Representations of Reeb spaces via simplified graphs and examples

TL;DR

This paper explores the representation of Reeb spaces, which are key topological tools, by simplified graphs, especially focusing on non-CW complex cases and providing illustrative examples.

Contribution

It advances the understanding of Reeb spaces beyond CW complexes by studying their graph representations and presenting new examples.

Findings

Reeb spaces can be represented by graphs even when not CW complexes.

The paper provides concrete examples of non-CW Reeb spaces.

It discusses the reconstruction of smooth functions from Reeb graphs.

Abstract

Reeb spaces of continuous real-valued functions on topological spaces are fundamental and strong tools in investigating the spaces. The Reeb space is the natural quotient space of the space of the domain represented by connected components of its level sets. They have appeared in theory of Morse functions in the last century and as important topological objects, they are shown to be graphs for tame functions on (compact) manifolds such as Morse(-Bott) functions and naturally generalized ones. Related general theory develops actively, recently, mainly by Gelbukh and Saeki. For nice Haudorff spaces and continuous functions there, they are "$1$-dimensional". We concentrate on Reeb spaces which are not CW complexes and study their representations by graphs and nice examples. Reconstructing nice smooth functions with given Reeb graphs is of related studies and pioneered by Sharko and followed by Masumoto, Michalak, Saeki, and so on. The author has also contributed to it.