Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting homology (via discrete Morse theory) is the same as that of $M$?

My question is motivated by a situation where I know the critical points of $f$, but not how they are connected by gradient flow lines. So I am interested in a solution which doesn't depend on that information. I imagine something like the following: Triangulate so that every critical point of $f$ lies on a simplex of the appropriate dimension, then define a discrete vector field by [some process]. This vector field has no nontrivial closed paths [for some reason]. By [some observation], the resulting homology is isomorphic to that of $M$.

For an expert in Morse theory, does this even sound plausible? Are there "well-known" methods or results which would fill in the gaps? Even better, does such a result exist somewhere already?

UPDATE: Mainly I am interested in computing the homology of $M$ without having complete information about the gradient flow of $f$. In particular, I have a specific smooth function on $\mathbb{R}^9$ coming from some data analysis. I would like to find the homology of the region $M = $ { $ f \le C $ }. I have a strategy for finding the critical points of $f$, but determining how they are connected by flow lines seems problematic. Converting to a discrete problem would, I hope, provide a way around this.

I'm also not worried about pathologies: $f$ has finitely many critical points, and $M$ is compact.

UPDATE 2: Thanks for the comments everyone! I will now go to think some more.

8more comments