Locally $n$-connected compacta and $U{V}^n$-maps

Abstract

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

16 pages