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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.

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  • $\begingroup$ Doesn't $f$ have critical simplices (not just points of $M$)? Because then I don't know what you mean. $\endgroup$
    – Chris Gerig
    Commented Jun 4, 2020 at 10:45
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    $\begingroup$ @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. $\endgroup$
    – mme
    Commented Jun 4, 2020 at 11:16
  • $\begingroup$ @MikeMiller That is exactly what it is shown in the reference I mentioned in my answer. $\endgroup$
    – Liviu Nicolaescu
    Commented Jun 4, 2020 at 14:44

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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.

For details see Chapter 11 of this paper. The faces of the barycentric subdivision of your simplicial complex are invariant sets of this flow, and on such a face the flow is depicted in Figure 2, p.16 of the above paper.

It took me a while to realize that in Morse theory the gradient flow associated to a Morse function is more important than the function itself. The function plays a sort of accounting role and the Morse condition restricts the nature of the stationary points of the gradient flows.

Remark A while ago I asked this question on MathOverflow that is related to the abundance of discrete Morse functions. They are extremely rare as opposed to the usual smooth Morse functions that are generic.

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  • $\begingroup$ 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. $\endgroup$
    – mme
    Commented Jun 24, 2020 at 14:07
  • $\begingroup$ @MikeMiller Given a triagulation the function that associates to each face its dimension is a discrete Morse function. However, if you randomly assign numbers to faces, then the probability that you get a discrete Morse function is small exponentially so as the number of faces increases. $\endgroup$
    – Liviu Nicolaescu
    Commented Jun 24, 2020 at 20:43

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