Locally n-connected compacta and UVⁿ-maps

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We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable LCⁿ-spaces. As a result, we show that for complete metrizable spaces the properties ALCⁿ, LCⁿ and WLCⁿ coincide to each other. We also provide the following spectral characterizations of ALCⁿ and cell-like compacta: A compactum X is ALCⁿ if and only if X is the limit space of a σ-complete inverse system S={Xα, pβα, α<β<τ} consisting of compact metrizable LCⁿ-spaces Xα such that all bonding projections pβα, as a well all limit projections pα, are UVⁿ-maps. A compactum X is a cell-like (resp., UVⁿ) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., UVⁿ) metric compacta.