How does the Framed Function Theorem simplify Cerf Theory?

Question

A handle decomposition of a manifold $M$ is a useful structure to carry around. It is induced by a Morse function $f\colon\, M\to \mathbb{R}$.

How are two handle decompositions of $M$ related? The space of Morse functions turns out not to be connected. But if one expands the space of Morse functions to include functions $g\colon\, M\to \mathbb{R}$ which are Morse at all but finitely degenerate times, at which birth-death singularities occur, then one obtains a connected space, via Cerf's Theorem. So any two Morse functions $F_{0,1}$ are connected by a 1-parameter family of generalized Morse functions $f_t$ with $t\in[0,1]$ devoid of the worst singularities, but which still have "mild" singularities of codimension 1. We have made progress. However, the global pattern of crossings and birth-death singularities as $t$ runs from $0$ to $1$ might still be quite involved. This pattern can be visualized as a curve in $I\times \mathbb{R}$ called a Cerf graphic (see e.g. Kirby-Gay). Our next goal is to simplify the Cerf graphic, to get rid of as many crossings and birth-death singularities as possible. This entails another expansion of our space of functions, this time allowing elliptic and hyperbolic umbilics, and swallowtails, which are worse sorts of singularities.

An alternative idea to embed the space of Morse functions in a connected space of functions, which does not involve swallowtails and umbilics, is to introduce framed functions. A framed function is a generalized Morse function (Morse at all but finitely many degenerate times, at which birth-death singularities occur), together with an orthonormal framing of the negative eigenspace at the Hessian of the function $f$ at each critical point of $f$. Amazingly, it turns out that the space of framed functions is connected and contractible (see e.g. this MO question). This is (one version of) the Framed Function Theorem.

It's silly, but I'm having difficulty understanding how everything now fits together. I'm interested in the context of TQFT's, in which I would like to slice a manifold with boundary into simple pieces using a height function (in particular, I'm working with compact manifolds rather than with closed manifolds). In order to do this, my Morse functions need to be especially nice, in that they should not have level surfaces crossing the boundary, because then slicing would create corners.

How does the Framed Function Theorem simplify Cerf Theory? Does it completely replace it (in the sense that I can use the Framed Function Theorem instead of Cerf's Theorem to prove the Kirby Theorem (or pretty-much anything else Cerf's Theorem is used for), for example), because I no longer need swallowtails and elliptic and hyperbolic umbilics?

I'm asking this question primarily in order to motivate myself to look more seriously at understanding framed functions. I'd like to understand how exactly they fit into the bigger story first.

Answer 1

The framed function theorem tells you that up to "contractible choice" a compact manifold admits a framed function: i.e., a function as you prescribe. Furthermore, a framed function is supposed to give you a cell structure on the manifold up to contractible choice.

Note: The framing data is there to give you an explicit coordinatization of the cells.

(Actually, Igusa only showed that the space of framed functions is $n$-connected, where $n$ is the dimension of the domain manifold, but ideas of Eliashberg are supposed to give contractibility).

This is supposed to give rise to an alternate proof of Waldhausen's celebrated theorem $$A(X) \simeq Q(X_+)\times \text{Wh}^{\text{diff}}(X).$$

One idea in the proof is roughly this: suppose for simplicity that $X =\ast$ is a point. Then there is a model (the "expansion space") for $\text{Wh}^{\text{diff}}(\ast)$ that is a moduli space of finite, contractible, based cell complexes. That is, a point in the expansion space is a point contractible finite cell complex and the topology is defined so that a perturbations are of three kinds: (1) sliding cells over each other, (2) a refinement of the partial ordering of the cells, and (3) an elementary expansion/contraction.

Now suppose that $p : E \to B$ is a smooth fiber bundle whose fibers are contractible manifolds $E_t$ (for $t\in B$; for example, a bundle of $h$-cobordisms of a disk is such a case). Then the framed function theorem implies that $E$ can be equipped with a fiberwise framed function $f: E \to \Bbb R$.

Now here is the difficult step: the framed function on each $f_t: E_t\to \Bbb R$ is supposed to give rise to a based cell structure on $E_t$ which varies continuously in $t$; these are supposed to amount to a parametrized family of cell complexes, i.e., a map $B \to \text{Wh}^{\text{diff}}(*)$.

Unfortunately, none of these ideas have appeared (there is a preprint by Igusa and Waldhausen which is supposed to do just this, but it never was released). In my Ph.D. thesis (written in 1989 under the direction of Igusa), I manage to give the details of the geometric construction of the family of cell complexes in the special case of a fiberwise framed Morse function, that is, I assumed there were no birth/death singularities in the family. The associated family of cell complexes had cell slides but no elementary expansions/collapses.