LINEAR CONDITIONS ON THE NUMBER OF FACES OF MANIFOLDS WITH BOUNDARY

The Euler equation and the Dehn-Sommerville equations are known to be the only (rational) linear conditions for f-vectors (number of simplic es at various dimensions) of triangulations of spheres. We generalize this fact to arbitrary triangulations, linear triangulations of manifo lds, and polytopal triangulations of Euclidean balls. We prove that fo r closed manifolds, the Euler equation and the Dehn-Sommerville equati ons remain the only linear conditions. We also prove that for manifold s with nonempty boundary, the Euler equation is the only Linear condit ion. These results are proved not only over Q. but also over B and Z/k Z. (C) 1997 Academic Press.