In Paul Selick's book Introduction to Homotopy Theory, he says that one can prove the CW-Approximation Theorem:

Given a topological space $Y$ there exists CW-complex $X$ and a map $f : X \rightarrow Y$ such that $f_\ast : \pi_n(X) \rightarrow \pi_n(Y)$ is an isomorphism for all $n$.

using the fact that

Let $f : X \rightarrow Y$ be a map between CW-complex. Then there exists $g : X \rightarrow Y$ such that $g \simeq f$ and $g$ is cellular.

However he does not give any detail and I don't see how one should proceed.

  • $\begingroup$ Paul Selick's book is very concise. It it not really a good "introduction". I would recommend considering another book for a learner. For example, Hatcher's text, which covers CW approximation on pg 352. (Personally I dislike Hatcher's style, but his book still comes highly recommended.) $\endgroup$ – Tyrone May 30 '18 at 9:31

In Hatcher in his book available here : http://www.math.cornell.edu/~hatcher/AT/AT.pdf describes "cellular approximation" that results in theorem 4.8 page 349, which is what you want to prove.

As usual with CW-complexes, this is prove by induction on the $n$-dimensional skeletons. The details can be quite complex depending on what you already know true about CW-complexes or not, I'm letting you read the text and ask another question if you don't understand.

  • $\begingroup$ Thanks for you answer. I know the proof of Hatcher's book. What I am really wondering is how to get the result from this proposition, as a "second proof" if you wish. $\endgroup$ – Oscar P. May 30 '18 at 10:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.