Homological dimension and dimensional full-valuedness

TL;DR

This paper explores the relationships between different homological dimensions of metric compacta, establishing conditions under which these dimensions coincide and demonstrating the dimensional full-valuedness of certain two-dimensional spaces.

Contribution

It proves that homological dimensions with respect to any field coincide for homologically locally connected spaces and extends the class of spaces known to be dimensionally full-valued.

Findings

Homological dimensions coincide under certain local connectivity conditions.

Two-dimensional lc^2 metric compacta satisfy the dimension formula for products.

Conditions weaker than lc^2 still ensure dimensional full-valuedness.

Abstract

There are different definitions of homological dimension of metric compacta involving either \v{C}ech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to different groups. It is shown that all homological dimensions of a metric compactum X with respect to any field coincide provided X is homologically locally connected with respect to the singular homology up to dimension n=dim X. We also prove that any two-dimensional lc^2 metric compactum X satisfies the equality dim(X times Y)=dim X+dim Y for any metric compactum Y. This improves the well known result of Kodama that every two-dimensional ANR is dimensionally full-valued. Actually, the condition X to be lc^2 can be weaken to the existence at every point x a neighborhood V of x such that the inclusion homomorphism H_k(V;S^1)\to H_k(X;S^1)$ is trivial for all k=1,2.