I apologize right away for the wall-o-text.
I'm participating in a cohomology reading course, and I'll be leading the class through the following proposition later this week, but I'm having a hard time trying to understand the proof.
Proposition 3.18/3.19 If a natural transformation between unreduced cohomology theories on the category of CW pairs is an isomorphism when the CW pair is $(point, \emptyset)$, then it is an isomorphism for all CW pairs.
Hatcher begins by defining the following maps (for a fixed CW complex $Y$):
$$h^{n}(X,A) = \bigoplus_{i}\left(H^{i}(X,A;R) \otimes_{R} H^{n-i}(Y;R)\right) \\ k^{n}(X,A) = H^{n}(X \times Y, A \times Y; R) \\ \mu: h^{n}(X,A) \rightarrow k^{n}(X,A)$$
His proof is below:
In the finite case, he references the five lemma several times, but doesn't mention the objects at all. Can somebody shed some light on what these figures are?
Also, I'm stuck in the infinite-dimensional case. His telescoping argument on the homology side of things is to consider $X \times [0, \infty)$ with the product cell structure, where $[0, \infty)$ is given the cell structure with integer points as vertices. He then consider the subcomplex $T = \bigcup_{i} X^{i} \times [i, \infty)$ and shows that $H_n(X) \cong H_n(T)$ for every $n$, but I don't see how this construction reduces it to the finite-dimensional case (I can post his entire proof of this lemma as well, if it would help).
Any and all help is greatly appreciated. Thank you.
I ended up getting some help over on Reddit. This question can be closed.
http://www.reddit.com/r/math/comments/238pik/help_understanding_a_proof_in_hatcher_cohomology/