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≥5.
On smooth functions with two critical values / Lerario, Antonio; Meroni, Chiara; Zuddas, Daniele. - In: REVISTA MATEMATICA COMPLUTENSE. - ISSN 1139-1138. - 37:3(2024), pp. 907-920. [10.1007/s13163-023-00484-z]
On smooth functions with two critical values
Lerario, Antonio;Meroni, Chiara;
2024-01-01
Abstract
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≥5.File | Dimensione | Formato | |
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