TITLE: Level sets of Morse functions
QUESTION [26 upvotes]: 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 $n=2$.

REPLY [14 votes]: 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.