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I'm studying Mayer-Vietoris sequences for relative homology groups using Hatcher's Algebraic Topology. In the proof (at page 152), the author first prove that the sequence $$0\to C_n(A\cap B, C\cap D)\stackrel{f}{\to} C_n(A,C)\oplus C_n(B,D) \stackrel{g}{\to} C_n(A\cup B,C\cup D)\to 0$$ is exact. I understand the given proof, but the method used seemed to be (in my opinion) a bit "exaggerated" to prove this, and, as far as I saw, Hatcher's book often gives straight-forward proofs, specially for exactness of sequences (maybe, he was just showing a fancy way to show the proposition, or maybe straigh-forward method is more complicated in this case or less enjoyable to do). So, I wondered if isn't there a straight-forward proof ot the exactness of the sequence.

Here is my attempt: firstly, notice that $f$ anf $g$ are well-defined and are homomorphisms because they are induced maps (from the second row maps of that big diagram in page 152), given by $f(\overline{x})=(\overline{x},-\overline{x})$ and $g(\overline{a},\overline{b}) = \overline{a+b}$. It's pretty clear that $f$ is injective, $g$ is surjective and $\mathrm{Im}(f)\subset\ker(g)$. To conclude the proof, I just need that $\ker(g)\subset\mathrm{Im}(f)$.

If $(a,b)\in\ker(g)$, $\overline{a+b}=\overline{0}$, so $a+b=\sum_i \alpha_i c_i+\sum_j \beta_j d_j$ (with $c_i\in C_n(C)$, $d_j\in C_n(D)$ and coefficients in $\mathbb{Z}$). Well, this is all that I got. Could someone help me? Thanks.

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First of all, the surjectivity of $g$ is not that obvious: you will need some kind of subdivision to prove it and this is actually the point where you will need some assumptions for your spaces.

For your actual question: $a+b=0$ means that each $c_i$ has to cancel against some $d_j$. But this can only happen if $c_i\subset C\cap D$.

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    Subdivision does not help. If you have an element of the codomain of g you need to produce a preimage, and producing a preimage of a subdivision does not do that. – 2017-03-07
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    Thanks for the help. But, actually, the exact sequence I wanted is the one that ends with $C_n(A+B,C+D):=C_n(A+B)/C_n(C+D)$ where, in Hatcher's notation, $C_n(A+B)$ stands for the sum of chains in $A$ and chains in $B$ (notation introduced at p. 124), not the chains of $A\cup B$ (what I agree that would be hard). This is why I thought that surjectivity of $g$ was clear (it would be just "decomposing" the sum in chains of $A$ and chains of $B$, wouldn't it?). – 2017-03-07