In this paper we prove the Upper Bound Conjecture (UBC) for
some classes of
(simplicial) homology manifolds: we show that the
UBC holds for all odd-dimensional homology manifolds and for all
-dimensional homology manifolds such that

and

where
are reduced Betti numbers of .
(This condition is satisfied by -dimensional homology manifolds with Euler c
haracteristic
when is even or
when
is odd,
and for those having vanishing middle homology.)
We prove an analog of the UBC for all other even-dimensional homology manifolds.

Kühnel conjectured that for every -dimensional combinatorial manifold with vertices, We prove this conjecture for all -dimensional homology manifolds with vertices, where or We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.