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World seems to be small. I have tried the approach you outlined with a former student of mine and we got stuck in the simplest case, the height function on a torus.The saddle point was (and is) a serious headache.

Let me mention our strategy maybe you get lucky. In a recent paper I showed that given any triangulated space together with a Morse-Forman function on it, there exists a genuine flow on this space with the following properties

1. The critical points are the exactly barycenters of all the faces.
2. The Conley index of the barycenter of a critical face is a sphere of the same dimension as the face.
3. The Conley index of the barycenter of a non-critical face is homotopically trivial

The flows are easily described and visualized and they belong to a class of flows I called tame. There examples of tame flows beyond the class generated by a Morse-Forman function and they are as well behaved. To get such flows on a triagulated space it suffices to label the vertices. The flow runs from the higher label to the lower label. We thought that using the Morse function for labeling would do the trick.

Given a Morse function on a manifold one could envisage constructing such a flow by choosing a sufficiently fine triangulation and trying to guess how the gradient flow would behave on each face. Things get hairy near the critical points. However I still believe this approach has legs.

As for associating a Morse-Forman function to a Morse function there are some recent results in this direction arXiv: 1010.0548 or arXiv: 0803.2616.

Remark. This in reply to your new question. There are as many stationary points as faces. The barycenter of each face is a stationary point. However, only few of these stationary points have nontrivial Conley index. I will refer to the other stationary points as ghosts, and the level sets containing ghosts as ghost levels.

The Conley index of a barycenter of a face is nontrivial if and only if that face is critical in the sense of Forman. In my paper mentioned in my answer I describe a general process (flip-flop) that generalizes the process of handle attachment. Crossing a ghost level does not change the homotopy type of the sublevel set.

The process of crossing of ghost level does not even change the simple homotopy type. If $\lbrace f=c\rbrace$ is a ghost level, then $\lbrace f\leq c+\varepsilon\rbrace$ is simple homotopic to the space obtained from $\lbrace f\leq c-\varepsilon\rbrace$ by attaching a cone over a contractibe subset $B\subset \lbrace f=c-\varepsilon\rbrace$.

The space $\lbrace f\leq c+\varepsilon \rbrace$ is $PL$ homeomorphic to the space obtained from $\lbrace f\leq c-\varepsilon\rbrace$ by a flip-flop which is essentially a blowdown followed by a blowup along clearly defined loci.

World seems to be small. I have tried the approach you outlined with a former student of mine and we got stuck in the simplest case, the height function on a torus.The saddle point was (and is) a serious headache.

Let me mention our strategy maybe you get lucky. In a recent paper I showed that given any triangulated space together with a Morse-Forman function on it, there exists a genuine flow on this space with the following properties

1. The critical points are the exactly barycenters of all the faces.
2. The Conley index of the barycenter of a critical face is a sphere of the same dimension as the face.
3. The Conley index of the barycenter of a non-critical face is homotopically trivial

The flows are easily described and visualized and they belong to a class of flows I called tame. There examples of tame flows beyond the class generated by a Morse-Forman function and they are as well behaved. To get such flows on a triagulated space it suffices to label the vertices. The flow runs from the higher label to the lower label. We thought that using the Morse function for labeling would do the trick.

Given a Morse function on a manifold one could envisage constructing such a flow by choosing a sufficiently fine triangulation and trying to guess how the gradient flow would behave on each face. Things get hairy near the critical points. However I still believe this approach has legs.

As for associating a Morse-Forman function to a Morse function there are some recent results in this direction arXiv: 1010.0548 or arXiv: 0803.2616.

Remark. This in reply to your new question. There are as many stationary points as faces. The barycenter of each face is a stationary point. However, only few of these stationary points have nontrivial Conley index. I will refer to the other stationary points as ghosts, and the level sets containing ghosts as ghost levels.

The Conley index of a barycenter of a face is nontrivial if and only if that face is critical in the sense of Forman. In my paper mentioned in my answer I describe a general process (flip-flop) that generalizes the process of handle attachment. Crossing a ghost level does not change the homotopy type of the sublevel set.

The process of crossing of ghost level does not even change the simple homotopy type. If $\lbrace f=c\rbrace$ is a ghost level, then $\lbrace f\leq c+\varepsilon\rbrace$ is simple homotopic to the space obtained from $\lbrace f\leq c-\varepsilon\rbrace$ by attaching a cone over a contractibe subset $B\subset \lbrace f=c-\varepsilon\rbrace$.

The space $\lbrace f\leq c+\varepsilon \rbrace$ is $PL$ homeomorphic to the space obtained from $\lbrace f\leq c-\varepsilon\rbrace$ by a flip-flop which is essentially a blowdown followed by a blowup along clearly defined loci.

World seems to be small. I have tried the approach you outlined with a former student of mine and we got stuck in the simplest case, the height function on a torus.The saddle point was (and is) a serious headache.

Let me mention our strategy maybe you get lucky. In a recent paper I showed that given any triangulated space together with a Morse-Forman function on it, there exists a genuine flow on this space with the following properties

1. The critical points are the exactly barycenters of all the faces.
2. The Conley index of the barycenter of a critical face is a sphere of the same dimension as the face.
3. The Conley index of the barycenter of a non-critical face is homotopically trivial

The flows are easily described and visualized and they belong to a class of flows I called tame. There examples of tame flows beyond the class generated by a Morse-Forman function and they are as well behaved. To get such flows on a triagulated space it suffices to label the vertices. The flow runs from the higher label to the lower label. We thought that using the Morse function for labeling would do the trick.

Given a Morse function on a manifold one could envisage constructing such a flow by choosing a sufficiently fine triangulation and trying to guess how the gradient flow would behave on each face. Things get hairy near the critical points. However I still believe this approach has legs.

As for associating a Morse-Forman function to a Morse function there are some recent results in this direction arXiv: 1010.0548 or arXiv: 0803.2616.

World seems to be small. I have tried the approach you outlined with a former student of mine and we got stuck in the simplest case, the height function on a torus.The saddle point was (and is) a serious headache.

Let me mention our strategy maybe you get lucky. In a recent paper I showed that given any triangulated space together with a Morse-Forman function on it, there exists a genuine flow on this space with the following properties

1. The critical points are the exactly barycenters of all the faces.
2. The Conley index of the barycenter of a critical face is a sphere of the same dimension as the face.
3. The Conley index of the barycenter of a non-critical face is homotopically trivial

The flows are easily described and visualized and they belong to a class of flows I called tame. There examples of tame flows beyond the class generated by a Morse-Forman function and they are as well behaved. To get such flows on a triagulated space it suffices to label the vertices. The flow runs from the higher label to the lower label. We thought that using the Morse function for labeling would do the trick.

Given a Morse function on a manifold one could envisage constructing such a flow by choosing a sufficiently fine triangulation and trying to guess how the gradient flow would behave on each face. Things get hairy near the critical points. However I still believe this approach has legs.

As for associating a Morse-Forman function to a Morse function there are some recent results in this direction arXiv: 1010.0548 or arXiv: 0803.2616.

Remark. This in reply to your new question. There are as many stationary points as faces. The barycenter of each face is a stationary point. However, only few of these stationary points have nontrivial Conley index. I will refer to the other stationary points as ghosts, and the level sets containing ghosts as ghost levels.

The Conley index of a barycenter of a face is nontrivial if and only if that face is critical in the sense of Forman. In my paper mentioned in my answer I describe a general process (flip-flop) that generalizes the process of handle attachment. Crossing a ghost level does not change the homotopy type of the sublevel set.

The process of crossing of ghost level does not even change the simple homotopy type. If $\lbrace f=c\rbrace$ is a ghost level, then $\lbrace f\leq c+\varepsilon\rbrace$ is simple homotopic to the space obtained from $\lbrace f\leq c-\varepsilon\rbrace$ by attaching a cone over a contractibe subset $B\subset \lbrace f=c-\varepsilon\rbrace$.

The space $\lbrace f\leq c+\varepsilon \rbrace$ is $PL$ homeomorphic to the space obtained from $\lbrace f\leq c-\varepsilon\rbrace$ by a flip-flop which is essentially a blowdown followed by a blowup along clearly defined loci.

World seems to be small. I have tried the approach you outlined with a former student of mine and we got stuck in the simplest case, the height function on a torus.The saddle point was (and is) a serious headache.)

Let me mention our strategy maybe you get lucky. In a recent paper I showed that given any triangulated space together with Morse-Forman function on it, there exists a genuine flow on this space with the following properties

1. The critical points are the exactly barycenters of all the faces.
2. The Conley index of the barycenter of a critical face is a sphere of the same dimension as the face.
3. The Conley index of the barycenter of a non-critical face is homotopically trivial

The flows are easily described and visualized and they belong to a class of flows I called tame. There examples of tame flows beyond the class generated by a Morse-Forman function and they are as well behaved. To get such flows on a triagulated space it suffices to label the vertices. The flow runs from the higher label to the lower label. We thought that using the Morse function for labeling would do the trick.

Given a Morse function on a manifold one could envisage constructing such a flow by choosing a sufficiently fine triangulation and trying to guess how the gradient flow would behave on each face. Things get hairy near the critical points. However I still believe this approach has legs.

As for associating a Morse-Forman function to a Morse function there are some recent results in this direction arXiv: 1010.0548 or arXiv: 0803.2616.

World seems to be small. I have tried the approach you outlined with a former student of mine and we got stuck in the simplest case, the height function on a torus.The saddle point was (and is) a serious headache.

Let me mention our strategy maybe you get lucky. In a recent paper I showed that given any triangulated space together with a Morse-Forman function on it, there exists a genuine flow on this space with the following properties

1. The critical points are the exactly barycenters of all the faces.
2. The Conley index of the barycenter of a critical face is a sphere of the same dimension as the face.
3. The Conley index of the barycenter of a non-critical face is homotopically trivial

The flows are easily described and visualized and they belong to a class of flows I called tame. There examples of tame flows beyond the class generated by a Morse-Forman function and they are as well behaved. To get such flows on a triagulated space it suffices to label the vertices. The flow runs from the higher label to the lower label. We thought that using the Morse function for labeling would do the trick.

Given a Morse function on a manifold one could envisage constructing such a flow by choosing a sufficiently fine triangulation and trying to guess how the gradient flow would behave on each face. Things get hairy near the critical points. However I still believe this approach has legs.

As for associating a Morse-Forman function to a Morse function there are some recent results in this direction arXiv: 1010.0548 or arXiv: 0803.2616.