Homological dimension and homogeneous ANR spaces

TL;DR

This paper explores the properties of homological dimension in metric compacta, establishing conditions under which these spaces are dimensionally full-valued, especially focusing on two-dimensional homogeneous ANR spaces.

Contribution

It introduces new properties of homological dimension and proves that two-dimensional lc^2 metric compacta are dimensionally full-valued, extending previous results for ANR spaces.

Findings

Two-dimensional lc^2 metric compacta are dimensionally full-valued.

Homological dimension properties help characterize dimensional full-valuedness.

Applications to homogeneous metric ANR-compacta are provided.

Abstract

The homological dimension $d_G$ of metric compacta was introduced by Alexandroff. In this paper we provide some general properties of $d_G$, mainly with an eye towards describing the dimensional full-valuedness of compact metric spaces. As a corollary of the established properties of $d_G$, we prove that any two-dimensional $lc^2$ metric compactum is dimensionally full-valued. This improves the well known result of Kodama that every two-dimensional $ANR$ is dimensionally full-valued. Applications for homogeneous metric $ANR$-compacta are also given.