  In accordance with the Bing-Borsuk conjecture \cite{bb}, we show that if $X$
is an $n$-dimensional homogeneous metric $ANR$ compactum and $x\in X$, then
there is a local basis at x consisting of connected open sets U such that the
homological properties of \bar U and bdU are similar to the properties of the
closed ball B^n in R^n and its boundary S^{n-1}. We discuss also the following
questions raised by Bing-Borsuk, where X is a homogeneous ANR-compactum with
dim X=n: Is it true that $X$ is cyclic in dimension n? Is it true that no
non-empty closed subset of X, acyclic in dimension n-$, separates X? It is
shown that both questions have positive answers simultaneously, and a positive
solution to each one of them implies a solution to another question of
Bing-Borsuk (whether every finite-dimensional homogenous metric AR-compactum is
a point).
