Simple polyhedra homeomorphic to Reeb spaces of stable fold maps

Abstract: Simple polyhedra are $2$-dimensional polyhedra and important objects in low-dimensional geometry and in the applications of {\it fold} maps, defined as smooth maps regarded as higher dimensional variants of Morse functions. For example, they are locally so-called {\it Reeb spaces} of (so-called stable) fold maps into the plane and represent the manifolds compactly. The Reeb space of a fold map is the space of all connected components of preimages of it and is a polyhedron whose dimension is same as that of the manifold of the target. Is a given simple polyhedron homeomorphic to the Reeb space of a suitable stable fold map? What are their global topologies like? Previously the author has challenged this for a specific case and presented fundamental construction and topological properties of the polyhedra as new results. The present paper extends some of these works and results and present results of new types.