Let $X$ is a topological space, and $A,B\subset X$ are such that $X=\mathrm{int}(A)\cup\mathrm{int} (B)$ , and $C=A\cap B$ . Then there is an exact sequence of homology groups:

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

Here, $i_*$ is induced by the inclusions $i:B\hookrightarrow X$ and $j_*$ by $j: A\hookrightarrow X$ , and $\partial_*$ is the following map: if $x$ is in $H_n(X)$ , then it can be written as the sum of a chain in $A$ and one in $B$ , $x=a+b$ . $\partial a=-\partial b$ , since $\partial x=0$ . Thus, $\partial a$ is a chain in $C$ , and so represents a class in $H_{n-1}(C)$ . This is $\partial_*x$ . One can easily check (by standard diagram chasing) that this map is well defined on the level of homology.