h-principle for loose Legendrian embeddings

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.