Branched surfaces homeomorphic to Reeb spaces of simple fold maps

TL;DR

This paper explores the topology of branched surfaces related to Reeb spaces of simple fold maps, introducing new constructions and studying their properties and embeddings into 3-manifolds.

Contribution

It provides explicit constructions of maps from branched surfaces to surfaces, extending smooth immersion classes and analyzing their global topological properties.

Findings

Constructed canonical maps from branched surfaces to surfaces.

Extended the class of smooth immersions to include these maps.

Analyzed embeddability of branched surfaces into 3-manifolds.

Abstract

Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds 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. They are the spaces of all connected components of preimages and natural quotient spaces of the manifolds of the domains. They are defined for general smooth maps and important topological objects in differential topology. They also play important roles in applied or applications of mathematics such as projections in data analysis and visualizations. The present paper concerns global topologies of branched surfaces and explicit construction of canonically obtained maps from the branched surfaces into surfaces of the targets via fundamental operations. The class of these induced maps extends the class of smooth immersions of compact surfaces into surfaces with no boundaries. It is also regarded as a variant of the class of so-called generic smooth maps between these surfaces. We study so-called "geography" of such maps as a natural, important and new study and also study global topological properties of the branched surfaces such as embeddability into $3$-dimensional closed and connected manifolds.