Local cohomological properties 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 cohomological properties of \overline U and bdU 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) is not trivial for some x\in X. This implies that every 3-dimensional homogeneous metric ANR compactum is dimensionally full-valued.