On homologically locally connected spaces

We provide some properties and characterizations of homologically $U{V}^n$-maps and $l{c}_G^n$-spaces. We show that there is a parallel between recently introduced by Cauty algebraic $ANR$’s and homologically $l{c}_G^n$-metric spaces, and this parallel is similar to the parallel between ordinary $ANR$’s and $L{C}^n$-metric spaces. We also show that there is a similarity between the properties of $L{C}^n$-spaces and $l{c}_G^n$-spaces. Some open questions are raised.