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I'm reading Milnor's Morse Theory and stumble upon a proof of a theorem below. My question in the end is only about topology.

$\textit{Theorem 3.5. }$ If $f$ is a differentiable function on a manifold $M$ with no degenerate critical points, and if each $M^a:=f^{-1}(-\infty, a]$ is compact, then $M$ has the homotopy type of a CW-complex, with one cell of dimension $\lambda$ for each critical points of index $\lambda$.

I try to summarize the part of the proof where I didn't understand as follows : with $M^{a_i}:=f^{-1}(-\infty, a_i]$, suppose that we have infinite sequences of homotopy equivalences $M^{a_i} \to K_i$ (where $K_i$ are CW-complexes),

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each extending the previous one. The claim is that if $K=\bigcup_{i=1}^{\infty} K_i$ equipped with direct limit topology, and $g: M \to K$ be the limit map, then $g$ induces isomorphisms of homotopy groups of all dimensions.

My Question: I have a very limited knowledge of Algebraic Topology so i can't understand how such claim is true. I don't even know what is "limit map" and how to working with direct limit. I really appreciate if somebody can give me some reference about this (so i can learn it by myself) and maybe a short explanation if possible on why $g$ induces such isomorphisms. I've found a similar question of this issue here but I still couldn't get accessible explanation. Thank you.

For those who want to see the full proof in the text, here it is.

Spoiler Warning! Spoiler Warning! Spoiler Warning!

Kelvin Lois
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  • Do you have some knowledge about CW-complexes? – Paul Frost Aug 29 '20 at 09:37
  • @PaulFrost Yes, I read Lee's Topological Manifolds, but nothing like Hatcher's appendix about sth like dominated CW-complexes. – Kelvin Lois Aug 29 '20 at 09:38
  • $K_{i+1}$ is obtained from $K_i$ by attaching cells. But it is not clear to me why $K_{i+1}$ is a CW-complex such that $K_i$ is a subcomplex of $K_{i+1}$. Are the attached cells of dimension $> \dim K_i$? – Paul Frost Aug 30 '20 at 22:09
  • @PaulFrost I don't think so, since the there's no restriction on the index of critical points (which correspond to the dimension of the attached cells) that will appear as we pass through a critical value. Why $K_i$ has to be a subcomplex of $K_{i+1}$? – Kelvin Lois Aug 30 '20 at 23:40
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    Ultimately, the claim boils down to the following fact: if $X = \bigcup_{i=0}^\infty X_i$ is the union of an increasing sequence of spaces $X_0\subseteq X_1\subseteq \cdots$, then the natural morphism $\operatorname{colim}_i \pi_n(X_i) \to \pi_n(X)$ is an isomorphism. This, in turn, boils down to the fact that, for such a space $X$, a map $K\to X$ from a compact $K$ must factor through some $X_i$. – Brian Shin Aug 31 '20 at 03:23
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    I have been careless about basepoints, and I have also forgotten that I need my spaces $X_i$ to be Hausdorff. – Brian Shin Aug 31 '20 at 03:45
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    @SiKucing This is required to show that $K$ is a CW-complex. I think it is true because the attaching maps go into appropriate skeleta of $K_i$. – Paul Frost Aug 31 '20 at 08:28
  • @BrianShin Would you write down the details (or point out some reference where i can look into) for that? – Kelvin Lois Aug 31 '20 at 11:27

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