Topological regular neighborhoods

Abstract

A theory of topological regular neighborhoods is described, which represents the full analogue in TOP of piecewise linear regular neighborhoods (or block bundles) in PL. In simplest terms, a topological regular neighborhood of a manifold M locally flatly embedded in a manifold Q (∂M = ∅ = ∂Q here) is a closed manifold neighborhood V which is homeomorphic fixing ∂V ∪ M to the mapping cylinder of some proper surjection ∂V → M . The principal theorem asserts the existence and uniqueness of such neighborhoods, for dim Q ≥ 6. One application is that a cell-like surjection of cell complexes is a simple homotopy equivalence (first proved for homeomorphisms by Chapman). There is a notion of transversality for such neighborhoods, and the theory also holds for locally tamely embedded polyhedra in topological manifolds. This work is a derivative of the work of Kirby-Siebenmann; its immediate predecessor is Siebenmann's "Approximating cellular maps by homeomorphisms" Topology 11 (1972), 271-294. This version of Part I is bare in spots and short on polish, but experts will find all necessary details. Part II is only sketched. ContentsPart I.

Approximately fibering a manifold over an aspherical one

“…The theory of MAFs is a "bundle theory", see [42], which plays a prominent role in the study of topological manifolds and controlled topology. For instance, Edwards [21] proved in the early 1970s that a locally flat submanifold of a topological manifold of dimension greater than five has a mapping cylinder neighborhood, where the projection map is a MAF.…”