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The second of the theorems you quoted is considerably harder to prove. The gist of the proof is as follows. Consider the closure $\overline{D(p)}$ of $D(p)$ in $M$. Then Lizhen Qin proves that it admits a resolution in the sense of semi-algebraic geometry. More precisely he constructs a compact space $\widehat{D(p)}$ and a continuous surjective map $\pi: \widehat{D(p)}\to\overline{D(p)}$ with the following properties.

$\bullet$ The space $\widehat{D(p)}$ is homeomorphic to a closed ball of dimension equal to the Morse index of $p$.

$\bullet$ The restriction of $\pi$ to the interior of $\widehat{D(p)}$ induces a homeomorphism onto $D(p)$.

The theorem requires that the gradient flow satisfy the Morse-Smale transversality condition wheras no such requirement is needed for the handle decomposition theorem. Moreover, the result is very sensitive to the behavior of the gradient flow near the critical points. In such a region the flow is a linear flow given by a symmetric matrix, the Hessian of $f$ at that particular point. If the eigenvalues are $\pm 1$ things are fine. For different eigenvalues things can go horribly wrong.

In Chap. 8 of my paper Tame flows I show that under appropriate conditions on the eigenvalues of the Hessians at the critical points the Morse-Smale condition is equivalent to the requirement that the stratification by unstable manifolds be a Whitney regular stratification. Moreover I give examples and pictures describing how the Whitney regularity be is destroyed if the spectra of the Hessian Hessians do not satisfy those constraints.

Another very good reference for these topics is Burghelea-Friedlander-Kappeler survey arXiv: 1101.0778. Burghelea has an alternate and much simpler argument for Lizhen Qin's result, and the paper arXiv: 1101.0778 is much more readable than Qin's.

In the shameless-plug department, I ought to mention the recent 2nd edition of my book An Invitation to Morse Theory. In Chapter 4 I discuss at length these issues without the tameness assumption.

2 clarified a statement; added 16 characters in body

The second of the theorems you quoted is considerably harder to prove. The gist of the proof is as follows. Consider the closure $\overline{D(p)}$ of $D(p)$ in $M$. Then Lizhen Qin proves that it admits a resolution in the sense of semi-algebraic geometry. More precisely he constructs a compact space $\widehat{D(p)}$ and a continuous surjective map $\pi: \widehat{D(p)}\to\overline{D(p)}$ with the following properties.

$\bullet$ The space $\widehat{D(p)}$ is homeomorphic to a closed ball of dimension equal to the Morse index of $p$.

$\bullet$ The restriction of $\pi$ to the interior of $\widehat{D(p)}$ induces a homeomorphism onto $D(p)$.

The theorem requires that the gradient flow satisfy the Morse-Smale transversality condition wheras no such requirement is needed for the handle decomposition theorem. Moreover, the result is very sensitive to the behavior of the gradient flow near the critical points. In such a region the flow is a linear flow given by a symmetric matrix, the Hessian of $f$ at that particular point. If the eigenvalues are $\pm 1$ things are fine. For different eigenvalues things can go horribly wrong.

In Chap. 8 of my paper Tame flows I show that under appropriate conditions on the eigenvalues of the Hessians at the critical points the Morse-Smale condition is equivalent to the requirement that the stratification by unstable manifolds be a Whitney regular stratification. Moreover I give examples and pictures describing how the Whitney regularity be destroyed if the spectra of the Hessian do not satisfy those constraints.

Another very good reference for these topics is Burghelea-Friedlander-Kappeler survey arXiv: 1101.0778. Burghelea has an alternate and much simpler argument for Lizhen Qin's result, and the paper arXiv: 1101.0778 is much more readable than Qin's.

In the shameless-plug department, I ought to mention the recent 2nd edition of my book An Invitation to Morse Theory. In Chapter 4 I discuss at length these issues without the tameness assumption.

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The second of the theorems you quoted is considerably harder to prove. The gist of the proof is as follows. Consider the closure $\overline{D(p)}$ of $D(p)$ in $M$. Then Lizhen Qin proves that it admits a resolution in the sense of semi-algebraic geometry. More precisely he constructs a compact space $\widehat{D(p)}$ and a continuous map $\pi: \widehat{D(p)}\to\overline{D(p)}$ with the following properties.

$\bullet$ The space is homeomorphic to a closed ball of dimension equal to the Morse index of $p$.

$\bullet$ The restriction of $\pi$ to the interior of $\widehat{D(p)}$ induces a homeomorphism onto $D(p)$.

The theorem requires that the gradient flow satisfy the Morse-Smale transversality condition wheras no such requirement is needed for the handle decomposition theorem. Moreover, the result is very sensitive to the behavior of the gradient flow near the critical points. In such a region the flow is a linear flow given by a symmetric matrix, the Hessian of $f$ at that particular point. If the eigenvalues are $\pm 1$ things are fine. For different eigenvalues things can go horribly wrong.

In Chap. 8 of my paper Tame flows I show that under appropriate conditions on the eigenvalues of the Hessians at the critical points the Morse-Smale condition is equivalent to the requirement that the stratification by unstable manifolds be a Whitney regular stratification. Moreover I give examples and pictures describing how the Whitney regularity be destroyed if the spectra of the Hessian do not satisfy those constraints.

Another very good reference for these topics is Burghelea-Friedlander-Kappeler survey arXiv: 1101.0778. Burghelea has an alternate and much simpler argument for Lizhen Qin's result, and the paper arXiv: 1101.0778 is much more readable than Qin's.

In the shameless-plug department, I ought to mention the recent 2nd edition of my book An Invitation to Morse Theory. In Chapter 4 I discuss at length these issues without the tameness assumption.