long exact sequence (of homology groups)

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:

$$\begin{array}{ccccccccccc}\hfill \mathrm{\cdots }\hfill & \hfill \to \hfill & \hfill H_n(A,B)\hfill & \hfill \stackrel{i_{*}}{\to }\hfill & \hfill H_n(X,B)\hfill & \hfill \stackrel{j_{*}}{\to }\hfill & \hfill H_n(X,A)\hfill & \hfill \stackrel{{\partial }_{*}}{\to }\hfill & \hfill H_{n-1}(A,B)\hfill & \hfill \to \hfill & \end{array}$$

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 elementof $H_{n-1}(A,B)$. This is ${\partial }_{*}a$.

When $B$ is the empty set, we get the long exact sequence of the pair $(X,A)$:

$$\begin{array}{ccccccccccc}\hfill \mathrm{\cdots }\hfill & \hfill \to \hfill & \hfill H_n(A)\hfill & \hfill \stackrel{i_{*}}{\to }\hfill & \hfill H_n(X)\hfill & \hfill \stackrel{j_{*}}{\to }\hfill & \hfill H_n(X,A)\hfill & \hfill \stackrel{{\partial }_{*}}{\to }\hfill & \hfill H_{n-1}(A)\hfill & \hfill \to \hfill & \end{array}$$

The existence of this long exact sequence follows from the short exact sequence

$$\begin{array}{ccccccccc}\hfill 0\hfill & \hfill \to \hfill & \hfill C_{*}(A,B)\hfill & \hfill \stackrel{i_{\mathrm{♯}}}{\to }\hfill & \hfill C_{*}(X,B)\hfill & \hfill \stackrel{j_{\mathrm{♯}}}{\to }\hfill & \hfill C_{*}(X,A)\hfill & \hfill \to \hfill & \hfill 0\hfill \end{array}$$

where $i_{\mathrm{♯}}$ and $j_{\mathrm{♯}}$ are the maps on chains induced by $i$ and $j$,by the Snake Lemma.