Surgery and stratified spaces

The past couple of decades has seen significant progress in the theory of stratified spaces through the application of controlled methods as well as through the applications of intersection homology. In this paper we will give a cursory introduction to this material, hopefully whetting your appetite to peruse more thorough accounts. In more detail, the contents of this paper are as follows: the first section deals with some examples of stratified spaces and describes some of the different categories that have been considered by various authors. For the purposes of this paper, we will work in either the PL category or a very natural topological category introduced by Quinn [Q4]. The next section discusses intersection homology and how it provides one with a rich collection of self dual sheaves. These can be manipulated by ideas long familiar to surgery theorists who have exploited Poincaré duality from the start. We will give a few applications of the tight connection between an important class of stratified spaces (Witt spaces), self dual sheaves, and K-theory; one last application will appear in the final section of the paper (where we deal with the classification of “supernormal” spaces with only even codimensional strata). Section three begins an independent direction, more purely geometric. We describe the local structure of topological stratified spaces in some detail, in particular explaining the teardrop neighborhood theorem ([HTWW], [H2]) and giving applications to isotopy theorems and the like. The last three sections describe the theory of surgery on stratified spaces, building on our understanding of teardrop neighborhoods, and some applications to classification problems (other applications can also be found in the survey [CW4]).