I refer to Hatcher's proof of Euler Characteristic. I don't understand why Hatcher mentions relative homology $C_n=H_n(X^n, X^{n-1})$.
I thought that once we prove that $\sum_n (-1)^n \text{rank}\, (C_n)=\sum_n (-1)^n \text{rank}\, H_n$, we are basically done with the proof? Since $\text{rank}\, C_n$ is the number of $n$-cells of $X$?
I may have missed an important point, thanks for any enlightenment.
Hatcher (pg 146):
Ignoring the very last line of the proof for the moment, what this proof demonstrates is an abstract statement in algebra that a priori has nothing whatsoever to do with $X$ or its cells, namely: for any chain complex $(C_n)$ with corresponding homology groups $(H_n)$, $$\sum_n (-1)^n \text{rank}(C_n) = \sum_n (-1)^n \text{rank} H_n $$ Having proved this equation, you are still very far from done, because you have proved nothing about $X$ yet. All you have proved is an abstract lemma in homological algebra.
To bring the CW complex $X$ into the picture, the idea is that there is a cellular chain complex with the following properties: its $n$th homology group is isomorphic to the absolute singular homology group $H_n(X)$; its $n$th chain group $C_n$ is the relative singular homology group $H_n(X_n,X_{n-1})$; and $\text{rank}(C_n) = \text{rank}(H_n(X_n,X_{n-1}))$ is equal to the number of $n$ cells. NOW you have proved something about $X$!