Alexandroff Manifolds and Homogeneous Continua

Abstract

AbstractWe prove the following result announced by the second and third authors: Any homogeneous, metric ANR-continuum is a -continuum provided dimGX = n ≥ 1 and , where G is a principal ideal domain. This implies that any homogeneous n-dimensional metric ANR-continuum is a Vn-continuum in the sense of Alexandroff. We also prove that any finite-dimensional cyclic in dimension n homogeneous metric continuum X, satisfying for some group G and n ≥ 1, cannot be separated by a compactum K with and dimGK ≤ n – 1. This provides a partial answer to a question of Kallipoliti–Papasoglu as to whether a two-dimensional homogeneous Peano continuum can be separated by arcs.