Strongly locally homogeneous generalized continua of finite cohomological dimension

Abstract

The notion of a Vn-continuum was introduced by Alexandroff [1] as a generalization of the concept of n-manifold. In this note we consider the cohomological analogue of Vn-continuum and prove that any strongly locally homogeneous generalized continuum X with cohomological dimension dimG⁡X=n is a generalized Vn-space with respect to the cohomological dimension dimG. In particular, every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. This provides a partial answer to a question raised in [12].An analog of the Mazurkiewicz theorem that no subset of covering dimension ≤n−2 cuts any region of the Euclidean n-space is also obtained for strongly locally homogeneous generalized continua X of cohomological dimension dimG⁡X=n.