On Legendrian cobordisms and generating functions

TL;DR

This paper explores Legendrian cobordisms in jet spaces, focusing on those compatible with generating functions, and establishes a group structure for Legendrian concordances within this framework.

Contribution

It introduces a group structure on Legendrian concordances equipped with generating functions and discusses homotopy notions in this context.

Findings

Legendrian concordances form a group under certain conditions.

Homotopy with respect to generating functions is well-defined.

Obstructions and structures related to Legendrian cobordisms are characterized.

Abstract

This note concerns Legendrian cobordisms in one-jet spaces of functions, in the sense of Arnol'd \cite{Arnold} -- 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.