If $X$ is a topological space, and $A$ and $B$ are subspaces with $X\supset A\supset B$ , then there is a long exact sequence:

$$\begin{CD} \cdots@>>>H_n(A,B)@>i_*>> H_n(X,B) @>j_*>>H_n(X,A)@>\partial_*>> H_{n-1}(A,B)@>>> \end{CD}$$

where $i_*$ is induced by the inclusion $i:(A,B)\hookrightarrow(X,B)$ , $j_*$ by the inclusion $j:(X,B)\hookrightarrow(X,A)$ , and $\partial$ is the following map: given $a\in H_n(X,A)$ , choose a chain representing it. $\partial a$ is an $(n-1)$ -chain of $A$ , so it represents an element of $H_{n-1}(A,B)$ . This is $\partial_*a$ .

When $B$ is the empty set, we get the long exact sequence of the pair $(X,A)$ : $$\begin{CD} \cdots@>>>H_n(A)@>i_*>> H_n(X) @>j_*>>H_n(X,A)@>\partial_*>> H_{n-1}(A)@>>> \end{CD}$$

The existence of this long exact sequence follows from the short exact sequence $$\begin{CD} 0@>>>C_*(A,B)@>i_\sharp>> C_*(X,B) @>j_\sharp>> C_*(X,A)@>>>0 \end{CD}$$ where $i_\sharp$ and $j_\sharp$ are the maps on chains induced by $i$ and $j$ , by the Snake Lemma.