Local cohomological properties of homogeneous ANR compacta

TL;DR

This paper investigates local cohomological properties of homogeneous ANR compacta, showing their local structure resembles Euclidean balls and boundaries, and characterizes when such spaces are dimensionally full-valued.

Contribution

It establishes local cohomological similarity to Euclidean spaces for homogeneous ANR compacta and characterizes dimensional fullness via homology groups.

Findings

Local basis at points resembles Euclidean balls and spheres.

Dimensionally full-valued spaces are characterized by non-trivial homology groups.

All 3-dimensional homogeneous metric ANR compacta are dimensionally full-valued.

Abstract

In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR compactum and x\in X, then there is a local basis at x consisting of connected open sets U such that the cohomological properties of \overline U and bdU are similar to the properties of the closed ball \mathbb B^n\subset\mathbb R^n and its boundary \mathbb S^{n-1}. We also prove that a metric ANR compactum X of dimension n is dimensionally full-valued if and only if the group H_n(X,X\setminus x) is not trivial for some x\in X. This implies that every 3-dimensional homogeneous metric ANR compactum is dimensionally full-valued.