h-Principle for Stratified Spaces

TL;DR

This paper extends the h-principle to stratified spaces, developing new sheaf-theoretic and bundle frameworks, and proves key theorems like holonomic approximation and a stratified Smale-Hirsch theorem.

Contribution

It introduces stratified continuous sheaves and smooth stratified bundles, generalizing the h-principle to stratified spaces in both frameworks.

Findings

Extended the h-principle to stratified spaces.

Proved a stratified holonomic approximation theorem.

Established a stratified Smale-Hirsch immersion theorem.

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.