Universal stratifications and a Bertini-type theorem

A stratification of the set of critical points of a map is universal in the class of stratifications satisfying the classical Thom and Whitney-a conditions if it is the coarsest among all such stratifications. We show that a universal stratification exists if and only if the ‘canonical subbundle’ of the cotangent bundle of the source of the map (constructed via operations introduced by Glaeser) is Lagrangian. The proof relies on a new Bertini-type theorem for singular varieties proved via an intriguing use of resolution of singularities. Many examples are provided, including those of maps without universal stratifications.