This is in some sense a follow-up to my question on submersions.
Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$. Assume that the codimension $k = n-m$ is negative.
I am interested in the topology of the point fibers $f^{-1}(y)\subseteq M$ for $y\in N$.
Sard's Theorem implies that for almost all $y\in N$ the fiber $f^{-1}(y)$ is a closed submanifold of $M$ of dimension $-k$. If $y$ is not a regular value, then $f^{-1}(y)$ is a priori just a compact subset of $M$ which needn't be a manifold. However one might hope that (under some conditions on the map $f$ which are satisfied in a dense subset of $C^\infty(M,N)$) it is close to being a manifold. For example, it might always be a pseudomanifold, or manifold with singularities.
Can anything general be said about the spaces $f^{-1}(y)\subseteq M$ occurring as fibers of generic maps?
I am particularly interested in the dimensions of the singular fibers (where dimension has to be interpreted appropriately). In all the examples I can picture (ie critical level sets of Morse functions) the dimensions of the singular fibers seem to be $\le -k$.
In fact I would really like to ask the following precise question about homotopy types:
Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$, with $m>n$. Is every fiber $f^{-1}(y)$ dominated by a CW-complex of dimension $\le m-n$?
The genericity assumption is there because I would like to be able to approximate an arbitrary map $g\colon M\to N$ by a map with singular fibers having the above nice property.
Probably what I am asking is covered in the literature on singularity theory or surgery theory, however I couldn't find any references on this particular point. Any answers or pointers would be appreciated.
Update: In this paper of Gromov (on page 3) it is stated that if $f\colon\thinspace M\to N$ is a smooth generic map with $m+1\ge n$ then for a critical value $y\in N$ the number of singularities of $f$ on the level set $f^{-1}(y)$ is at most $n$. (A weaker claim is made on page 115 of Golubitsky and Guillemin (Lemma 1.9), namely that the same conclusion holds if $f$ is infinitesimally stable = stable).
This implies that $f^{-1}(y)$ has the structure of a smooth $(m-n)$-dimensional submanifold of $M$ away from a finite number of isolated singular points. So my questions now become:
Does this imply that $f^{-1}(y)$ has the homotopy type of a CW-complex of dimension $m-n$?
Can anyone supply a proof or a reference for the statement of Gromov quoted above?
Update II: Thanks to Tom Goodwillie, who has answered both of the questions in my update. However I'm still left wondering about the two original questions. My gut feeling is that the Hawaiian earing (or some related pathology) should not appear as the fiber of a generic smooth map, and perhaps the fibers should generically be manifolds with singularities, or something similar.
But I'd be happy to be proved wrong!
Best Answer
Fibers of a generic smooth map are polyhedra (so in particular CW-complexes) by the triangulation conjecture of Thom, proved by Andrei Verona [Stratified Mappings - Structure and Triangulability, Springer LNM vol. 1102]. I'm not sure that the full strength of the triangulation conjecture is really needed here - hopefully Ryan's projected answer will clarify this. A more general triangulation conjecture was proved by Masahiro Shiota [Thom’s conjecture on triangulations of maps, Topology, 39 (2000), 383–399], who also has further results on this subject on the arXiv.
Verona's theorem says that any proper, topologically stable smooth map between smooth manifolds $M$ and $N$ is "triangulable", i.e. equivalent to a PL map by a topological change of coordinates in $M$ and in $N$. On the other hand the set of proper topologically stable smooth maps $M\to N$ is dense in $C^\infty(M,N)$ by the Thom-Mather theorem.
So don't worry, there are no Hawaiian earrings or other nightmares hiding in the fibers. To ensure that nothing obstructs easy sleep, we want the fibers of a generic smooth map $M^m\to N^n$ to be polyhedra of dimension $\le\max(m-n,0)$. Indeed, forgetting about the "polyhedra" part for a moment, we know the dimension estimate from multijet transversality (as sketched by Tom). We can now achieve "polyhedra" and "of dimension $\le\max\(m-n,0)$" to hold simultaneously because maps generic in two ways are still generic. (That is, the intersection of two open and dense subsets of $C^\infty(M,N)$ is open and dense. I guess this needs $M$ to be compact, and the general case follows by writing $M$ as a union of an increasing chain of compact submanifolds and applying Baire's theorem.)