
C
n
−
1
(
X
)
→
C
n
(
X
)
{\displaystyle d^{n}:C^{n-1}(X)\to C^{n}(X)}
. Then elements of
ker
d
{\displaystyle {\text{ker }}d}
are cocycles. Elements of
im
d
{\displaystyle {\text{im }}d}
are coboundaries. If
φ
{\displaystyle \varphi }
is a cocycle, then
d
∘
φ
=
φ
∘
∂
=
0
{\displaystyle d\circ \varphi =\varphi \circ \partial =0}
, which means cocycles vanish on boundaries.
See also
Čech cohomology
Cocycle condition