A note on asymptotic behaviors and topological properties of two smooth real-valued functions and several graphs associated to them

TL;DR

This paper investigates the topological and asymptotic properties of graphs derived from two smooth, non-intersecting real-valued functions and their Reeb spaces, extending understanding of these structures beyond classical Morse functions.

Contribution

It explores the Reeb spaces of non-proper, smooth functions in the plane, providing new insights into their topological and asymptotic behaviors, especially for non-analytic functions.

Findings

Reeb spaces can be characterized for certain non-proper smooth functions.

Asymptotic behaviors of these graphs reveal new topological features.

The study extends classical Morse theory to broader classes of functions.

Abstract

This is a note on the graphs of two smooth real-valued functions in the plane with no intersection and the natural map onto the region surrounded by them with the canonical projection to the line composed, yielding its Reeb space. The Reeb space of a real-valued function on a topological space is the set of all connected components of all level sets and topologized naturally. Such spaces have been fundamental and strong tools in theory of Morse functions and its generalization and variants, since the former half of the 20th century. They are graphs for tame functions such as Morse(-Bott) functions. The author has launched and has been studying this problem since 2020s, interested in Reeb spaces of smooth or non-analytic non-proper functions. For smooth closed manifolds and nice compact spaces, topological properties and combinatorial ones on Reeb spaces have been investigated by Gelbukh, Saeki, and so on.