8 Applications of Klee ’ s Dehn - Sommerville relations

We use Klee’s Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f -vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of oddand even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn-Sommerville relations and strengthen Kalai’s result on the number of their edges.