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.
Best Answer
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 $H_{n-1}(A_\bullet)$.