We specify a result of Yokoi cite{yo} by proving that if $G$ is an abelian group and $X$ is a homogeneous metric $ANR$ compactum with $dim_GX=n$ and $check{H}^n(X;G) eq 0$, then $X$ is an $(n,G)$-bubble. This implies that any such space $X$ has the f
ollowing properties: $check{H}^{n-1}(A;G) eq 0$ for every closed separator $A$ of $X$, and $X$ is an Alexandroff manifold with respect to the class $D^{n-2}_G$ of all spaces of dimension $dim_Gleq n-2$. We also prove that if $X$ is a homogeneous metric continuum with $check{H}^n(X;G) eq 0$, then $check{H}^{n-1}(C;G) eq 0$ for any partition $C$ of $X$ such that $dim_GCleq n-1$. The last provides a partial answer to a question of Kallipoliti and Papasoglu cite{kp}.
If $g$ is a map from a space $X$ into $mathbb R^m$ and $z otin g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines $Pi^1subsetmathbb R^m$ containing $z$ such that $|g^{-1}(Pi^1)|geq 2$. We prove that for any $n$-dimensional metric compactum $X$ the f
unctions $gcolon Xtomathbb R^m$, where $mgeq 2n+1$, with $dim P_{2,1,m}(g,z)leq 0$ for all $z otin g(X)$ form a dense $G_delta$-subset of the function space $C(X,mathbb R^m)$. A parametric version of the above theorem is also provided.
