Orientations in Legendrian Contact Homology and Exact Lagrangian Immersions

Abstract

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 Z_2 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 Z_2 otherwise.