Locally $n$-connected compacta and $UV^n$-maps

TL;DR

This paper develops a framework to relate properties of metrizable ANR-spaces to locally n-connected spaces, providing spectral characterizations of ALC^n and cell-like compacta via inverse systems and UV^n-maps.

Contribution

It introduces a machinery for transferring properties from ANR to LC^n spaces and characterizes ALC^n and cell-like compacta through inverse systems with UV^n-maps.

Findings

Complete metrizable spaces with ALC^n, LC^n, and WLC^n properties coincide.

Spectral characterizations of ALC^n and cell-like compacta via inverse systems.

Identification of UV^n-maps as key components in the structure of these spaces.

Abstract

We provide a machinery for transferring some properties of metrizable $ANR$-spaces to metrizable $LC^n$-spaces. As a result, we show that for complete metrizable spaces the properties $ALC^n$, $LC^n$ and $WLC^n$ coincide to each other. We also provide the following spectral characterizations of $ALC^n$ and cell-like compacta: A compactum $X$ is $ALC^n$ if and only if $X$ is the limit space of a $\sigma$-complete inverse system $S=\{X_\alpha, p^{\beta}_\alpha, \alpha<\beta<\tau\}$ consisting of compact metrizable $LC^n$-spaces $X_\alpha$ such that all bonding projections $p^{\beta}_\alpha$, as a well all limit projections $p_\alpha$, are $UV^n$-maps. A compactum $X$ is a cell-like (resp., $UV^n$) space if and only if $X$ is the limit space of a $\sigma$-complete inverse system consisting of cell-like (resp., $UV^n$) metric compacta.