An impediment to polyhedrality

AbstractA triangulated 3-sphere is said to be polyhedral provided it is isomorphic to the boundary of some convex 4-dimensional polytope. We show that a certain complex of six triangles when embedded in a 3-sphere in a certain way will prevent polyhedrality of the sphere. We also show that spheres with this subcomplex are not invertible, and that this complex together with an additional triangle prevents the existence of a dual diagram