Singular Vertices and the Triangulation Space of the D-sphere

By a sequence of numerical experiments we demonstrate that generic triangulations of the $D-$sphere for $D>3$ contain one {\it singular} $(D-3)-$simplex. The mean number of elementary $D-$simplices sharing this simplex increases with the volume of the triangulation according to a simple power law. The lower dimension subsimplices associated with this $(D-3)-$simplex also show a singular behaviour. Possible consequences for the DT model of four-dimensional quantum gravity are discussed.