Singularity from Eulerian Viewpoint

This is a survey on our work generalizing the classical Dehn-Sommerville equations (analogous to Poincaré duality, see [10]) for f -vectors of triangulations of manifolds without boundary to general polyhedra. Our key observation is that the exact data needed for the generalization is the classification of points of polyhedra by the Euler characteristics of their links. From this viewpoint, a point is regular if the Euler characteristic of the link is the same as the sphere of the appropriate dimension. For any polyhedron, the extent to which the classical Dehn-Sommerville equations are still valid is measured by the size of the singularities from Eulerian viewpoint.