Alexandroff Manifolds and Homogeneous Continua

2012/08/31 by Karassev, Alexandre, Todorov, Vladimir, Valov, Vesko
#54F45 (Primary) 54F15 (Secondary) #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1208.6345

Abstract

We prove the following result announced in Todorov and Valov: Any homogeneous, metric ANR-continuum is a Vⁿ_G-continuum provided dim_GX=n≥ 1 and \checkHⁿ(X;G)≠ 0, where G is a principal ideal domain. This implies that any homogeneous n-dimensional metric ANR-continuum with \checkHⁿ(X;G)≠ 0 is a Vⁿ-continuum in the sense of Alexandroff (1957). We also prove that any finite-dimensional homogeneous metric continuum X, satisfying \checkHⁿ(X;G)≠ 0 for some group G and n≥ 1, cannot be separated by a compactum K with \checkH^n-1(K;G)=0 and dim_G K≤ 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.

Related

• Homogeneous ANR-spaces and Alexandroff manifolds
• On generalized Vⁿ-continua
• Alexandroff type manifolds and homology manifolds
• Generalized Cantor manifolds and homogeneity
• Generalized Cantor manifolds and indecomposable continua