Reeb spaces of functions being analytic on dense subsets and their graph structures

TL;DR

This paper explores the topological and combinatorial properties of Reeb spaces of real-valued functions that are analytic on dense subsets, focusing on their graph structures and vertex definitions.

Contribution

It introduces new constructions of such functions and analyzes their Reeb spaces, emphasizing their graph-like structures and vertex concepts.

Findings

Reeb spaces of these functions are often graphs.

Explicit examples of functions with analytic properties are constructed.

Natural definitions of vertices in Reeb spaces are discussed.

Abstract

Reeb spaces of real-valued functions on manifolds are the spaces of all connected components (contours) of level sets and endowed with the natural quotient topology. They have been fundamental and strong tools in investigating manifolds via smooth functions with mild critical points since the birth of fundamental theory of Morse functions in the 20th century. We are concerned with topologies and combinatorics of them. Following an explicit note on explicit Reeb spaces of explicit functions which are real analytic (on dense sets) and seem to be simplest and most fundamental, edited by the author himself. We investigate other construction of examples of such functions and their Reeb spaces. Reeb spaces are naturally graphs in considerable cases and as another work, we also discuss natural definitions of vertices for them.