In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical points and gluing/compactifying appropriately). Has something similar been proven for Discrete Morse Theory?



Using Discrete Morse Theory to represent hom classes
In section 11 of [this paper](http://www3.nd.edu/~lnicolae/tameflow.pdf) I show that a discrete Morse function on a simplicial complex leads to a dynamical description of Forman's theory. More precisely there is a canonical flow associated to the function such that the (open) faces of the barycentric subdivision are invariant sets. The stationary points of this flow are the barycenters of the original simplicial complex. 


The discrete Morse function defines a continuous function on topological space defined by the simplicial complex which is affine on the faces of the barycentric subdivision and is a Lyapunov function for the above flow, i.e., it decreases along the trajectories of the flow. 


The key property of the flow, as far as its connection with Forman's theory is concerned, is the following: a barycenter of a face $F$, that is, a stationary point of the flow, has nontrivial Conley index if and only if the corresponding face is critical in Forman's sense. The unstable manifold of this point is the interior of the face $F$ and the (homotopic) Conley index is the homotopy type of the pair $[F,\partial F]$. We see that this is the same as the homotopy type of a pointed sphere of the same dimension as $F$.


Using the finite volume flow technology of Harvey-Lawson one can then obtain a chain homotopy from the simplicial chain complex to the Forman complex.


