Self-indexing Morse functions on non-compact manifolds

Question

Hi,

given a compact manifold M we can always alter a given Morse function f to a self-indexing one (i.e., one where every critical point c has $f(c) = \operatorname{index}(c)$) - a proof of this may be found in, e.g., "Lectures on the h-Cobordism Theorem".

But what about non-compact manifolds? Is it always possible to find a self-indexing Morse function on a non-compact manifold?

Or stated in a slightly another way, which is the one I actually need: Given a non-compact, connected manifold M, does there always exist a handle decomposition $M_0 \subset M_1 \subset \cdots \subset M_m = M$, where all $M_k$ are connected and $M_k$ is constructed from $M_{k-1}$ by attaching (possibly infinitely many) k-handles?

Thanks in advance.

Answer 1

Since Ryan has reawakened this question, let me add a few remarks. One way to get a handle structure on a manifold $M$ is to start with a smooth triangulation of $M$, so I will assume such a triangulation exists. A nice neighborhood of the 0-skeleton then gives a collection of 0-handles. Enlarge this to a nice neighborhood of the 1-skeleton by adding 1-handles. Then enlarge to a nice neighborhood of the 2-skeleton by adding 2-handles, and so on. To a handle structure constructed in this way it's clear one can associate a self-indexing morse function. The handle structure does not quite satisfy all the conditions imposed in the original question since $M_0$, the union of the 0-handles, is not connected, but requiring $M_0$ to be connected would force it to be just a single 0-handle, hence there could only be finitely many 1-handles attached to this 0-handle, then only finitely many 2-handles, etc., forcing $M$ to be compact. This is assuming that $M_0$ is to be obtained from the empty manifold $M_{-1}$ by attaching 0-handles. If $M_{-1}$ is not required to be empty then one could just do the stupid thing of taking $M_{-1}=M$ and attaching no handles at all.

This perhaps reduces the question to the existence of a smooth triangulation. Open sets in Euclidean space certainly have smooth triangulations, for example. If I recall correctly, one proof of the existence of smooth triangulations for closed manifolds involves embedding the manifold in a Euclidean space of high enough dimension, choosing a suitably fine triangulation of this Euclidean space and intersecting the simplices with the manifold and then somehow subdividing the resulting cells into simplices. This should work just as well for noncompact manifolds that have proper embeddings into Euclidean space.

Answer 2

On non-compact manifolds I think one must accept that proper Morse functions carry far less topological data than their analogous brethren on compact manifolds.

A silly illustration: let $N$ be a compact topologically-fascinating manifold, and consider the projection map $pr_2: N \times R \to R$. We see that $pr_2$ is a proper Morse function having no critical points on the noncompact $N \times R$. Moreover, the condition of having no critical points is surely an open condition, hence all Morse functions $f$ in a neighborhood of $pr_2$ (say, in compact-open topology on $C^\infty$) will have no critical points. In otherwords, the "passing through critical points means adding handles"-machinery is worthless (at least for these $f$ near $pr_2$). The value in having a proper Morse function $f$ with no critical points on some noncompact $M$ is to tell us exactly that $M$ splits as $N \times R$ (i.e. the normal bundle of a fibre is trivial). But I think it can tell us no more than that: the topology of the $N$ factor remains `hidden' from $f$.

Moreover, if one does not restrict to proper Morse functions, then I think one must expect even less! For illustration, just think of a closed manifold $M$ with some Morse function $f$ and let $Z$ be the set of critical points. Then $f$ restricts to a nonproper critical-point-free Morse function on the noncompact $M \setminus Z$. But examples show that $M \setminus Z$ does not split an $R$-factor as before. Moreover, I'll be damned if I see any way to leverage $f$ to yield any topological data on $M \setminus Z$.

Finally, one cannot ignore that our gradient flows (which, as we all learned from Milnor, are singlehandedly responsible for our much-valued deformation retracts) may not be globally defined in the non-compact case. This is probably the most important point.

In brief, I think Morse functions (proper or not) on non-compact manifolds are almost topologically useless. It should not be taken for granted that (i) a given Morse function has critical points, and (ii) that the given Morse function has enough critical points to sufficiently illuminate the topological structure of our non-compact manifold.