There is an exercise in Hatcher (Section 2.2) which asks:
For a CW pair $(X,A)$ show there is a relative cellular chain complex formed by the groups $H_i(X_i,X_{i−1} ∪ A_i)$, having homology groups isomorphic to $H_n(X,A)$.
The following link is Hatcher's book:
http://pi.math.cornell.edu/~hatcher/AT/AT.pdf
To solve this problem, I think I should draw a huge diagram as in p.139, by augmenting some sequences, but I have no idea what sequence do I have to use.
Any hints??
Best Answer
First consider the homology of the pair $(X^i \cup A^{i+1}, X^{i-1} \cup A^i)$. This is a CW pair, so by excision $$H_i(X^i \cup A^{i+1}, X^{i-1} \cup A^i) \cong \tilde{H}_i(X^i \cup A^{i+1} / X^{i-1} \cup A^i) \cong \tilde{H}_i (X^i / X^{i-1} \cup A^i) \cong H_i(X^i, X^{i-1} \cup A^i)$$where the second isomorphism holds because adding the $(i+1)$-cells of $A$ will have no effect on $H_i$ since we're collapsing the $i$-skeleton of $A$ to a point in the quotient (this is basically a relative version of Lemma 2.34 in your book).
This means that we can substitute this into the long exact sequence of the pair $(X^i \cup A^{i+1}, X^{i-1} \cup A^i)$ to get
$$\cdots \to H_i(X^{i-1} \cup A^i) \to H_i(X^i \cup A^{i+1}) \to H_i(X^i, X^{i-1} \cup A^i) \to H_{i-1}(X^{i-1} \cup A^i) \to \cdots.$$
But $X^i \cup A^{i+1}$ fits into another long exact sequence, too: the long exact sequence of the pair $(X^{i+1} \cup A^{i+2}, X^i \cup A^{i+1})$, which is given by
$$\cdots \to H_{i+1}(X^{i+1} \cup A^{i+2}) \to H_{i+1}(X^{i+1}, X^i \cup A^{i+1}) \to H_i(X^i \cup A^{i+1}) \to \cdots$$
where we've identified the relative group in the middle by the same isomorphism above with $i$ replaced by $i+1$.
so define the map $H_{i+1}(X^{i+1}, X^i \cup A^{i+1}) \to H_i(X^i, X^{i-1}\cup A^i)$ to be the composition $H_{i+1}(X^{i+1}, X^i \cup A^{i+1}) \to H_i(X^i \cup A^{i+1}) \to H_i(X^i, X^{i-1} \cup A^i)$, where the first map is from the second long exact sequence above, and the second map is from the first long exact sequence above.
The remaining steps are to show that the sequence of these maps forms a chain complex and that the homology of this chain complex yields the homology of the pair $(X,A)$.