h-principle for loose Legendrian embeddings

TL;DR

This paper explains Murphy's h-principle for loose Legendrian embeddings, showing their flexible nature in contact topology and demonstrating applications like non-loose embeddings and a Gromov nonsqueezing version.

Contribution

It provides a comprehensive exposition and proof of Murphy's h-principle for loose Legendrian embeddings, highlighting their flexibility and applications.

Findings

Existence of non-loose Legendrian embeddings

Proof of Gromov's nonsqueezing theorem for loose charts

Accessible exposition of microlocal sheaf theory background

Abstract

This article provides an exposition of Emmy Murphy's work on loose Legendrian embeddings. After a brief review of the rudiments of contact topology, we state and discuss some foundational results from the theory of h-principles, providing many relevant examples from contact topology on the way. We then proceed to prove Murphy's h-principle for loose Legendrian embeddings. We also provide an accessible exposition of some background material from microlocal sheaf theory. As applications, we demonstrate the existence of non-loose Legendrian embeddings, and prove a version of Gromov's nonsqueezing theorem for loose charts.