Legendrian contact homology in 𝑃×ℝ

Abstract

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × R P\times \mathbb {R} , where P P is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of R n \mathbb {R}^n and, more generally, invariants of self transverse immersions into R n \mathbb {R}^n up to restricted regular homotopies. When n = 3 n=3 , this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.