Using Discrete Morse Theory to represent hom classes

Question

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?

Answer 1

In section 11 of this paper 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.

Answer 2

The answer to your question as stated is no.

What discrete Morse theory gives you, starting from a finite regular CW complex $X$ and a discrete Morse function $f:X \to \mathbb{R}$ (with discrete vector field $V$) is a chain complex generated by the critical cells

$$ \cdots \stackrel{\partial_{k+1}}{\longrightarrow} M_k \stackrel{\partial_k}{\longrightarrow} M_{k-1} \stackrel{\partial_{k-1}}{\longrightarrow} \cdots \stackrel{\partial_1}{\longrightarrow} M_0$$

whose homology is isomorphic to that of $X$ via a chain map $\phi:(X,d) \to (M,\partial)$ which is given by $\phi = (1 + dV + Vd)^N$ for some suitably huge $N$. Now if all the $\partial_*$ are trivial (i.e., if you have a so-called perfect Morse function), then all is well: the homology generators here are just the chains themselves, and in Section 7 of the original paper

R Forman, Morse theory for cell complexes, Adv. Math 90-145, 1998

Forman shows that $M_k$ consists precisely of the $\phi$-invariant $k$-dimensional chains in $X$. There is no direct recipe, as far as I'm aware, of generating equivalence classes of $\phi$-invariant chains which represent the same integral homology class in $X$ when the boundary operators $\partial_*$ end up being nontrivial.