Smooth functions with simple structures on 3-dimensional closed manifolds with prescribed Reeb graphs and preimages of regular values
Naoki Kitazawa · arXiv (Cornell University) · 2021
The present paper gives a new answer to so-called realization problems of graphs as Reeb graphs of smooth functions with prescribed preimages of regular values having nice structures. The Reeb space of a smooth function is the quotient space of the manifold of the domain induced from the following equivalence relation; two points in the manifold are defined to be equivalent if and only if they are points of a same connected component of a same preimage. Reeb spaces are in considerable cases graphs (Reeb graphs). These topological objects with preimages represent the manifolds of the domains well and important tools in geometry. Recently they play important roles in applications of mathematics such as visualizations. Realization problems ask us whether we can construct smooth functions with prescribed Reeb graphs and preimages of regular values. Especially, studies on construction respecting preimages are essentially started by the author.