Topological regular neighborhoods

TL;DR

This paper develops a comprehensive theory of topological regular neighborhoods in high-dimensional manifolds, establishing their existence, uniqueness, and applications in topology, analogous to PL regular neighborhoods.

Contribution

It introduces the full theory of topological regular neighborhoods in TOP, extending PL concepts to topological manifolds with new existence and uniqueness results.

Findings

Existence and uniqueness of topological regular neighborhoods for dim Q>7.

Application to cell-like surjections being simple homotopy equivalences.

Extension of the theory to locally tamely embedded polyhedra.

Abstract

This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. Organizers of the Workshops in Geometric Topology (http://www.uwm.edu/~craigg/workshopgtt.htm) with the support of the National Science Foundation have facilitated the preparation of electronic versions of these articles to make them publicly available. Preparation of the first "Suspensions of homology spheres" was completed in 2006. A more complete introduction to the series can be found in that article (arXiv:math/0610573). In the current paper, 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 (BdM and BdQ empty) is a closed manifold neighborhood V which is homeomorphic fixing BdV and M to the mapping cylinder of some proper surjection BdV-->M. The principal theorem asserts the existence and uniqueness of such neighborhoods, for dim Q>7. 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 polyhedral in topological manifolds.