Branched surfaces homeomorphic to Reeb spaces of smooth maps of a class
Naoki Kitazawa · arXiv (Cornell University) · 2021
Classes of branched surfaces extend the classes of surfaces satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are negative are regarded as branched surfaces, the classes of which are defined in suitable manners. They are defined as the spaces of all connected components of preimages in general situations, which are natural quotient spaces of the domains. They are important topological objects in differential topology and also play important roles in applied or applications of mathematics such as projections in data analysis, visualizations, and so on. The present paper concerns global topologies of these branched surfaces and explicit construction via fundamental operations. The author has believed that global topologies of Reeb spaces of suitable smooth maps are important to understand and obtained several Reeb spaces with information on global algebraic topological or differential topological properties. The present study concentrates on $2$-dimensional cases. This is a new work mainly motivated by systematic theory of so-called Morse functions and its higher dimensional variants. Besides mathematics including the singularity theory of differentiable maps and its applications to (differential) topology, this must be meaningful in applied or applications11 of mathematics as before.