For every 
,
the kernel of 
is called the group of cycles,
 |
(1)
|
The letter 
is short for the German word for cycle, "Zyklus." The image 
is contained in the group of cycles because
, and is called
the group of boundaries,
 |
(2)
|
The quotients 
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 cycle 
in 
with 
, is mapped to a cycle 
in 
. Similarly, a boundary 
in 
gets mapped to a boundary 
in 
. Consequently, the map between homologies 
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.
