Simple polyhedra homeomorphic to Reeb space of stable fold maps
Naoki Kitazawa · arXiv (Cornell University) · 2021
Simple polyhedra are $2$-dimensional polyhedra and important objects in low-dimensional geometry and in the applications of fold maps, defined as smooth maps regarded as higher dimensional variants of Morse functions, they are also important since they are locally so-called Reeb spaces of (so-called stable) fold maps into the plane and represent the manifolds compactly. The Reeb space of a fold map is defined as the space of all connected components of preimages of the map and is a polyhedron whose dimension is same as that of the manifold of the target. The present paper challenges the problem asking whether a given simple polyhedron is homeomorphic to the Reeb space of a suitable stable fold map. In a previous work, the author has challenged this for a specific case and presented fundamental construction and topological properties of the polyhedra as related new results. The present paper extend these works and results.