how to obtain a generalized Morse function out of a fiber bundle?

Question

I already asked this question in MSE but did not get any answer/comment yet.

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth function $f:E\to\mathbb{R}$ such that, when restricted to the fibers, is a generalized Morse function (i.e. a smooth function with only non degenerate critical points and birth-death singularities). For the proof of this fact he refers to "On the homotopy type of the space of generalized Morse functions (Topology, Vol 23, No2., 1984)". There he proves that if $N$ is an $n$-dimensional smooth manifold, then there is an $n$-connected map between the space of such functions on $N$ and certain infinite loop space $\Omega^{\infty}S^{\infty}(BO\wedge N_+)$.

My question is, how come does the connectivity of the latter map imply the existence of the generalized Morse function $f$?. Perhaps more generally, is there a canonical way to assign such a function to the total space of a fiber bundle?

Answer 1

It's basically this:

Associated with a manifold $N$ are two spaces, call them $F_1$ and $F_2$. The first is the space of generalized Morse functions on $N$ and the second is basically $\Omega^\infty S^\infty(BO\wedge N_+)$. These are sufficiently canonical (functorial w.r.t. diffeomorphisms) that, if you have a fiber bundle $E\to B$ with fiber $N$, you get two more fiber bundles with base $B$, having fibers $F_1$ and $F_2$ respectively. Also, there is a canonical map $F_1\to F_2$, yielding a map of bundles. The $F_2$-bundle has a canonical section.

Igusa shows that the map $F_1\to F_2$ is $n$-connected, where $n=dim(N)$. If $dim(B)\le n$ then it follows that the section of the $F_2$-bundle is homotopic to one which comes from a section of the $F_1$-bundle. A section of the $F_1$-bundle is what you want: a function on $E$ that is fiberwise generalized Morse.