  Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty,
a]$ of a Morse function $f: M \to R$, where $M$ is a smooth finite-dimensional
manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$
and give conditions under which the topology of $f^{-1}(a)$ changes when
passing a critical value. We show that for a general class of functions, which
includes all exhaustive Morse function, the topology of a regular level
$f^{-1}(a)$ always changes when passing a single critical point, unless the
index of the critical point is half the dimension of the manifold $M$. When $f$
is a natural Hamiltonian on a cotangent bundle, we obtain more precise results
in terms of the topology of the configuration space. (Counter-)examples and
applications to celestial mechanics are also discussed.
