Show that the Mayer-Vietoris sequence is natural, i.e. If \(X\mapsvia{f} Y\) where \(f(A) \subset C\) and \(f(B) \subset D\), then this commutes: \[ \begin{CD} H_n(X) @>>> H_n(A\intersect B) @>>> H_n(A) \oplus H_n(B) @>>> H_{n-1}(X)\\ @VVf_*V @VVf_*V @VVf_*V @VVf_*V\\ H_n(Y) @>>> H_n(C\intersect D) @>>> H_n(C) \oplus H_n(D) @>>> H_{n-1}(Y)\\ \end{CD} \]
Solution
1. The diagram printed in the retained source has the Mayer–Vietoris groups in the wrong order.
The source displays \[ H_n(X)\longrightarrow H_n(A\cap B)\longrightarrow H_n(A)\oplus H_n(B)\longrightarrow H_{n-1}(X), \] which is not a segment of the Mayer–Vietoris long exact sequence. The correct segment is \[ H_n(A\cap B)\xrightarrow{(i_*,-j_*)}H_n(A)\oplus H_n(B) \xrightarrow{k_*+\ell_*}H_n(X)\xrightarrow{\partial}H_{n-1}(A\cap B). \] Thus the intended assertion is the naturality of this correct sequence.
2. Let \(S_\bullet(A+B)\subset S_\bullet(X)\) be the subcomplex generated by singular simplices whose images lie in \(A\) or in \(B\). There is a short exact sequence\[ 0\to S_\bullet(A\cap B)\xrightarrow{(i_\sharp,-j_\sharp)} S_\bullet(A)\oplus S_\bullet(B) \xrightarrow{k_\sharp+\ell_\sharp}S_\bullet(A+B)\to0. \]
The first map is injective. A pair \((\alpha,\beta)\) lies in the kernel of \(k_\sharp+\ell_\sharp\) exactly when \(\alpha=-\beta\) as a chain of \(X\); such a chain is supported in both \(A\) and \(B\), hence comes uniquely from a chain \(\gamma\in S_\bullet(A\cap B)\) as \((\gamma,-\gamma)\). Surjectivity holds because every generator of \(S_\bullet(A+B)\) lies wholly in \(A\) or wholly in \(B\).
3. The analogous sequence for \(Y=C\cup D\) fits into a commutative diagram of short exact sequences under \(f\).
Because \(f(A)\subset C\) and \(f(B)\subset D\), also \(f(A\cap B)\subset C\cap D\), and the induced singular-chain maps preserve all three subcomplexes. Naturality of inclusions gives \[ (f_\sharp\oplus f_\sharp)(i_\sharp,-j_\sharp) =(i'_\sharp,-j'_\sharp)f_\sharp, \] and \[ f_\sharp(k_\sharp+\ell_\sharp) =(k'_\sharp+\ell'_\sharp)(f_\sharp\oplus f_\sharp). \] Hence the chain-level diagram commutes.
4. Passing to homology gives a commutative ladder of long exact sequences, including the connecting homomorphisms.
The long exact homology sequence of a short exact sequence of chain complexes is natural: a morphism of short exact sequences induces a morphism of the associated long exact sequences. Applying this to 3 gives commutativity at every term, including \[ \begin{CD} H_n(S_\bullet(A+B)) @>{\partial}>> H_{n-1}(A\cap B)\\ @V{f_*}VV @VV{f_*}V\\ H_n(S_\bullet(C+D)) @>{\partial'}>> H_{n-1}(C\cap D). \end{CD} \]
5. The inclusions \(S_\bullet(A+B)\hookrightarrow S_\bullet(X)\) and \(S_\bullet(C+D)\hookrightarrow S_\bullet(Y)\) induce isomorphisms on homology, naturally with respect to \(f\).
This is the small-simplices theorem for excisive covers. Barycentric subdivision makes every singular simplex homologous, naturally, to a sum of simplices lying in one member of the cover. Therefore the inclusions are chain-homotopy equivalences onto homology and commute with the map induced by \(f\).
6. Under the identifications in 5, the ladder in 4 is exactly the Mayer–Vietoris naturality ladder. Thus the intended Mayer–Vietoris sequence is natural.
4. The maps at the other terms are the homology maps induced by the restrictions of \(f\), so every square in the corrected Mayer–Vietoris diagram commutes.