Long exact sequence in relative homology
= Long exact sequence in relative homology
For a pair $A\subseteq X$, the <short exact sequence of chain complexes> $0\to C_*(A)\to C_*(X)\to C_*(X,A)\to0$ induces
$$
\cdots\to H_n(A)\to H_n(X)\to H_n(X,A)\xrightarrow{\partial}H_{n-1}(A)\to\cdots.
$$