Regular neighborhoods in topological manifolds.

Abstract

Regular neighborhoods have proved to be a very useful tool in the theory of PL manifolds. In this paper we want to make a very easy construction of regular neighborhoods in the topological category. F. E. A. Johnson [6] has constructs regular neighborhoods in the topological category, but only in the case of nonintersection with the boundary. R. D. Edwards [2] has announced a very general construction of regular neighborhoods; see also [3]. The present construction has the advantage of allowing a “relative” version , (Theorem 13), in the sense that if L is a complex, K is a subcomplex, and L is locally tamely embedded in a topological manifold V , then one may find a regular neighborhood ofK in V , intersecting L in a regular neighborhood of K in L, in the usual PL sense. This is used in [10] to prove embedding theorems for topological manifolds. In [11] we have a proof that the opposite procedure is possible; namely a spine of a topological manifold. We should emphasize that the regular neighborhoods we obtain are mapping cylinder neighborhoods; i. e. if K ⊂ N where N is a regular neighborhood of K, then there is a map π : ∂N → K such that N is homeomorphic to the mapping cylinder of π (Theorem 15). Let K be a compact topological space with a given simple homotopy structure; i. e. of the homotopy type of a finite CW-complex, with the homotopy equivalence specified up to torsion.