On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds which may not be orientable

TL;DR

This paper extends previous work on realizing prescribed graphs as Reeb graphs of smooth functions to include non-orientable 3-manifolds, broadening the understanding of Reeb space structures.

Contribution

It generalizes the construction of smooth functions with prescribed Reeb graphs to non-orientable 3-manifolds, expanding the class of manifolds where realization results apply.

Findings

Constructed explicit smooth functions on non-orientable 3-manifolds with prescribed Reeb graphs.

Extended realization results to include non-orientable cases.

Provided new insights into the topology of Reeb spaces for non-orientable manifolds.

Abstract

The Reeb space of a smooth function is a topological and combinatoric object and fundamental and important in understanding topological and geometric properties of the manifold of the domain. It is the graph and a topological space endowed with a natural topology. This is defined as the quotient space of the manifold of the domain where the equivalence relation is as follows: two points in the manifold are equivalent if and only if they are in a same connected component of a level set or a preimage. In considerable cases they are graphs (Reeb graphs): if the function is a so-called Morse(-Bott) functions for example, then this is the graph such that a point is a vertex if and only if the corresponding connected component of the level set contains some singular points. The author previously constructed explicit smooth functions on suitable 3-dimensional connected, closed and orientable manifolds whose Reeb graphs are isomorphic to prescribed graphs and whose preimages are as prescribed types. This gives a new answer to so-called realization problems of graphs as Reeb graphs of smooth functions of suitable classes. The present paper concerns a variant in the case where the 3-dimensional manifolds may not be non-orientable extending the result before. \end{abstract}