Alexandroff Manifolds and Homogeneous Continua

Abstract

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V_G^n$-continuum provided ${\dim }_{G}X=n\ge 1$ and ${\check{H}}^n(X;G)\ne 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with ${\check{H}}^n(X;G)\ne 0$ is a $V^n$-continuum in the sense of Alexandroff (1957). We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying ${\check{H}}^n(X;G)\ne 0$ for some group $G$ and $n\ge 1$, cannot be separated by a compactum $K$ with ${\check{H}}^{n-1}(K;G)=0$ and ${\dim }_{G}K\le n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu (2007) whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.