On smooth functions with two critical values

TL;DR

This paper proves that every smooth closed manifold admits a special type of smooth function with only two critical values, called Reeb functions, and explores their topological implications across different dimensions.

Contribution

It introduces Reeb functions with prescribed critical sets and uses them to analyze manifold topology, extending classical results to higher dimensions.

Findings

Existence of Reeb functions on all smooth closed manifolds.

Characterization of 3-manifolds with specific Heegaard splittings via Reeb functions.

Extension of results to dimensions five and higher.

Abstract

We prove that every smooth closed manifold admits a smooth real-valued function with only two critical values. We call a function of this type a \emph{Reeb function}. We prove that for a Reeb function we can prescribe the set of minima (or maxima), as soon as this set is a PL subcomplex of the manifold. 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 $n\geq 5$.