(also nonabelian homological algebra)
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 $A_\bullet \to B_\bullet \to C_\bullet$ of chain complexes, then the connecting homomorphisms induce morphisms $\delta_n : H_n(C) \to H_{n-1}(A)$ 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 $H_\bullet(-)$ of a mapping cone inclusion on chain complexes.
In the case that $\mathcal{A} \simeq R$Mod for some ring $R$, 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 .
Let $R$ be a commutative ring and let $\mathcal{A} = R$Mod. Write $Ch_\bullet(\mathcal{A})$ for the category of chain complexes in $\mathcal{A}$.
Let
be a short exact sequence in $Ch_\bullet(\mathcal{A})$.
For $n \in \mathbb{Z}$, define a group homomorphism
called the $n$th connecting homomorphism of the short exact sequence, by sending
where
$c \in Z_n(C)$ is a cycle representing a given homology group;
$\hat c \in C_n(B)$ is any lift of that cycle to an element in $B_n$, which exists because $p$ is a surjection (but which no longer needs to be a cycle itself);
$[\partial^B \hat c]_A$ is the $A$-homology class of $\partial^B \hat c$ which is indeed in $A_{n-1} \hookrightarrow B_{n-1}$ by exactness (since $p(\partial^B \hat c) = \partial^C p(\hat c) = \partial^C c = 0$) and indeed in $Z_{n-1}(A) \hookrightarrow A_{n-1}$ since $\partial^A \partial^B \hat c = \partial^B \partial^B \hat c = 0$.
Def. 1 is indeed well defined in that the given map is independent of the choice of lift $\hat c$ involved and in that the group structure is respected.
To see that the constructon is well-defined, let $\tilde c \in B_{n}$ be another lift. Then $p(\hat c - \tilde c) = 0$ and hence $\hat c - \tilde c \in A_n \hookrightarrow B_n$. This exhibits a homology-equivalence $[\partial^B\hat c]_A \simeq [\partial^B \tilde c]_A$ since $\partial^A(\hat c - \tilde c) = \partial^B \hat c - \partial^B \tilde c$.
To see that $\delta_n$ is a group homomorphism, let $[c] = [c_1] + [c_2]$ be a sum. Then $\hat c \coloneqq \hat c_1 + \hat c_2$ is a lift and by linearity of $\partial$ we have $[\partial^B \hat c]_A = [\partial^B \hat c_1] + [\partial^B \hat c_2]$.
Under chain homology $H_\bullet(-)$ the morphisms in the short exact sequence together with the connecting homomorphisms yield the homology long exact sequence
Consider first the exactness of $H_n(A) \stackrel{H_n(i)}{\to} H_n(B) \stackrel{H_n(p)}{\to} H_n(C)$.
It is clear that if $a \in Z_n(A) \hookrightarrow Z_n(B)$ then the image of $[a] \in H_n(B)$ is $[p(a)] = 0 \in H_n(C)$. Conversely, an element $[b] \in H_n(B)$ is in the kernel of $H_n(p)$ if there is $c \in C_{n+1}$ with $\partial^C c = p(b)$. Since $p$ is surjective let $\hat c \in B_{n+1}$ be any lift, then $[b] = [b - \partial^B \hat c]$ but $p(b - \partial^B c) = 0$ hence by exactness $b - \partial^B \hat c \in Z_n(A) \hookrightarrow Z_n(B)$ and so $[b]$ is in the image of $H_n(A) \to H_n(B)$.
It remains to see that
the image of $H_n(B) \to H_n(C)$ is the kernel of $\delta_n$;
the kernel of $H_{n-1}(A) \to H_{n-1}(B)$ is the image of $\delta_n$.
This follows by inspection of the formula in def. 1. We spell out the first one:
If $[c]$ is in the image of $H_n(B) \to H_n(C)$ we have a lift $\hat c$ with $\partial^B \hat c = 0$ and so $\delta_n[c] = [\partial^B \hat c]_A = 0$. Conversely, if for a given lift $\hat c$ we have that $[\partial^B \hat c]_A = 0$ this means there is $a \in A_n$ such that $\partial^A a \coloneqq \partial^B a = \partial^B \hat c$. But then $\tilde c \coloneqq \hat c - a$ is another possible lift of $c$ for which $\partial^B \tilde c = 0$ and so $[c]$ is in the image of $H_n(B) \to H_n(C)$.
Of course the situation for cochain cohomology is formally dual to this situation. For convenience we repeat the statement for dual chains:
Let $A^\bullet \to B^\bullet \to C^\bullet$ be a short exact sequence of cochain complexes.
For $[c]_C \in H^n(C)$ the class of a closed element $c$, by surjectivity of $B \to C$ there is an element $\hat c \in B$ mapping to it. This need not be closed anymore, but of course $d_B \hat c$ is. By the fact that $B \to C$ is a chain map we have that the image of $d_B \hat c$ in $C$ vanishes. Therefore by the exactness of the sequence the element $d_B \hat c$ may be regarded as a closed element of $A$. The cohomology class $[d_B \hat c]_A$ of this is what the connecting homomorphism
assigns to $[c]_C$:
This is indeed well defined, in that it is independent of the choice of $\hat c$: for $\hat c'$ another choice, we have that the difference $\hat c - \hat c'$ is in the kernel of $B \to C$ hence is in $A$. Then $d_B \hat c' = d_B \hat c + d_A(\hat c - \hat c')$. Hence $[d_B \hat c]_A = [d_B \hat c']_A$.
Let $0 \to A_\bullet \stackrel{f}{\to} B_\bullet \stackrel{g}{\to} C_\bullet \to 0$ be a short exact sequence of chain complexes in some abelian category $\mathcal{A}$. Then for all $n \in \mathbb{Z}$ there are natural connecting homomorphisms $\partial : H_n(C) \to H_{n-1}(A)$ 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.
The connecting homomorphism of the long exact sequence in homology induced from short exact sequences of the form
is called a Bockstein homomorphism.
The connecting homomorphism in a long exact sequence in homology induced from a short exact sequence $A_\bullet \stackrel{f}{\to} B_\bullet \to C_\bullet$ is equivalently the image under the homology group functor of the homotopy cofiber sequence induced by $f$. This is discussed in detail at mapping cone in the section homology exact sequences.
For instance section 1.3 of