Let $M$ be a smooth manifold and $K$ a triangulation of $M$, so $K$ is a regular CW-complex and in particular a simplicial complex. Assume that $M$ is compact so $K$ is finite. Let $f\colon K \to \mathbb{R}$ be a discrete Morse function (in the sense of Forman). Is is possible to define a smooth Morse function $f'\colon M \to \mathbb{R}$ with the same critical points as $f$ (and satisfying a correspondence between the indexes of the critical points)? Is it possible to do it in "an algorithmic way" (I mean that the proof is constructive)?

As far as I know, the converse was addressed by Gallais and Benedetti, am I right?

I apologize in advance if the questions are to vague or the answers are well-known. Thanks in advance for your time.

simplices(not just points of $M$)? Because then I don't know what you mean. $\endgroup$ – Chris Gerig Jun 4 at 10:45