Homogeneous ANR-spaces and Alexandroff manifolds

Abstract

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 $\check{H}^n(X;G)\neq 0$, then $X$ is an $(n,G)$-bubble. This implies that any such space $X$ has the following properties: $\check{H}^{n-1}(A;G)\neq 0$ 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 $\dim_G\leq n-2$. We also prove that if $X$ is a homogeneous metric continuum with $\check{H}^n(X;G)\neq 0$, then $\check{H}^{n-1}(C;G)\neq 0$ for any partition $C$ of $X$ such that $\dim_GC\leq n-1$. The last provides a partial answer to a question of Kallipoliti and Papasoglu \cite{kp}.