On Reeb graphs induced from smooth functions on $3$-dimensional closed manifolds with finitely many singular values II
Naoki Kitazawa · arXiv (Cornell University) · 2021
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 domain where the equivalence relation is as follows: two points in the domain is 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 and so on, 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 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 an extension in the case where the $3$-dimensional manifolds may not be non-orientable.