A 4-dimensional 1-LCC Shrinking Theorem

AbstractThis 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