### Abstract:

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.