  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 K\"uhnel'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 K\"uhnel's conjecture
providing upper bounds on other Betti numbers of odd- and 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.
