Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic assumption that $(f,g)$ satisfies Morse-Smale transversality. What additional conditions does one need to impose in this situation to ensure that the CW complex is regular? (Regular means the attaching maps are homeomorphisms.)

It is true that every CW complex is homotopy equivalent to a simplicial complex, and those are all regular, but I'm asking for regularity on the nose.

(Note: I originally asked the question on MSE.)

EDIT: I'm particularly interested in the case when Morse functions give rise to pseudo-regular CW complexes, by which I mean the incidence numbers of the cells all lie in $\{0,\pm 1\}$. By this definition, regular implies pseudo-regular, and pseudo-regular allows for a trivial boundary operator on the Morse-Smale complex.

Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic assumption that $(f,g)$ satisfies Morse-Smale transversality. What additional conditions does one need to impose in this situation to ensure that the CW complex is regular? (Regular means the attaching maps are homeomorphisms.)

It is true that every CW complex is homotopy equivalent to a simplicial complex, and those are all regular, but I'm asking for regularity on the nose.

(Note: I originally asked the question on MSE.)

EDIT: I'm particularly interested in the case when Morse functions give rise to pseudo-regular CW complexes, by which I mean the incidence numbers of the cells all lie in $\{0,\pm 1\}$. By this definition, regular implies pseudo-regular, and pseudo-regular allows for a trivial boundary operator on the Morse-Smale complex.

Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic assumption that $(f,g)$ satisfies Morse-Smale transversality. What additional conditions does one need to impose in this situation to ensure that the CW complex is regular? (Regular means the attaching maps are homeomorphisms.)

It is true that every CW complex is homotopy equivalent to a simplicial complex, and those are all regular, but I'm asking for regularity on the nose.

(Note: I originally asked the question on MSE.)

Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic assumption that $(f,g)$ satisfies Morse-Smale transversality. What additional conditions does one need to impose in this situation to ensure that the CW complex is regular? (Regular means the attaching maps are homeomorphisms.)

It is true that every CW complex is homotopy equivalent to a simplicial complex, and those are all regular, but I'm asking for regularity on the nose.

(Note: I originally asked the question on MSE.)

EDIT: I'm particularly interested in the case when Morse functions give rise to pseudo-regular CW complexes, by which I mean the incidence numbers of the cells all lie in $\{0,\pm 1\}$. By this definition, regular implies pseudo-regular, and pseudo-regular allows for a trivial boundary operator on the Morse-Smale complex.