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 ${\dim }_{G}X=n$ and ${\check{H}}^n(X;G)\ne 0$, then $X$ is an $(n,G)$-bubble. This implies that any such space $X$ has the following properties: ${\check{H}}^{n-1}(A;G)\ne 0$ for every closed separator $A$ of $X$, and $X$ is an Alexandroff manifold with respect to the class $D_G^{n-2}$ of all spaces of dimension ${\dim }_G\le n-2$. We also prove that if $X$ is a homogeneous metric continuum with ${\check{H}}^n(X;G)\ne 0$, then ${\check{H}}^{n-1}(C;G)\ne 0$ for any partition $C$ of $X$ such that ${\dim }_{G}C\le n-1$. The last provides a partial answer to a question of Kallipoliti and Papasoglu \cite{kp}.

10 pages