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 f from s onto a complete separable ARX such
that the collection of nondegenerate values Nf of f is a singleton whose
pre-image under f is a 1-dimensional AR widely embedded in s, and (ii)
there is a fine homotopy equivalence g from s onto a complete separable
ARY such that Ng is a Cantor set 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.
