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Topology and its Applications


This paper contains several shrinking theorems for decompositions of 4-dimensional manifolds. Let f : M → X be a closed, cell-like mapping of a 4-manifold M onto a metric space X and let Y be a closed subset of X such that X - Y is a 4-manifold and Y is locally simply co-connected in X. The main result states that f can be approximated by homeomorphisms if Y is a 1-dimensional ANR. The techniques of the proof also show that f can be approximated by homeomorphisms in case Y is an arbitrary 0-dimensional closed subset. Combining the two results gives the same conclusion in case Y contains a closed, 0-dimensional subset C such that Y - C is a 1-dimensional ANR. The construction in the paper also gives a proof of a taming theorem for 1-dimensional ANRs.

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