Exact sequences and connecting homomorphisms

Consider a degreewise short exact sequence $0\to A\xrightarrow{i}B\xrightarrow{p}C\to0$ of cochain complexes of modules. Choose a cycle $c$ in degree $n$ of the quotient and lift it to $b$ in $B^n$. Although its image is closed, the lift itself may not be closed. The image of $d_Bb$ under $p$ is zero, so exactness uniquely writes that differential as $i(a)$ for an element of $A^{n+1}$.

The element $a$ is closed: applying $i$ to its differential gives the square of the differential of $b$, hence zero, and $i$ is injective. We define the connecting map to send the class of $c$ to the class of $a$, with this positive lift-and-differentiate convention. For chain complexes the same construction lowers degree instead. Merely switching the location of an index without changing the surrounding convention is a common source of an incorrect boundary formula.

A second lift differs from the first by $i(u)$ for some $u$ in $A^n$. Its differential changes $a$ by $d_Au$, leaving the cohomology class unchanged. If the representative $c$ is changed by a boundary, lift the element whose differential supplies that boundary. Adding its differential to $b$ changes the representative downstairs but does not change $d_Bb$. Combining these two observations proves that the construction is well defined on cohomology classes.

Addition of lifts proves additivity, and a commutative diagram of short exact sequences carries every step of the construction to the corresponding step below it. This is naturality of the connecting map. For arbitrary abelian categories, the element argument is replaced by kernels and pullbacks or a diagram chase. Here modules permit an explicit calculation, but the proof depends on exactness and compatible differentials rather than on a chosen basis or finite generation.

At the quotient term, exactness says that a quotient cohomology class comes from a closed element of $B$ precisely when its connecting class vanishes. Indeed, if $a=d_Au$, replacing $b$ by $b-i(u)$ kills its differential. The converse follows by choosing a closed lift. This proves one key junction of the long exact sequence directly and gives the connecting map its practical meaning: it tests whether a desired lift can satisfy the extra equation of being closed.

Degreewise split exactness does not make every connecting map zero: chosen splittings might fail to respect differentials. A splitting by cochain maps does make the connecting map vanish, because it supplies closed lifts of all cycles. Even then, a vanishing connecting map alone need not establish a specified chain-level splitting. When using a long exact sequence, identify both the group and the actual homomorphism; an unknown multiplication map can change a cokernel even if all surrounding ranks are known.