On smooth functions with two critical values

We prove that every smooth closed connected manifold admits a smooth real-valued function with only two critical values such that the set of minima (or maxima) can be arbitrarily prescribed, as soon as this set is a finite subcomplex of the manifold (we call a function of this type a Reeb function). In analogy with Reeb’s Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension 3, we give a characterization of manifolds having a Heegaard splitting of genus g in terms of the existence of certain Reeb functions. Similar results are proved in dimension \(n\ge 5\).