Local homological properties and cyclicity of homogeneous ANR-compacta

Abstract:

In accordance with the Bing-Borsuk conjecture, 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 $\overline U$ and $bd \overline U$ are similar to the properties of the closed ball $\mathbb B^n\subset \mathbb R^n$ and its boundary $\mathbb S^{n-1}$. We discuss also the following questions raised by Bing-Borsuk [Ann. of Math. (2) 81 (1965), 100–111], 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 $n-1$, separates $X$?

It is shown that both questions simultaneously have positive or negative answers, 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).