Orientations in Legendrian Contact Homology and Exact Lagrangian Immersions

Ekholm, Tobias; Etnyre, John; Sullivan, Michael G.

Orientations in Legendrian Contact Homology and Exact Lagrangian Immersions

Authors

Ekholm, Tobias

Etnyre, John

Sullivan, Michael G.

Description

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.
60 pages, 4 figures