Legendrian contact homology in $P \times \mathbb{R}$

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 includes 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.