Linear conditions on the number of faces of manifolds with boundary (Q1364327)

The Dehn-Sommerville equations (which incorporate the Euler equation) are known to span the space of linear (strictly speaking, affine) relations which hold between the numbers of faces of triangulations of spheres. In this paper, the authors generalize this fact in several different ways: to arbitrary triangulations, linear triangulations of manifolds, and polytopal triangulations of balls. Among their results, they show that, for triangulations of a manifold \(M\) without boundary, the Dehn-Sommerville equations again span the space of such linear relations, whereas if \(M\) has a boundary, then the Euler relation alone holds. Moreover, all this remains true over the integers and their finite quotient rings (the latter concerns possible torsion), as well as the rationals.