Geometry/Topology Seminar

"Cellular computation of Legendrian contact homology in demension 2."This is joint work with Mike Sullivan. We consider a Legendrian surface, $L$, in $R^5$ (or more generally in the 1-jet space of a surface). Such a Legendrian can be conveniently presented via its front projection which is a surface in $R^3$ that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to $L$ by starting with a cellular decomposition of the base projection (to $R^2$) of $L$ that contains the projection of the singular set of $L$ in its 1-skeleton. A collection of generators is 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. Our motivation is to give a cellular computation of the Legendrian contact homology DGA of $L$. In this setting, the construction of Legendrian contact homology was carried out by Etnyre-Ekholm-Sullvan with the differential defined by counting holomorphic disks in $C^2$ with boundary on the Lagrangian projection of $L$. Equivalence of our DGA with LCH may be established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.