Generally, a connecting homomorphism is a morphism of the kind produced by the snake lemma.
Specifically, when the double complex that goes into the snake lemma is regarded as part of a short exact sequence of chain complexes, then the connecting homomorphisms induce morphisms on the homology groups of these chain complexes which exhibit the corresponding long exact sequence in homology of the form
This long exact sequence is the image under chain homology
of the long homotopy fiber sequence of chain complexes induced by the short exact sequence. Hence the connecting homomorphism is the image under of a mapping cone inclusion on chain complexes.
For long (co)homology exact sequences
In the case that Mod for some ring , the construction of the connecting homomorphism for homology long exact sequences is easily described in terms of elements and checking its properties is elementary, see In terms of elements below. By the embedding theorems the general case can be reduced to this case. But there is also a general abstract description without recourse to elements, which we discuss further below in General abstract construction .
In terms of elements
Let be a commutative ring and let Mod. Write for the category of chain complexes in .
be a short exact sequence in .
For , define a group homomorphism
called the th connecting homomorphism of the short exact sequence, by sending
is a cycle representing a given homology group;
is any lift of that cycle to an element in , which exists because is a surjection (but which no longer needs to be a cycle itself);
is the -homology class of which is indeed in by exactness (since ) and indeed in since .
Def. 1 is indeed well defined in that the given map is independent of the choice of lift involved and in that the group structure is respected.
To see that the constructon is well-defined, let be another lift. Then and hence . This exhibits a homology-equivalence since .
To see that is a group homomorphism, let be a sum. Then is a lift and by linearity of we have .
Under chain homology the morphisms in the short exact sequence together with the connecting homomorphisms yield the homology long exact sequence
Consider first the exactness of .
It is clear that if then the image of is . Conversely, an element is in the kernel of if there is with . Since is surjective let be any lift, then but hence by exactness and so is in the image of .
It remains to see that
the image of is the kernel of ;
the kernel of is the image of .
This follows by inspection of the formula in def. 1. We spell out the first one:
If is in the image of we have a lift with and so . Conversely, if for a given lift we have that this means there is such that . But then is another possible lift of for which and so is in the image of .
Let be a short exact sequence of chain complexes in some abelian category . Then for all there are natural connecting homomorphisms such that we have a long exact sequence of the form
in chain homology.
Applying the snake lemma to the commuting diagram
shows that the rows in the commuting diagram
are exact sequences. Therefore applying the snake lemma to this, once more, yields the desired long exact sequence.
Relation to homotopy fiber sequences
The connecting homomorphism in a long exact sequence in homology induced from a short exact sequence is equivalently the image under the homology group functor of the homotopy cofiber sequence induced by . This is discussed in detail at mapping cone in the section homology exact sequences.
For instance section 1.3 of