Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it induces a CW decomposition for $M$.

### A Morse function induces a handle decomposition

Denote by $X(M;f;s)$ the manifold $M$ with an $s$--handle attached by $f\colon\,(\partial D^s)\times D^{n-s}\to M$.

Theorem: Let f be a $C^\infty$ function on $M$ with no critical points on $f^{-1}[-\epsilon,\epsilon]$ except $k$ nondegenerate ones on $f^{-1}(0)$, all of index $s$. Then $f^{-1}[-\infty,\epsilon]$ is diffeomorphic to $X(f^{-1}[-\infty,-\epsilon];f_1,\ldots,f_k;s)$ (for suitable f_{i}).

**Historical note**: This was stated by Smale in 1961, with proof outline. Milnor's *Morse Theory*, Theorem 3.2 states and proves a weaker, homotopy version of the theorem, where there is only one handle in play. I asked about a proof of this theorem in this MO question, and it turned out that the first complete proof appeared in Palais, simplified lated by Fukui [Math. Sem. Notes Kobe Univ. 3 (1975), no. 1, paper no. X, pp. 1-4]. There's an alternative proof given in Appendix C to Madsen-Tornehave.

**Discussion**: Roughly, the theorem states that passing a critical point of a Morse function corresponds to attaching a handle. Thus, a Morse function induces a handle decomposition for $M$.

### A Morse function induces a CW decomposition

Let $f$ be a Morse function on $M$. Choosing a complete Riemannian metric on $M$ determines a stratification of $M$ into cells $D(p)$ (the unstable (descending) manifold for a critical point $p$ of $f$) in which two points lie in the same stratum if they are on the same unstable manifold. Each $D(p)$ is homeomorphic to an open cell, but the closure $\overline{D(p)}$ can be complicated.

Theorem: The union of compactified unstable manifolds $\bigcup \overline{D(p)}$ gives a CW decomposition of $M$ that is homeomorphic to $M$.

**Historical note**: A nice discussion of this theorem may be found in Bott's excellent Morse Theorem Indomitable, page 104. Milnor's *Morse Theory* derives a homotopy version of this statement (Theorem 3.5) from the homotopy version of the statement that a Morse function induces a handle decomposition (Theorem 3.2). The theorem seems to have been first proven by Kalmbach, and was recently strengthened to give the explicit characteristic maps by Lizhen Qin (understanding his papers is the motivation for my question).

The two statements given above look to me as though they should be very similar, especially since the homotopy version of the second follows directly from the homotopy version for the first in Milnor's book. But briefly searching through the literature makes it seem that they are virtually independant- papers proving one aren't even cited in papers proving the other, and the proofs look to me to be entirely unconnected. I don't understand why, probably because I'm having difficulty breaking free from the intuitive picture of the proof in Milnor's book, which works fine up to homotopy.

Question: Can you give an example, or intuition, for a case in which one of the above theorems is difficult but the other is easy? Is there an example for a compact finite-dimensional manifold with a Morse function such that the handlebody decomposition can be read straight off the Morse function, but the reading off the CW decomposition takes substantial extra work? Or the converse?

Stated differently, where does the "up to homotopy proof" on page 23 of Milnor conceptually collapse when we are working up to diffeomorphism instead of up to homotopy?

An application of topological equivalence to Morse theory(published as doi.org/10.1007/s11784-020-00843-z) $\endgroup$13more comments