On generalized Vⁿ-continua

2023/03/29 by Karassev, A., Krupski, P., Todorov, V. +1
#FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Primary 54F45 #Secondary 55M10

paper · doi:10.48550/arxiv.2303.16373

Abstract

The notion of a Vⁿ-continuum was introduced by Alexandroff \citeps 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 dim_G X=n is a generalized Vⁿ-space with respect to the cohomological dimension dim_G. 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 \citetv. 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 dim_G X=n.

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