I was studying Whitehead's Theorem following Algebraic Topology by Allen Hatcher (freely available in his webpage). The theorem is stated as follows:
If a map $f\colon X \to Y$ between connected CW complexes induces isomorphisms $f_* : π_n (X) → π_n (Y)$ for all $n$, then $f$ is a homotopy equivalence. In case $f$ is the inclusion of a subcomplex $X \hookrightarrow Y$ , the conclusion is stronger: $X$ is a deformation retract of $Y$.
I can somehow "see an intuitive justification" of the theorem as follows:
Given that CW complexes are built using attaching maps whose domains are spheres, it is perhaps not too surprising that homotopy groups of CW complexes carry a lot of information. Moreover, given a cellular map which induces isomorphisms on homotopy groups, it could seem reasonable that this map is an homotopy equivalence. Since for any continuous map between CW complexes we can consider its cellular approximation and both maps are homotopic, the theorem follows.
However, when I study the proof of the theorem step by step I get lost in the details. And by no means I am able to catch the idea behind the proof. I will try to be more explicit:
1) What is the intuition behind the compression lemma, stated below?
Let $(X,A)$ be a CW pair and let $(Y,B)$ be any pair with $B \neq \emptyset$. For each $n$ such that $X \to A$ has cells of dimension $n$, assume that $\pi_n(Y,B,y_0)=0$ for all $y_0\in B$. Then every map $f\colon (X,A) \to (Y,B)$ is homotopic $rel A$ to a map $X \to B$.
2) Is there any geometric idea behind the proof of Whitehead's Theorem? How do you come up with the idea of using the compression lemma?
Thanks in advance!