  We introduce and investigate the notion of (strong) $K^n_G$-manifolds, where
$G$ is an abelian group. One of the result related to that notion (Theorem 3.4)
implies the following partial answer to the Bing-Borsuk problem \cite{bb},
whether any partition of a homogeneous metric $ANR$-space $X$ of dimension $n$
is cyclic in dimension $n-1$: If $X$ is a homogeneous metric $ANR$ compactum
with $\check{H}^{n}(X;G)\neq 0$, then $\check{H}^{n-1}(M;G)\neq 0$ for every
set $M\subset X$, which is cutting $X$ between two disjoint open subsets of
$X$. Another implication of Theorem 3.4 (Corollary 3.6) provides an analog of
the classical result of Mazurkiewicz \cite{ma} that no region in $\mathbb R^n$
can be cut by a subset of dimension $\leq n-2$. Concerning homology manifolds,
it is shown that if $X$ is arcwise connected complete metric space which is
either a homology $n$-manifold over a group $G$ or a product of at least $n$
metric spaces, then $X$ is a Mazurkiewicz arc $n$-manifold. We also introduce a
property which guarantees that $H_k(X,X\setminus x;G)=0$ for every $x\in X$ and
$k\leq n-1$, where $X$ is a homogeneous locally compact metric $ANR$.
