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Every compact two dimensional manifold admits a Morse function such that any its regular level set is at most two circles. I am interested in a generalization of that phenomenon.

Does there exist a finite collection of compact manifolds of dimension $n$ such that any compact manifold of dimension $n+1$ admits a Morse (variant - strong Morse, that is all its critical values are pairwise distinct) function $f$ such that for any regular value $c$ the level set $f^{-1}(c)$ is a disjoint union of manifolds from the collection.

I am especially interested in the case $n=2$.

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I am not sure, but I believe that if you take the 3-manifold obtained as the mapping torus of a sufficiently non trivial surface homeomorphism, the morse functions there will have as preimage the same surface. This would make a counter-example. However, it seemed more clear when I started to write it.... –  rpotrie Jul 4 '10 at 23:27
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No such collection exists for $n=2$. This follows the construction in my paper "Small 3-manifolds of large genus". The result of the paper is that for any $g$, there are closed orientable irreducible non-Haken 3-manifolds $M$ of genus $\geq g$. It follows from techniques of Scharlemann-Thompson, based on work of Casson-Gordon, that for any Morse function on $M$, there must be a regular value level surface with a component of genus $\geq g$.

I would suspect this is also false in dimensions $n\geq 3$. In particular, I would conjecture that for any finite collection of closed $n$-manifolds, there exists a congruence arithmetic hyperbolic $n+1$-manifold $M$ such that the components of regular level sets of any Morse function on $M$ do not lie in this collection. I believe it may be possible to use a generalization of methods of Lackenby to prove this.

Addendum: I'll add some details explaining Scharlemann-Thompson's method. First some background. It is natural to consider an extension of Morse functions to manifolds with boundary, such that each boundary component must be a component of a regular level set of the function. If you cut a manifold along some component of a regular level set of a Morse function, then the restriction gives a Morse function on the resulting manifold with boundary. A Morse function on a 3-manifold has critical points of index $0,1,2,3$. Negating the Morse function exchanges critical points of index $i$ with index $3-i$. A handlebody of genus $g$ is an orientable 3-manifold which admits a Morse function with only one critical point of index $0$ and $g$ critical points of index $1$. Alternatively, it is obtained from a ball ($0$-handle) by adding $g$ 1-handles. Therefore it has a single boundary component of genus $g$. A genus 3 handlebody Negating the Morse function, we see that a handlebody is also a manifold with a Morse function with a single critical point of index $3$, and $g$ critical points of index $2$.

If one has a self-indexing Morse function $f$ of a connected 3-manifold, then the level surface $f^{-1}(1.5)$ is a Heegaard splitting, which divides the manifold into two genus $g$ handlebodies (one may always cancel excess index $0$ and $3$ critical points to assume that there is only one of each).

A natural generalization of a handlebody is a compression body, which is a 3-manifold with boundary admitting a Morse function with critical points of index $0$ and $1$ (or $2$ and $3$). A connected compression body with a self-indexed Morse function with index $0$ and $1$ critical points has a "lower boundary" which consists of the boundary components with minimal Morse value, and "upper boundary" consisting of components with maximal Morse value. The upper boundary is obtained from the lower boundary and a collection of spheres for each index 0 critical point by adding tubes for each 1-handle. The term compression body arises from the fact that the lower boundary is obtained from the upper boundary by compression (adding 2-handles), and possibly removing some spheres. For example, for a handlebody, the lower boundary is empty, and the upper boundary is a genus $g$ surface.

We now show how to obtain a decomposition of a 3-manifold into compression bodies from a Morse function, which are separated by surfaces $\Sigma_{\pm}$. For convenience, assume $f$ is generic, so that it has at most one critical point at each level, and has $n$ critical points. We may take numbers $r_0 < c_1 < r_1 < c_1 < r_2 < \cdots < r_{n-1} < c_n < r_n$, so that $r_i$ is a regular level of $f$, and $c_i$ is a critical level between $r_{i-1}$ and $r_i$. Note that $f^{-1}(r_0)=f^{-1}(r_n) =\emptyset$. Then $\Sigma_+\cup \Sigma_{-} \subset f^{-1}(\{r_0, \ldots, r_n \})$. Each complementary piece $f^{-1}([r_i,r_{i+1}])$ is a compression body with a single handle. We let $C_0$ be the union of components of $\{f^{-1}([r_{i-1},r_i])\}$ which contain an index $0, 1$ or no critical point, and we let $C_1$ be the union of the components containing an index $2$ or $3$ critical point. Therefore $C_0, C_1$ are compression bodies. The boundary of the compression body $C_0$ has two subsurfaces, the "lower boundary" $\Sigma_-$ and "upper boundary" $\Sigma_+$. Negating the Morse function on $C_1$, we see that it has the same upper boundary and lower boundary as $C_0$.

Intuitively, the surface $\Sigma_+$ is a union of components of regular level surfaces of $f$ which lie above a critical point of index $0$ or $1$, and below a critical point of index $2$ or $3$. Similarly, $\Sigma_-$ is obtained from the components of regular level surfaces of $f$ which lie above a critical point of index $2$ or $3$, and below a critical point of index $0$ or $1$. Then $\Sigma_-$ consists of components of the "thin levels" of $f$, and $\Sigma_+$ consists of the "thick levels".

If one applies this process to a self-indexing Morse function of a closed manifold, one obtains a Heegaard splitting. In general, one obtains what Scharlemann-Thompson call a generalized Heegaard splitting, which is a decomposition into two (possibly disconnected) compression bodies, so that the upper and lower boundaries of the two compression bodies are identified. For manifolds with boundary, the boundary is required to be a subset of the lower boundaries of the compression bodies.

Now comes the key observation of Scharlemann-Thompson: Suppose one has components $W$ of $C_0$ and $X$ of $C_1$ such that $W\cap X = F$ is a subsurface of $\Sigma_+$. Suppose also that there is a $1$-handle $e_1$ of $W$ such that $e_1\cap F$ is disjoint from $e_2\cap F$, where $e_2$ is a $2$-handle of $X$. Then one may reorder the handles so that $e_2$ is attached first to $W-e_1$, and then one attaches $e_1$ to $W-e_1\cup e_2$, and then attaches $X-e_2$. This splits up two compression bodies into four compression bodies. What happens in this process is that the components of the surfaces in the new generalized Heegaard splitting have non-increasing genus. They define a complexity which is the sum of $2 genus-1$ for each component at a given level. However, we would like to keep track of the complexity of each piece, so we just keep track of a lex-ordered sorted tuple of $\{2 genus(S)-1\}$ for each component $S\subset \Sigma_+$. We notice that this is non-increasing under this operation of exchanging orders of handles: it may increase the number of components, but decreases the lex-ordering of the complexity function. I won't go through all the cases, but if $(e_1\cap F)\cup (e_2\cap F)$ is non-separating in $F$ and intersects the same component of $F$ such that $genus(F)=g$, then $F$ is replaced with two surfaces of genus $g-1$. This has the effect of decreasing the lex order of the sorted tuple of complexities.

From the perspective of Morse functions, we've chosen a Morse function $f'$ on $W\cup X$ so that $e_1$ and $e_2$ correspond to critical points of $f'$ and so that the ascending manifold of $e_1$ is disjoint from the descending manifold of $e_2$. Then a standard operation in Morse theory allows one to move the level of the index 2 critical point below the level of the index 1 critical point. The advantage of considering compression bodies here is that one doesn't have to keep track of all Morse functions, but can consider all handles at once. Scharlemann and Thompson's philosophy is that one should "push down" index 2 critical points as low as possible, up to handle slides, and one should "push up" index 1 critical points as high as possible.

We also need to consider what happens if a $2$-handle is attached to a $0$-handle (or a $3$-handle is attached to a $1$-handle). If we assume that $M$ is irreducible, then attaching a 2-handle to a ball ($0$-handle) gives a manifold with two $S^2$ boundary components. Since $M$ is irreducible, one of these spheres bounds a ball to one side. Thus, we may replace that ball with a single $0$-handle. In the process, we may have gotten rid of many components of $\Sigma_{\pm}$ lying inside the ball, so this does not increase complexity (and in particular, does not increase the genus of any component of $\Sigma_+$), and decreases the total number of critical points.

By performing these operations as many times as possible, we may assume that we have reached a generalized Heegaard splitting of an orientable irreducible non-Haken 3-manifold $M$ with several properties. We don't have any $2$-handles attached to $0$-handles, or $3$-handles to $1$-handles (and therefore no component of $\Sigma_-$ is a 2-sphere). For each component $F$ of $\Sigma_+$, in the adjacent components of $C_0$ and $C_1$ above and below, we don't have any $2$-handle $e_2$ with disjoint intersection of $e_2\cap F$ with a 1-handle $e_1\cap F$ below. Such a generalized Heegaard splitting is called strongly irreducible by Casson-Gordon. What Casson and Gordon prove is that for a strongly irreducible Heegaard splitting, $\Sigma_-$ is an incompressible surface. Thus, if $M$ is irreducible and non-Haken (doesn't contain any incompressible surface), we see that $\Sigma_-$ must be empty. But this implies that $\Sigma_+$ is a Heegaard splitting of $M$. Since our operations never increase the genus of a component of $\Sigma_+$, we see that there must have been a component of a regular level set of the original Morse function which had genus $\geq g$.

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Agol -- could you please give some more details on how one deduces the existence of a genus $g$ level surface from the papers of Casson- Gordon and Scharlemann-Thompson? –  algori Jul 5 '10 at 0:18
    
@algori: I'll try to post some more details tomorrow, but it shouldn't be immediately obvious how it follows from their results. One needs a slightly different complexity than the one Scharlemann-Thompson use. –  Ian Agol Jul 5 '10 at 0:43
    
Agol, could you please (in addition to algori's question) also explain what are standard properties of Morse functions mentioned in your answer? –  Petya Jul 5 '10 at 0:47
    
Agol -- yes, it would be very interesting to see the details. At the moment this argument looks mysterious to me. –  algori Jul 5 '10 at 1:00
    
I have many questions on your answer. The first one: you definition of a Morse function on a manifold with boundary contradicts to your example: "If you cut a manifold along some component of a regular level set, then you get a Morse function on the resulting manifold with boundary." (Counter-example is standard Morse function on 2-torus and one circle from a two-component level set). Secondly - I do not understand the definition of $\Sigma_+$. Do you start from an arbitrary Morse function $f$ or you suppose that all critical values of index 1 are below critical values of index 2? –  Petya Jul 5 '10 at 20:51
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I posted a short paper to the arXiv, Group width, which answers this question at least for nonsimply connected manifolds. I think the same question for simply connected manifolds still deserves an answer.

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