On valences of polyhedra

Abstract

The paper deals with the problem of realizing polyhedral maps by polyhedra. Here, a polyhedral map is a two-dimensional cell-complex whose underlying point set is a closed topological manifold in some finite dimensional real space R d and a polyhedron is a polyhedral map with the property that the two-dimensional cells are convex polygons. A polyhedron realizes a polyhedral map if the corresponding cell complexes are isomorphic. The central problem is to characterize those polyhedral maps which can be realized by polyhedra. The present paper gives necessary combinatorial conditions and states various unsolved problems.