H-Principle for Stratified Spaces

Abstract

Abstract We extend Gromov and Eliashberg–Mishachev’s $h-$principle on manifolds to stratified spaces. This is done in both the sheaf-theoretic framework of Gromov and the smooth jets framework of Eliashberg–Mishachev. The generalization involves developing 1. the notion of stratified continuous sheaves to extend Gromov’s theory, 2. the notion of smooth stratified bundles to extend Eliashberg–Mishachev’s theory. A new feature is the role played by homotopy fiber sheaves. We show, in particular, that stratumwise flexibility of stratified continuous sheaves along with flexibility of homotopy fiber sheaves furnishes the parametric $h$-principle. We extend the Eliashberg–Mishachev holonomic approximation theorem to stratified spaces. We also prove a stratified analog of the Smale–Hirsch immersion theorem.