1-dimensional phenomena in cell-like mappings on 3-manifolds (Q1093956)

Two 1-dimensional phenoma are studied here. The first pertains to the 3- manifold domain M of a cell-like map \(f: M\to Y\), and it gives rise to an infinite 1-skeleton \(\Sigma\) in M on which f is 1-1; if, in addition, the nondegeneracy set of f misses a dense subset of each arc in \(\Sigma\), then Y admits a natural embedding in \(M\times R^ 1\). The second involves the range: if Y is a 3-manifold except possibly at points of a 1-complex \(\Gamma\) topologically embedded in Y as a closed subset, then f can be approximated by a new cell-like map \(p: M\to Y\) whose nondegeneracy set has embedding dimension 1 and \(f\times Id: M\times R^ 1\to Y\times R^ 1\) can be approximated by homeomorphisms. As a result, then \(Y\times R^ 1\) is a 4-manifold.