Connecting homomorphism and boundary map any relation? In the Snake Lemma, there is a connecting homomorphism that links one kernel to another cokernel. In quite a few sources it is written using the symbol $\partial$.
Something like this:
$\ker c\xrightarrow{\partial}\text{coker}\ a$
Given that the boundary map in homology is also usually denoted as $\partial$, I am curious is there a relationship between them? (in the case that the objects in the Snake Lemma are Chain Complexes)
Or is just a case of using the same symbol?
Thanks.
 A: The construction of the connecting map in homology is a direct consequence/application of the Snake Lemma.  Given a ses of chain complexes and chain maps $0 \to A_{\bullet} \to B_{\bullet} \to C_{\bullet} \to 0$ you apply the Snake Lemma to the diagram with top row
$$
A_n/\mathrm{im}(d) \to B_n/\mathrm{im}(d) \to C_n/\mathrm{im}(d) \to 0
$$
and bottom row
$$
0 \to Z_{n-1}(A_\bullet) \to Z_{n-1}(B_\bullet) \to Z_{n-1}(C_\bullet)  
$$
(vertically connected by differentials).  The lemma gives the connecting homomorphism at once since the relevant cokernel on the left vertical is $H_{n-1}(A_\bullet)$.  
A: For the long exact sequence of the pair, the connecting homomorphism is related to the boundary homomorphism in a fairly direct way.  Let $(X,A)$ be a pair of spaces, and consider the short exact sequence of chain complexes
$$0\to C_n(A)\xrightarrow{i} C_n(X)\xrightarrow{q} C_n(X,A)\to 0$$
Consider the following portion of the long exact sequence
$$H_n(A)\xrightarrow{i_*} H_n(X)\xrightarrow{q_*} H_n(X,A)\xrightarrow{\partial} H_{n-1}(A)$$
Treating $C_n(A)$ as a subset of $C_n(X)$, and $C_n(X,A)$ as cosets of $C_n(A)$ in $C_n(X)$, the maps can be defined as follows:
\begin{align*}
i_*(\sigma+\partial C_{n+1}(A))&=\sigma+\partial C_{n+1}(X)\\
q_*(\sigma+\partial C_{n+1}(X))&=\sigma+\partial C_{n+1}(X)+C_n(A)\\
\partial(\sigma+\partial C_{n+1}(X)+C_n(A))&=\partial\sigma +\partial C_n(A)
\end{align*}
where in the the first two cases, $\partial\sigma=0$, and in the third case, $\partial\sigma\in C_n(A)$.
Notice that the definition of the connecting homomorphism is literally the boundary homomorphism applied to the coset.
