If $X$ is a space with a pair of subspaces $A, B \subset X$ such that $X$ is the union of the interiors of $A$ and $B$, then there is a long exact sequence of homology groups
$\displaystyle \ldots\to H_n(A\cap B) \quad \xrightarrow{\Phi}\quad H_n(A) \oplus H_n(B) \quad \overset{\Psi}{\to} \quad H_n(X) \quad \overset{\partial}{\to} \quad H_{n-1} (A \cap B)\to \ldots$
The explicit action of the boundary map $\partial : H_n(X)\to H_{n-1}(A \cap B)$ escapes me. Allen Hatcher, in his book, writes
A class $\alpha \in H_n(X)$ is represented by a cycle $z$, and by barycentric subdivision or some other method we can choose $z$ to be a sum $x+y$ of chains in $A$ and $B$, respectively.
I think I can understand how $\partial \alpha$ can then be identified with $H_{n-1}(A \cap B)$, but can someone make clear to me how we choose such a sum? If you can go through all the steps taking an element in $H_n(X)$ to $H_{n-1}(A\cap B)$ I will be even more grateful.