Is there a discrete Cerf theory?

Question

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condition that critical points be non-degenerate. The theory has proven quite useful in that it has allowed Morse theoretic arguments to be used in discrete settings, and it forms a part of computational topology.
My attention was recently drawn to the technique as a result of reading a preprint of Conant, Schneiderman, and Teichner. Therefore, this question might well be hopelessly naïve (it's also possible that it's open).

Is there a discrete version of Cerf Theory? Are there at least partial results in this direction? Conversely, is it known that no such theory can exist?

My motivation is that I would imagine that a discrete proof of Kirby's Theorem, among other results which use Cerf Theory, might prove quite valuable in quantum topology (I'd love such a result at my fingertips!). I know that this doesn't (yet?) exist, or I would surely have heard about it. Additionally, a topological "machine" can only be used by a computer if it requires only finite information.

Answer 1

I realize that I am several months late to the Cerf theory party, but this paper of Chari and Joswig might be of interest to the original poster and certainly deserves a mention in the context of this question. At the very least, their construction provides various interesting topological avenues of investigating the relationship between two discrete Morse functions on the same complex.

Start with a simplicial complex $\Delta$ and let $M(\Delta)$ be the set of all possible discrete Morse functions $\mu:\Delta \to \mathbb{R}$. Note that two such functions $\mu$ and $\mu'$ are considered equivalent if they induce the same discrete vector field. We can assume without loss of generality that $M(\Delta)$ has been quotiented by this obvious equivalence relation: $\mu \sim \mu'$ if and only if $\mu(\sigma) > \mu(\tau) \leftrightarrow \mu'(\sigma) > \mu'(\tau)$ for each facet relation $\sigma \prec \tau$ in $\Delta$. Thus, we may as well assume that $M(\Delta)$ is simply the collection of acyclic partial matchings on $\Delta$.

It turns out that $M(\Delta)$ itself can be canonically endowed with the structure of a simplicial complex. Each facet relation $\sigma \prec \tau$ of $\Delta$ is a vertex, and a $D$-dimensional simplex spans $D+1$ facet relations $\sigma_d \prec \tau_d$ if and only if pairing $\sigma_d$ to $\tau_d$ for each $1 \leq d \leq D+1$ creates an acyclic matching on $\Delta$.

The reason that $M(\Delta)$ is interesting with regards to your question, is that each simplex corresponds to a unique acyclic matching on $\Delta$. So, among other things you can try, just impose a Morse function $f:M(\Delta) \to \mathbb{R}$ on $M(\Delta)$ itself so that the two acyclic matchings you wish to compare are critical cells. Now each gradient path between these critical cells corresponds to a "deformation" from one to the other in the set of acyclic matchings on $\Delta$.

Answer 2

The following simple statement can be made about straight-line homotopies of discrete Morse functions.

A discrete Morse function $f$ with gradient vector field $V_f$ is called flat if $f(\sigma) = f(\tau)$ whenever $(\sigma,\tau) \in V_f$. (For any discrete Morse function there is an equivalent flat function, i.e., having the same sublevel complexes). This definition is also due to Forman (Witten–Morse theory for cell complexes).

Now let $f$ and $g$ be two flat Morse functions with gradient vector fields $V_f$ and $V_g$, respectively. Then $f_t=(1-t)f+tg$ is a flat Morse function with gradient vector field $V=V_f\cap V_g$ for every $t$ with $0\leq t\leq 1$. The proof is straightforward from the definitions.

This means, intuitively, that in the described scenario all changes in the critical set of $f_t$ happen only at $t=0$ and $t=1$. This is due to the use of flat Morse functions. For general discrete Morse functions $f$ and $g$, the function $f_t$ is not always a discrete Morse function, and this can happen not only at discrete values of $t$ but on whole intervals.