The Mayer-Vietoris Sequence

 
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 $\partial(d)$.