A combinatorial condition for the existence of polyhedral 2-manifolds

Abstract

LetP denote a polyhedral 2-manifold, i.e. a 2-dimensional cell-complex inR d (d≧3) having convex facets, such that set (P) is homeomorphic to a closed 2-dimensional manifold. LetE be any subset of odd valent vertices ofP, andcE its cardinality. Then for the numbercP(E) of facets containing a vertex ofE the inequality 2cP(E)≧cE+1 is proved. This local combinatorial condition shows that several combinatorially possible types of polyhedral 2-manifolds cannot exist.