Part II Lecture Notes: Algebraic Topology, James Lingard
Let $X$ be a simplicial complex which is the union $A \cup B$ of two subcomplexes. Then
$$
\cdots \longrightarrow H_{i}(A \cap B) \longrightarrow H_{i}(A) \oplus H_{i}(B) \longrightarrow H_{i}(X) \longrightarrow H_{i-1}(A \cap B) \longrightarrow \cdots
$$
is a long exact sequence. What are the three homomorphisms?
• The homomorphism $H_{i}(A \cap B) \rightarrow H_{i}(A) \oplus H_{i}(B)$ is the obvious pair of inclusions.
• The homomorphism $H_{i}(A) \oplus H_{i}(B) \rightarrow H_{i}(X)$ is given by $(x, y) \mapsto x-y$.
• The homomorphism $H_{i}(X) \rightarrow H_{i-1}(A \cap B)$ is the "boundary map" constructed as follows.
\[\begin{CD}
0@>>>C_i(A\cap B)@>>>C_i(A)\oplus C_i(B)@>>>C_i(X)@>>>0\\
&@VV\partial V@VV\partial V@VV\partial V\\
0@>>>C_{i-1}(A\cap B)@>>>C_{i-1}(A)\oplus C_i(B)@>>>C_{i-1}(X)@>>>0\\
\end{CD}\]
For any element of $H_{i}(X)$, take a representative cycle $z \in Z_{i}(X)$ and choose an element $d \in C_{i}(A) \oplus C_{i}(B)$ which maps to $z$. Then the equivalence class of $z$ maps to the equivalence class of the (unique) inverse image $c \in Z_{i-1}(A \cap B)$ for $\partial(d)$.