Transversal homotopy theory of Whitney stratified manifolds

Abstract

We modify the theory of homotopy groups to obtain invariants of Whitney stratified spaces by considering smooth maps which are transversal to all strata, and smooth homotopies through such maps. Using this idea we obtain transversal homotopy monoids with duals for any Whitney stratified space. Just as in ordinary homotopy theory we may also define higher categorical invariants of spaces. Here instead of groupoids we obtain categories with duals. We concentrate on examples involving the sphere, stratified by a point and its complement, and complex projective space stratified in a natural way. We also suggest a definition for n-category with dual, which we call a Whitney category. This is defined as a presheaf on a certain category of Whitney stratified spaces, that resticts to a sheaf on a certain subcategory. We show in detail that this definition matches the accepted notion of n-category with duals, at least for small n. It also allows us to prove a version of the Tangle Hypothesis, due to Baez and Dolan, which states that ``The n-category of framed codimension k-tangles is equivalent to the free k-tuply monoidal n-category with duals on one object."