Local homological properties and cyclicity of homogeneous ANR compacta

2016/05/11 by Valov, Vesko
#55M10 #55M15 #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1605.03263

Abstract

In accordance with the Bing-Borsuk conjecture \citebb, we show that if X is an n-dimensional homogeneous metric ANR compactum and x∈ X, then there is a local basis at x consisting of connected open sets U such that the homological properties of U and bdU are similar to the properties of the closed ball Bⁿ in Rⁿ and its boundary S^n-1. We discuss also the following questions raised by Bing-Borsuk, where X is a homogeneous ANR-compactum with dim X=n: Is it true that X is cyclic in dimension n? Is it true that no non-empty closed subset of X, acyclic in dimension n-, separates X? It is shown that both questions have positive answers simultaneously, and a positive solution to each one of them implies a solution to another question of Bing-Borsuk (whether every finite-dimensional homogenous metric AR-compactum is a point).

Related

• Local cohomological properties of homogeneous ANR compacta
• Local structure of homogeneous ANR-spaces
• Homogeneous locally compact spaces
• Homological dimension and homogeneous ANR spaces
• Homogeneous metric ANR-compacta