Fine homotopy equivalences from s onto complete separable AR's are constructed that are analogs of certain cell-like maps defined on Euclidean space. In particular, (i) there is a fine homotopy equivalence / from s onto a complete separable ARX such that the collection of nondegenerate values N f of / is a singleton whose pre-image under / is a 1-dimensional AR widely embedded in s, and (ii) there is a fine homotopy equivalence g from s onto a complete separable AR Y such that N g is a Cantor

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... t and every nondegenerate fiber of g is a tame Z-set in s. Neither X nor Y is homeomorphic to s but both become homeomorphic to s upon multiplication by a certain complete 1-dimensional AR.