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Abstract

In accordance with the Bing–Borsuk conjecture, we show that if $X$ is an $n$-dimensional homogeneous metric ANR continuum and $x\in X$, then there is a local basis at $x$ consisting of connected open sets $U$ such that the cohomological properties of $\overline U$ and ${\rm bd}\,U$ are similar to the properties of the closed ball $\mathbb B^n\subset \mathbb R^n$ and its boundary $\mathbb S^{n-1}$. We also prove that a metric ANR compactum $X$ of dimension $n$ is dimensionally full-valued if and only if the group $H_n(X,X\setminus x;\mathbb Z)$ is not trivial for some $x\in X$. This implies that every $3$-dimensional homogeneous metric ANR compactum is dimensionally full-valued.