  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.
