In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR continuum and x ∈ X, then there is a local basis at x consisting of connected open sets U such that the cohomological properties of Ū and bd U are similar to the properties of the closed ball ⁿ ⊂ ℝⁿ and its boundary . We also prove that a metric ANR compactum X of dimension n is dimensionally full-valued if and only if the group Hₙ(X,X∖x;ℤ) is not trivial for some x ∈ X. This implies that every 3-dimensional homogeneous metric ANR compactum is dimensionally full-valued.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm93-12-2015,
author = {V. Valov},
title = {Local cohomological properties of homogeneous ANR compacta},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {257-270},
zbl = {06575011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm93-12-2015}
}
V. Valov. Local cohomological properties of homogeneous ANR compacta. Fundamenta Mathematicae, Tome 233 (2016) pp. 257-270. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm93-12-2015/