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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".

1

# Pair consisting of a compact manifold and Morse function

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.