Exotic Stratifications Münster June 19, 2008

We discuss joint work with B. Hughes, S. Weinberger and B. Williams [7] on some examples of stratified spaces with two strata. We will describe the structure of neighborhoods of the lower stratum in the whole space in general and give some specific calculations when the lower stratum is a circle. Using some known K-theory calculations we produce examples where the neighborhood “improves” under crossing the stratified space with a high dimensional torus and other examples where it does not. 1. Stratified spaces A manifold stratified space in the sense of Quinn with two strata is a pair (X,B) with B and X−B manifolds together with some homotopytheoretic gluing data. I want to describe a theory of neighborhoods of B in X and work out some examples (the exotic stratifications of the title but perhaps better referred to as non-exotic). 2. Neighborhood theories Tubular neighborhood theorem for smooth embeddings is model. (1) From a smooth embedding B → W we construct a vector bundle over B of dimension k, the codimension of the embedding. (2) Vector bundles over B of dimension k are classified by maps of B into a classifying space. (3) There is a smooth embedding of B into the total space of any vector bundle. (4) There is an embedding of the total space of the vector bundle into W which is unique up to isotopy. (5) All dimension k vector bundles over B occur as a normal bundle to some codimension k embedding. Milnor [10] discovered microbundles and proved that the analogues of (2) and (5) hold for topological (and PL) embeddings. He also proved (1) and (4) hold “stably”, that is for B → W × R for some k. Kister [8] proved micro-bundles are fibre bundles. 1 2 LAURENCE R. TAYLOR Hirsch, Browder and Rourke & Sanderson produced examples showing (1) and (4) need not hold without stabilization. Rourke & Sanderson did produce a topological theory of neighborhoods, for locally-flat embeddings. There is a map from the classifying space for dimension k micro-bundles to the corresponding classifying space for neighborhoods and so the normal micro-bundle question is reduced to a lifting question. In an apparently different direction Edwards proved that locallyflat embeddings have mapping-cylinder neighborhoods. In other words there is a map f : N → B such that the mapping-cylinder of f embeds in W as a neighborhood of B. Edwards showed that the map f is a manifold approximate fibration.