  We provide some properties and characterizations of homologically $UV^n$-maps
and $lc^n_G$-spaces. We show that there is a parallel between recently
introduced by Cauty algebraic $ANR$'s and homologically $lc^n_G$-metric spaces,
and this parallel is similar to the parallel between ordinary $ANR$'s and
$LC^n$-metric spaces. We also show that there is a similarity between the
properties of $LC^n$-spaces and $lc^n_G$-spaces. Some open questions are
raised.
