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?




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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.




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



Discrete Morse function from smooth one
This is a rapidly developing area, and there are many short-cuts if all you want to do is compute the homology of sub-level sets of $f$. To answer your main question, as Liviu has already mentioned: **there is no standard way**. However, you should be able to compute the homology you desire as follows.


My impression is that you have three jobs, in chronological order:


1. Construct a simplicial complex $X\_M$ homologically faithful to $M$.
2. Construct a discrete Morse function $\mu:X\_M \to \mathbb{R}$ which approximates $f$, and
3. Compute homology of everything in sight.


First, the easiest way to build a simplicial approximation if you know your $M$ is to embed it in some suitable $\mathbb{R}^n$ and sample the hell out of it. Since you are working on data analysis, this should not be too drastic a step. Given a point sample $P$ coming from a submanifold of Euclidean space, for each radius $\epsilon$ you can construct a Cech complex of radius $\epsilon$ around $P$. Precise bounds on how many points $P$ should have and how large $\epsilon$ can be in order for the Cech complex to recover the homology of $M$ with high confidence are available in the work of Niyogi, Smale and Weinberger [here](http://people.cs.uchicago.edu/~niyogi/papersps/NiySmaWeiHom.pdf) in the case when $P$ is uniformly sampled. These bounds are in terms of the injectivity radius of the embedding of $M$ into Euclidean space, and of course once these bounds are satisfied it doesn't hurt to add your known critical points to $P$. You have your homologically faithful Cech complex $X\_M$.


Next, for 2, you can easily infer a discrete Morse function on an entire simplicial complex just from knowing its values on the vertices using the [work](http://ftp4.de.freesbie.org/pub/misc/EMIS/journals/EM/expmath/volumes/14/14.4/King.pdf) of King, Knudson and Mramor. You may be required to perturb $f$ slightly so that its restriction to $P$ is injective, but this is easy and generically true. You have $\mu$!


And finally, I have written [software](http://www.math.rutgers.edu/~vidit/perseus.html) to handle 3 if you already have a $\mu:X\_M \to \mathbb{R}$: you can input a filtered simplicial complex and compute not just homology at each sub-level set of $\mu$ but the persistent homology across all level-sets in the case of field coefficients. Meaning, instead of just knowing the homology of the subcomplexes $X\_M^c$ consisting of all simplices with $\mu$-value less than or equal to $c$, you also recover the morphism on homology groups induced by including $X\_M^c$ into $X\_M^d$ whenever $c \leq d$.


All the best with your computations.


