Chain homology is given by the homology groups of a chain complex, defined using its homology
cycles and homology boundaries. For every
, the group
kernel of 
is called the group of homology
cycles,
 |
(1)
|
The letter 
is short for the German word for cycle, "Zyklus." The image 
is contained in the group of homology cycles
because 
,
and is called the group of homology
boundaries,
 |
(2)
|
The quotient groups 
are the homology groups
of the chain.
Given a short exact sequence of chain complexes
 |
(3)
|
there is a long exact sequence in homology
 |
(4)
|
In particular, a homology cycle 
in 
with 
is mapped to a homology
cycle 
in 
.
Similarly, a homology boundary 
in 
gets mapped to a homology
boundary 
in 
.
Consequently, the map between homology
groups 
is well-defined. The only map
which is not that obvious is 
, called the connecting
homomorphism, which is well-defined by the snake lemma.
Proofs of this nature are (with a modicum of humor) referred to as diagram chasing.
