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 $ 2k$-dimensional homology manifolds $ \Delta$ such that

$\displaystyle \beta_k(\Delta)\leq \sum\{\beta_i(\Delta) \,: i\neq k-2,k,k+2$    and $\displaystyle 1\leq i\leq{2k-1} \}, $

where $ \beta_i(\Delta)$ are reduced Betti numbers of $ \Delta$. (This condition is satisfied by $ 2k$-dimensional homology manifolds with Euler c haracteristic $ \chi\leq 2$ when $ k$ is even or $ \chi\geq 2$ when $ k$ 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 $ 2k$-dimensional combinatorial manifold with $ n$ vertices, $ (-1)^k(\chi(\Delta)-2)\leq {n-k-2 \choose k+1}/ {2k+1 \choose k}.$ We prove this conjecture for all $ 2k$-dimensional homology manifolds with $ n$ vertices, where $ n \geq 4k+3$ or $ n\leq 3k+3.$ We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.