A note on Reeb spaces of some explicit real analytic functions

TL;DR

This paper explores the Reeb spaces of explicit real analytic functions, showing they can be more complex than finite graphs, and extends previous studies on the topology of level sets and real algebraic constructions.

Contribution

It presents explicit examples of real analytic functions whose Reeb spaces are not finite graphs, expanding understanding of their topological complexity.

Findings

Reeb spaces of certain real analytic functions are more complex than finite graphs.

Constructed explicit examples of real analytic functions with non-finite graph Reeb spaces.

Extended previous work on the topology of level sets and real algebraic methods.

Abstract

Reeb spaces of smooth functions are fundamental and strong tools in understanding manifolds via smooth functions with mild critical points. They are defined as the natural spaces of all connected components of level sets. They are also important objects in related studies. Realization of graphs as Reeb spaces of smooth functions of certain nice classes is of such studies. In this paper, we present Reeb spaces of explicit real analytic functions which are not finite graphs. Related problems were started by Sharko, in 2006, who has studied smooth functions with critical points represented by certain elementary polynomials, and followed by a study of Masumoto and Saeki, which is on smooth functions on closed surfaces under an extended situation, and a study of Michalak, which is on Morse functions on closed manifolds. The author has contributed to this by respecting topologies of level sets, and real algebraic construction.