On homologically locally connected spaces

TL;DR

This paper explores properties and characterizations of homologically locally connected spaces and maps, drawing parallels with algebraic $ANR$'s and classical $LC^n$-spaces, and discusses open questions in the field.

Contribution

It establishes new connections between homologically $UV^n$-maps, $lc^n_G$-spaces, and algebraic $ANR$'s, highlighting similarities with classical topological concepts.

Findings

Parallel between algebraic $ANR$'s and homologically $lc^n_G$-metric spaces.

Similarity between properties of $LC^n$-spaces and $lc^n_G$-spaces.

Open questions proposed for further research.

Abstract

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.