Abstract: We consider Legendrian contact homology of a Legendrian surface, L, in ℝ5 (or more generally in the 1−jet space of a surface). Such a Legendrian, L, can be conveniently presented via its front projection which is a surface in ℝ3 that is immersed except for certain standard singularities.
Legendrian contact homology associates a differential graded algebra (DGA) to L. In this setting, the construction of the DGA was carried out by Etnyre−Ekholm−Sullvan with the differential defined by counting holomorphic disks in ℂ2 with boundary on the Lagrangian projection of L. Subsequent work of Ekholm, allows for the differential to be computed with a count of certain gradient flow trees replacing the holomorphic disks. This simplifies matters by replacing a PDE problem with an ODE problem. However, the required gradient flow trees are still complicated global objects, so that computing the differential in this manner for a given Legendrian is far from algorithmic.
I will discuss work in progress with Mike Sullivan. The goal is to give a computation of the DGA of L by starting with a cellular decomposition of the base projection (to ℝ2) of L that contains the projection of the singular set of L in its 1−skeleton. Generators are associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell.