Legendrian Contact Homology in P X R

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