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.