# Smooth Morse function from Forman's discrete Morse function

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?

• Doesn't $f$ have critical simplices (not just points of $M$)? Because then I don't know what you mean. – Chris Gerig Jun 4 at 10:45
• @Chris A critical k-dimensional simplex should correspond to (part of) the stable manifold of an index k critical point by analogy to handlebody decompositions. – Mike Miller Jun 4 at 11:16
• @MikeMiller That is exactly what it is shown in the reference I mentioned in my answer. – Liviu Nicolaescu Jun 4 at 14:44

You can do the next best think. To a Forman-Morse function $$f$$ one can associate a flow on the manifold whose stationary points are precisely the barycenters of the faces of your simplicial decomposition. The Conley index of the barycenter of a critical face has the homotopy type of a sphere of the dimension of that face. The Conley index of the barycenter of a non-critical face is homotopically trivial.
Additionally, one can construct a continuous function $$\tilde{f}$$ on the manifold that decreases along the trajectories of this flow and whose value at a barycenter is equal to the value of $$f$$ on the corresponding face. As Mike Miller correctly pointed out, a critical face is filled out by the trajectories exiting the barycenter.
• Do you know that this is impossible for smooth Morse functions, let's say for the trivial discrete Morse function $\mu(\sigma) = \dim \sigma$? That is, is there some smooth triangulation of a smooth manifold $M$ so that there is not a smooth function $f$ on $M$ so that the barycenters of the simplices are the critical points of $f$, and the unstable manifold at a critical point is the interior of the corresponding simplex? This may be a naive question, I don't know. – Mike Miller Jun 24 at 14:07