On smooth functions with two critical values

2022/06/14 by Lerario, Antonio, Meroni, Chiara, Zuddas, Daniele · 2 citations
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2206.06955

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 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≥ 5.

Cited by

• Smooth functions which are Morse on preimages of values not being local extrema and constructing natural functions of the class on connected sums of manifolds admitting these functions
• On manifolds homeomorphic to spheres

Related

• Reeb spaces of smooth functions on manifolds
• Reeb spaces of smooth functions on manifolds II
• Smooth functions which are Morse on preimages of values not being local extrema and constructing natural functions of the class on connected sums of manifolds admitting these functions
• Real algebraic functions on closed manifolds whose Reeb spaces are given graphs
• On Reeb graphs induced from smooth functions on closed or open manifolds