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 highly connected$n$-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 it 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 (say, with section) whose fibers are contractible manifolds $E_t$ (for $t\in B)$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 B$\Bbb R$.

Now here is the difficult step: the framed function on each $f_t: M_t\to E_t\to \Bbb R$ is supposed to give rise to a based cell structure on $M_t$ 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. 2 added 790 characters in body Note: The framing data is there to give you an explicit coordinatization of the cells. This is supposed to give rise to an alternate proof of Waldhausen's celebrated theorem $$A(X) \simeq Q(X_+)\times \text{Wh}^{\text{diff}}(X)$$. text{Wh}^{\text{diff}}(X).$$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. That is, a point it 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 (say, with section) 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 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 t; these are supposed to amount to a parametrized family of cell slides and elementary expansions and contractions. 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. 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. (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).