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Consider the following situation:

Let $M$ and $M'$ be two closed manifolds and suppose $f:M\to \mathbb{R}$ and $f':M'\to \mathbb{R}$ are smooth morse functions on $M$ and $M'$ respectively. We say the pair $(M,f)$ and $(M',f')$ are equivalent if there is a smooth diffeomorphism $\phi:M\to M'$ so that $f'\circ \phi=f$ and consider equivalence classes $[(M,f)]$.

My questions are:

1) Has this been studied at all? and if so

2) Is there any sort classification result for (oriented) surfaces $M$?

What I'm looking for in 2) is that any equivalence class $[(M^2, f)]$ contains some geometrically nice example...for instance an immersed surface in $\mathbb{R}^3$ with the morse function given by restriction of the $z$ coordinate function. I'm mostly interested in examples where $f$ has as few critical points as the topology of $M$ allows.

I apologize if this is trivial as it came up while I was playing around with an idea for a different problem and (like most aspects of topology) is sadly not something I'm very familiar with.

If it helps, the example I have in my head is to take a genus-2 surface in $\mathbb{R}^3$ that looks like a figure 8 and morse function given by restricting the $z$ coordinate and contrast it with the genus-2 surface in $\mathbb{R}^3$ that looks like $\infty$ (i.e. the first one on its side) with Morse function given by restricting the $z$ coordinate. These two pairs shouldn't be equivalent as the number of components of level curves of the first morse function is at most 2 while the second has level curves of the morse function with 3 components (I apologize for the crummy graphics). This is in spite of both morse functions having the same number of critical points (six). I was wondering how many other examples of morse functions there were with 6 critical points that weren't equivalent to these two.

Edit:

I felt I should add...heuristically what I am interested in is to what extent can one use a Morse function to say whether the handles (in a surface) are "next to each other" or "on top of one another".

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The paper "Counting Morse functions on the 2-sphere" by Liviu Nicolaescu (arxiv.org/abs/math/0512496) has a notion of equivalence ("geometric equivalence") similar to yours, although he allows composing with a diffeomorphism of $\mathbb{R}$ as well. The results of his paper indicate that at least with his notion of equivalence, any equivalence class of Morse functions on the 2-sphere can be represented by the height function of an embedded surface. I have no idea if the techniques of this paper generalize to higher genus surfaces, however. –  Evan Jenkins Nov 2 '10 at 4:24
    
I'm pretty sure I've heard before that in fact any Morse function can be made to be a height function (if you give yourself enough dimensions). I know someone I could ask for a reference, if you're interested. –  Aaron Mazel-Gee Nov 3 '10 at 15:40
    
Any Morse function can be made into a height function of an embedding if you give your self enough dimensions. Just pick an embedding of M into $\mathbb{R}^N$ for large N, then look at the map into $$\mathbb{R}^N \times \mathbb{R}$$ given by crossing the embedding with the Morse function. It is still an embedding. QED. –  Chris Schommer-Pries Nov 3 '10 at 17:13
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1 Answer

up vote 3 down vote accepted

You might have a look at Hatcher and Thurston's paper "A presentation for the mapping class group of a closed orientable surface". They use Morse functions on the surface to facilitate the derivation of their presentation. On page 223 and following, they discuss how to form a graph from a Morse function on a surface, which is the leaf space of the level sets of the Morse function. This graph, together with its map to $\mathbb{R}$, uniquely determines the Morse function on the surface (up to graph isomorphism preserving the function). In particular, one need only know the values of the Morse function at the vertices of the graph (corresponding to the critical points of the Morse function on the surface) since it is monotonic on each of the edges. The space of such functions will therefore be parameterized by a subset of a vector space, satisfying certain inequalities. As they indicate on p. 224, the surface with the Morse function may be recovered from the graph (with its Morse function) by embedding it in $\mathbb{R}^3$ and taking the boundary of a regular neighborhood (after a small perturbation).

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