How did we know to invent homological algebra?

Solution 1:

Have you read Weibel's History of Homological Algebra?

Solution 2:

As I know, "ancient" topologists had studied Betti numbers $\beta_n(A)$ (the ranks of abelian groups $H_n(A)$) and then they noticed the connection between sequences $\{\beta_n(A)\}$ and $\{\beta_n(B)\}$.

Solution 3:

In my opinion, the relative singular homology group $H_n(B, A)$ is naturally a geometric object. As Stefan H. says in the comments, the idea of considering the boundary of a relative cycle in $(A, B)$ as a cycle of one dimension less in $B$ is quite natural.

But why is this connected to the short exact sequence? A quotient of chain groups (the relative group) ought to be related to a quotient of the underlying spaces.

In the relative group $C_n(B, A)$, the condition for a chain to be a cycle is relaxed from the boundary being $0$ to the boundary being a chain in the subspace $A$. Under mild assumptions on the pair $(A, B)^\dagger$, the quotient map $B \to B/A$ induces an isomorphism $$ H_n(B, A) \overset{\sim}{\longrightarrow} \tilde{H}_n(B/A). $$ When the subspace $A$ is quotiented to a point, the boundary of a chain in $B$ maps to that point in $H_n(B/A)$, or $0$ in the reduced group.

$^\dagger$ $A$ is closed and is a deformation retract of a neighborhood $A'$ in $B$