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. (Actually, Igusa only showed that the space of framed functions is highly connected, 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 $\text{Wh}^{\text{diff}}(\ast)$ given by that is a moduli space of finite, contractible, based cell complexes. Now suppose that $p : E \to B$ is a smooth fiber bundle whose fibers are contractible manifolds $E_t$ (for $t\in B)$. Then the framed function theorem implies that $E$ can be equipped with a fiberwise framed function $f: E \to B$.
Now here is the difficult step: the framed function on each $f_t: M_t\to \Bbb R$ is supposed to give rise to a based cell structure on $M_t$ which varies continuously in $t$ where the corresponding movements in the moduli space correspond to cell slides and elementary expansions and contractions.
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).