Give an elementary derivation for the Mayer–Vietoris sequence in simplicial homology for a \(\Delta\)-complex \(X\) decomposed as the union of subcomplexes \(A\) and \(B\).
Solution
1. Let \(C_n(X)\), \(C_n(A)\), \(C_n(B)\), \(C_n(A \cap B)\) be the simplicial chain groups.
setup.
2. There is a short exact sequence of chain complexes \[0 \to C_n(A \cap B) \xrightarrow{(i_*, j_*)} C_n(A) \oplus C_n(B) \xrightarrow{k_* - \ell_*} C_n(A + B) \to 0,\] where \(i, j\) are the inclusions of \(A \cap B\) into \(A\) and \(B\), and \(k, \ell\) are the inclusions of \(A\) and \(B\) into \(X\) (with \(C_n(A + B) = C_n(X)\) since \(A \cup B = X\)).
the map \(C_n(A \cap B) \to C_n(A) \oplus C_n(B)\) is injective (a simplex in \(A \cap B\) is determined by its images in \(A\) and \(B\)), and the map \(C_n(A) \oplus C_n(B) \to C_n(X)\) is surjective (every simplex of \(X\) lies in \(A\) or \(B\)), with exactness in the middle (a pair \((a, b)\) maps to \(0\) iff \(a = b\) as chains, i.e. iff it comes from \(A \cap B\)).
3. This short exact sequence of chain complexes induces a long exact sequence in homology: \[\cdots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to H_{n-1}(A \cap B) \to \cdots.\] Proof: the standard long exact sequence in homology associated to a short exact sequence of chain complexes.
4. This is the Mayer–Vietoris sequence.
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5. Q.E.D.
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