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Abstract. A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × 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 R n and, more generally, invariants of self transverse immersions into 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.

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We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n + 1)-space from Z2 to Z. We demonstrate how the Z-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into C n and again including orientations strengthens the results. More precisely, we prove that the number of double points of an exact Lagrangian immersion of a closed manifold M whose associated Legendrian embedding has good DGA is at least half of the dimension of the homology of M with coefficients in an arbitrary field if M is spin and in Z2 otherwise.

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We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R 3 and a relative version of Eliashberg and Hofer's contact homology. We use this translation to transport the idea of "coherent orientations" from the contact homology world to Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's differential graded algebra invariant to an algebra over Z[t, t −1 ] with a full Z grading. JBE is partially supported by an NSF Postdoctoral Fellowship (Grant # DMS-0072853). LLN is partially supported by grants from the NSF and DOE. JMS is partially supported by an NSF Graduate Student Fellowship and an ARCS Fellowship.

We establish a long exact sequence for Legendrian submanifolds L ⊂ P × R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H * maps to linearized contact cohomology CH * , which maps to linearized contact homology CH * , which maps to singular homology. In particular, the sequence implies a duality between Ker(CH * → H * ) and CH * / Im(H * ). Furthermore, this duality is compatible with Poincaré duality in L in the following sense: the Poincaré dual of a singular class which is the image of a ∈ CH * maps to a class α ∈ CH * such that α(a) = 1.The exact sequence generalizes the duality for Legendrian knots in R 3 [26] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7].