Naturality condition for connecting homomorphisms?

Question

I've been reading about the Mayer-Vietoris sequence, but I don't follow a certain naturality condition.

Suppose two spaces can be written as $X=X_1^\circ\cup X_2^\circ$ and $Y=Y_1^\circ\cup Y_2^\circ$. If $f$ is a continuous map such that $f(X_1)\subset Y_1$ and $f(X_2)\subset Y_2$, then the following composites are the same: $$ H_n(X)\stackrel{D}{\to}H_{n-1}(X_1\cap X_2)\stackrel{g_*}{\to}H_{n-1}(Y_1\cap Y_2) $$ and $$ H_n(X)\stackrel{f_*}{\to}H_n(Y)\stackrel{\Delta}{\to}H_{n-1}(Y_1\cap Y_2) $$

where $D$ and $\Delta$ are the connecting homomorphisms, and $g$ is the restriction of $f$.

From what I understand, the connecting homomorphisms $D=dh^{-1}_*q_*$ where $d$ is the connecting homomorphism for $(X_1,X_1\cap X_2)$, and $h\colon (X_1,X_1\cap X_2)$ and $q\colon (X,\emptyset)\to (X,X_2)$ are inclusions. Similarly, $\Delta=d'h'^{-1}_*q'_*$ Writing out what the maps do to some class $z+B_n(X)$, I end up with $d'(h'^{-1}qf(z)+B_n'(Y))$ and $g_*(dh^{-1}q(z)+B_{n-1}(X_1\cap X_2))$, but it's not clear to me that these are the same.

Answer 1

The theorem you are looking for is the naturality of the connecting homomorphism in long exact sequences of homology groups.

Theorem. Let $$ \require{AMScd} \begin{CD} 0 @>>> A_* @>>> B_* @>>> C_* @>>>0\\ & @VV{\alpha_*}V @VV{\beta_*}V @VV{\gamma_*}V\\ 0 @>>> D_* @>>> E_* @>>> F_* @>>>0 \end{CD} $$ be a commutative diagram of chain complexes and chain maps with exact rows. We have an induced map between the long exact sequences of homology groups, in particular $$ \begin{CD} H_n(C_*) @>{\partial}>> H_{n-1}(A_*) \\ @V{\gamma_n^h}VV & @VV{\alpha_{n-1}^h}V \\ H_n(F_*) @>{\tilde\partial}>> H_{n-1}(D_*) \\ \end{CD} $$ commutes. In other words, the connecting homomorphism is natural.

In your case, you start with a diagram $$ \require{AMScd} \begin{CD} 0 @>>> C_*(X_1\cap X_2) @>>> C_*(X_1)\oplus C_*(X_2) @>>> C_*(X_1+X_2) @>>>0\\ & @VVV @VVV @VVV\\ 0 @>>> C_*(Y_1\cap Y_2) @>>> C_*(Y_1)\oplus C_*(Y_2) @>>> C_*(Y_1+Y_2) @>>>0\\ \end{CD}. $$ The rows here are the short exact sequences of chain complexes that induce the Mayer-Vietoris sequences for $X$ and $Y$.

Figure out the vertical maps and convince yourself the 2 squares commute, then the naturality of the connecting homomorphism is exactly the commuting square you are looking for.