On the smooth Whitney fibering conjecture

Abstract

We improve upon the first Thom–Mather isotopy theorem for Whitney stratified sets. In particular, for the more general Bekka stratified sets we show that there is a local foliated structure with continuously varying tangent spaces, thus proving the smooth version of the Whitney fibering conjecture. A regular wing structure is also shown to exist locally, for Bekka stratifications. The proofs involve integrating carefully chosen controlled distributions of vector fields. As an application of our main theorem, we show the density of the subset of strongly topologically stable mappings in the space of all smooth quasi-proper mappings between smooth manifolds, an improvement of a theorem of Mather.