Orientations in Legendrian Contact Homology and Exact Lagrangian Immersions
Joint work with Tobias Ekholm and Michael G. Sullivan,
IJM 16 (2005), no. 5, 453-532.

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.

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