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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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Level sets of Morse functions

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 $c$ $f^{-1}(c)$ is a disjoint union of manifolds from the collection.

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