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.
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