Splitting manifold approximate fibrations
Manifold approximate fibrations and their mapping cylinders play the role in the theory of topological manifolds that fiber bundles and tubular neighborhoods play in the theory of smooth manifolds. Manifold approximate fibrations are even more prominent in the study of generalized manifolds (ANR homology manifolds) and manifold (topologically) stratified spaces. This paper establishes important splitting properties of manifold approximate fibrations for generalized manifolds and manifold stratified spaces. However, the results are new and interesting for topological manifolds and their submanifolds. The results have already proved useful for transversality theorems for submanifolds of generalized manifolds [see \textit{J. Bryant}, Topology Appl. 94, 7-12 (1999; Zbl 0929.57012)]. For an example of the type of result proved here, let \(X\) be a closed generalized \(n\)-manifold, \(n\geq 5\), satisfying the disjoint disks property, and tamely embedded (codimension \(\geq 3\) and \(1\)-LCC) in a topological manifold \(M\). Suppose further that \(Y\) is a compact tame submanifold (topological or generalized) of \(X\). Then \(X\) has a mapping cylinder neighborhood in \(M\) such that the map \(p:W\to X\) giving the mapping cylinder is a manifold approximate fibration and \(p|_{p^{-1}(y)}:p^{-1}(Y)\to Y\) is also a manifold approximate fibration (that \(p\) is a manifold approximate fibration is trivial; the important new point is to be able to arrange for \(p\) to be a manifold approximate fibration over \(Y\)).
scientific article; zbMATH DE number 1801891