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Self-indexing Morse functions on non-compact manifolds

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?

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I don't think that the two points of view you suggest are equivalent, unless I misunderstand what you might mean by More function; I would guess you mean a smooth proper map with isolated fold singularities? – Oscar Randal-Williams Jul 12 at 10:08
Without assuming proper there always exists a regular function on every open manifold. As noted by Oscar, the two points of view are different. – Daniele Zuddas Jul 12 at 11:02
I don't need the Morse function to be proper since this would imply that there are only finitely many critical points (assuming that the Morse function is self-indexing). Sure, on every manifold does exist a Morse function, such that we get a handle decomposition $M_0 \subset M_1 \subset \cdots \subset M_\infty = M$. My question is if we can rearrange these handles such that they are glued "in the right order". – AlexE Jul 12 at 11:22

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.