A polyhedrcxl 2-muGfold is a 2-manifold th At is the union of convex polygons, called its facets, such that the intersection of any two facets is either empty, a vertex of each facet, or an edge of each facet. Polyhedral 2-manifolds may be viewed as generalizations of 3-dimensional convex polytopes. One property that convex polytopes have is that each vertex is convex, that is, there is a plane that intersects the set of facets that meet the vertex such that the intersection is the boundary of a convex polygon.
Any polyhedral 2-manifold of genus greater than or equal to 1 must have ,a nonconvex vertex. This was first mentioned in print by Altshuler [l]; however, it probably has been known before because every such manifold must have a saddle point, and a saddle point is nonconvex.
In her thesis, J. Simutis mentions the possibility that every toroidal polytopte (i.e., polyhedral 2-manifold of genus 1) has at least six nonconvex vertices [3]. In this paper we construct a toroidal polytope with only five nonconvex vertices and prove that every toroidal polytope has at least four nonconvex vertices.
It might seem that the number of nonconvex vertices that a polyhedral 2-manifold must have increases with the genus; however, we show that this is not so. We construct polyhedral 2-manifolds of every positive genus that have at most seven nonconvex vertices.
A 2-cell complex C is a collection of convex k-dimenr ional polytopes -1s k s the faces of C, s (i) the intersection of any two faces of C is a face of both faces, (ii) any face of a face of C is a face of C'.