On Legendrian cobordisms and generating functions

Abstract

This note concerns Legendrian cobordisms in one-jet spaces of functions, in the sense of Arnol'd [2] -consisting of big Legendrian submanifolds between two smaller ones. We are interested in such cobordisms which fit with generating functions, and wonder which structures and obstructions come with this notion. As a central result, we show that the classes of Legendrian concordances with respect to the generating function equipment can be given a group structure. To this construction we add one of a homotopy with respect to generating functions.

Augmentations and immersed Lagrangian fillings

“…Interestingly, in $$J^1S^1$$ it is often the case that cobordant Legendrians may become non‐cobordant once they are equipped with augmentations. We note that for long Legendrian knots in $$\mathbb {R}^3$$ a concordance group of similar spirit, incorporating quadratic at infinity generating families as additional equipment rather than augmentations, is considered recently in the work of Limouzineau [26].…”