A chain complex is a sequence of maps
 |
(1)
|
where the spaces 
may be Abelian groups or modules.
The maps must satisfy 
.
Making the domain implicitly understood, the maps are denoted by 
, called the boundary
operator or the differential. Chain complexes are an algebraic tool for computing
or defining homology and have a variety of applications.
A cochain complex is used in the case of cohomology.
Elements of 
are called chains. For each 
, the kernel of 
is called the group of cycles,
 |
(2)
|
The letter 
is short for the German word for cycle, "Zyklus." The image 
is contained in the group of cycles because
. It is called
the group of boundaries.
 |
(3)
|
The quotients 
are the homology groups of the chain.
For example, the sequence
 |
(4)
|
where every space is 
and each map is given by multiplication by 4 is a chain complex. The cycles at each
stage are 
and the boundaries are 
.
So the homology at each stage is the group of two elements 
. A simpler example is given by a linear
transformation 
,
which can be extended to a chain complex by the zero vector space and the zero
map. Then the nontrivial homology groups are 
and 
.
The terminology of chain complexes comes from the calculation for homology of geometric objects in a topological space,
like a manifold. For example, in the figure above, let
and 
denote the points, and 
and 
denote the oriented segments, which are the chains. The boundary
of 
is 
, and the boundary of 
is 
.
The group 
is the free Abelian group 
and the group 
is the free Abelian group 
. The boundary
operator is
 |
(5)
|
The other groups 
are the trivial group, and the other maps are the
zero map. Then 
is generated by 
and 
is the trivial subgroup. So 
is the rank one free Abelian
group isomorphic to 
.
The zero-dimensional case is slightly more interesting. Every element of 
has no boundary and so is in 
while the boundaries 
are generated by 
. Hence, 
is also isomorphic to 
. Note that the result is not affected by how the circle is
cut into pieces, or by how many cuts are used.
