Exercise 11, chapter 4, section 1 of 'Algebraic Topology' by Hatcher

Question

I'm stuck in front of this exercise:

Let $X$ be a CW-complex. If there exists an increasing sequence of sub-complexes $X_1\subset X_2\subset\cdots$ such that $X=\bigcup X_i$ and each $X_i$ is contractible in $X_{i+1}$, i.e. the inclusion $X_i\to X_{i+1}$ is null-homotopic, then $X$ is contractible.

I thought I have solved it, but then I realized that I never used the CW-complex structure of $X$, so for sure it was wrong. Anyway, I'm sure I only miss to understand how to use the CW-structure to conclude but I'm stuck in a loop of arguments from which I cannot go out. If you can suggest which is the right way to start, I will be very happy (and grateful;)).

Thank you very much!

Answer 1

I really was not able to find out a solution from ali's hint. Anyway, I could find a solution by myself. The idea is to proceed directly, with the 'hand', and observe that $X$ is contractible if and only if $\pi_n(X)=0$ for every $n$ (if you want to be precise, you can show this claim by yourself using Whitehead's theorem). If $f:S^n\to X$ represents an element in $\pi_n(X)$, the image of $S^n$ in $X$ is compact and by a result about CW-complexes (Hatcher, Appendix) it is contained in some finite subcomplex. Now observe that if $X_i$ is contractible in $X_{i+1}$, then it is contractible in $X$ as well. Finally, use the hypothesis to conclude that $f(S^n)$ is contractible in $X$, i.e. every $[f]\in\pi_n(X)$ is trivial.

I wanted to write this because of this remark: I think the hypothesis of this exercise are a little weak. It is completely licit taking the following increasing sequence $$\{pt\}\subset X$$ of subcomplexes. Moreover, the only inclusion is null-homotopic. Since this can be done for every CW-complex, the exercise makes us able to show that each CW-complex is contractible!?!? Of course it is not true and in fact my solution doesn't work in this situation (and you may pay attention to this in following my idea). But we must at least exclude these degenerate cases...

Answer 2

you can use this lemma and then induction$$$$ lemma:if (X,A) is a cw_pair and $f:X\to Y$ ,F is a homotopy from $f|_{A\,}$to $\,g\,$ you can extend it to a homotopy F' from $X\times I\,$to $Y$ such that $F'|_{A}=F$(for example you can extend the nullhomotopy of id:$X_i\to X$ to a homotopy of$\,id:X\to X\,$to $f_i$ such that $f_i(X_i)=x_0$ $$$$ for the proof of lemma first prove it for $(D^n,S^n)$ and then use induction and cw_structure.