Local homological properties and cyclicity of homogeneous ANR compacta

TL;DR

This paper investigates local homological properties of homogeneous ANR compacta, confirming conjectures about their cyclicity and separation properties, and establishing conditions under which such spaces are points.

Contribution

It proves that homogeneous ANR compacta of dimension n are cyclic in dimension n and do not have closed subsets that separate the space, advancing the Bing-Borsuk conjecture.

Findings

Homogeneous ANR compacta have local bases with properties similar to Euclidean balls.

Such spaces are cyclic in their top dimension.

No non-empty closed subset of these spaces separates them.

Abstract

In accordance with the Bing-Borsuk conjecture \cite{bb}, we show that if $X$ is an $n$-dimensional homogeneous metric $ANR$ compactum and $x\in X$, then there is a local basis at x consisting of connected open sets U such that the homological properties of \bar U and bdU are similar to the properties of the closed ball B^n in R^n 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).