Cell-like images and UV groups

Let X and A be compact metric spaces. The main problem studied in this paper is that of finding conditions under which a map f : A → X can be lifted to a cell-like space; i.e., conditions are sought under which there exist a cell-like continuum Z and continuous maps g : Z → X and f ′ : A → Z such that g ◦ f ′ = f . A theorem is proved which spells out technical conditions on the embedding of A into X and on the homotopy pro-proups of X under which such a lifting exists. The main corollary asserts the following: If X is UVk+1, dimA ≤ k, and f : A → X is continuous, then there exist a cell-like continuum Z and maps g : Z → X and f ′ : A → Z such that g ◦ f ′ = f . An example is constructed which shows that the hypothesis cannot be weakened to UVk. The theorem and example are both applied to the problem of calculating UVm groups.