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Topology and its Applications


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{hook}: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{hook}′:A→Z such that g{ring operator}f{hook}′=f{hook}. A theorem is proved which spells out technical conditions on the embedding of A into X and on the homotopy pro-groups of X under which such a lifting exists. The main corollary asserts the following: If X is UVk+1, dim A≤k, and f{hook}′:A→X is continuous, then there exist a cell-like continuum Z and maps g:Z→X and f{hook}′:A→Z such that g{ring operator}f{hook}′= f{hook}. 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.

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