I am interested in the properties of connectedness of level sets of Morse functions. Let $M$ a compact smooth $n$-manifold, and $1\leq k<n$. Is it possible to construct $k$ Morse functions $f\_1,\cdots,f\_k:M\rightarrow [0,1]$ on $M$ such that for $\mathscr{L}^k$ almost every $t\in [0,1]^k$,
$$
\bigcap\_{j=1}^k f\_j^{-1}\{t\_j\}
$$
is a connected $(n-k)$ submanifold? (of course $\mathrm{Im} f\_j=[0,1]$) The case $k=n-1$ is of primary interest for me, and even $n=2$ would be interesting. For example, il we take a $2$-sphere, if $x\_3$ if the third coordinate in $\mathbb{R}^3$ then except in $-1$ and $1$, we have a connected curve, but if we take for example $f(x)=(x\_3-\frac{1}{2})^2$, then $f^{-1}\{t\}$ is the union of two circles, the equator if $t=\frac{1}{2}$ or the empty set, so we cannot choose arbitrary Morse functions in this construction, if it is possible.



Construction of appropriate Morse functions
For $n>1$ and $k=1$ this is possible (contrary to a comment above). Choose a Morse function that has a single minimum and a single maximum (I assume your $M$ is connected!). If $n=2$, assume also that you pass all of the index $1$ critical points simultaneously (ie each the index $1$ critical level is one point). Then the level set just above the index $1$ critical level is still connected, even in dimension $2$ (paradoxically, the hardest case for this argument). So for $n=2$ we are done.


It's probably easier to visualize this case ($k=1$) in terms of handles; I'm just saying to add the $1$-handles simultaneously in dimension $2$. Recall that passing an index $d$ critical point means that you do a surgery on a $d-1$ sphere, and this doesn't disconnect the level set unless $d=n-1$. So after passing the critical levels of index $< n-1$, you have a connected level set. On the other hand, if there's only one $n$-handle (one maximum) then for $n>2$ the level set after any $n-1$ handle is still connected.


You might be able to do the case $n=3$, $k=2$ by similar naive arguments, ie choose a Morse function on your 3-manifold where there is only one index $1$ (resp. $2$) critical level, and then look for a family of Morse functions on the family of surfaces sweeping out your 3-manifold, each with connected level sets. 


