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!

  • 1
    $\begingroup$ I like to look at the compression lemma as a lifting problem drawing a square diagram and and thinking of $A\hookrightarrow X$ as a cofibration on the left vertical, and $B\rightarrow Y$ a fibration on the right vertical. The relative map $f$ fills the horizontals, and the compression lemma is the same as finding a diagonal arrow making the diagram commute appropriately (either strictly or up to homotopy). The lemma appears in Arkowtiz's "Introduction to Homotopy Theory" as Lemma 4.5.7 on pg 142 in this form. I'd suggest checking out his proof for a second viewpoint. $\endgroup$ – Tyrone Oct 28 '17 at 5:46
  • $\begingroup$ Thanks for your comment. Maybe, I was looking for a more geometric flavoured approach.... but your comment is enlightening anyway :). $\endgroup$ – D1811994 Oct 28 '17 at 12:26

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