NEIGHBORHOODS OF STRATA IN MANIFOLD STRATIFIED SPACES Supported in part by NSF Grant DMS-9971367.

Abstract

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.2000 Mathematics Subject Classification. Primary 57N80, 57N40; Secondary 55R65, 58A35.1. Introduction. The foundations of differential topology include the tubular neighborhood theorem: a smooth submanifold of a smooth manifold has a neighborhood that is the total space of a disc bundle over the submanifold. For locally flat topological submanifolds, the best result about neighborhoods (in high dimensions) is due to Edwards [3]: the submanifold has a mapping cylinder neighborhood given by a manifold approximate fibration (see also [14]).For stratified spaces, the stratifications of Whitney are considered to be the correct theory in the smooth category. For Whitney stratified spaces, the tubular neighborhood theorem of Thom [23] and Mather [16], [17] says that each stratum has a neighborhood that is the total space of a bundle over the stratum, and the fiber of the bundle is the cone on the stratified link (see Goresky and MacPherson [4] for an exposition). As is the case for submanifolds, the structure on the neighborhoods is not part of the definition, and the proof of their existence is non-trivial.In the topological category, Quinn [19] has introduced a natural stratification theory. The purpose of this paper is to establish the existence of a type of tubular neighborhood for strata in Quinn's stratified spaces, or manifold stratified spaces.