On Legendrian cobordisms and generating functions

Abstract

This note concerns Legendrian cobordisms in one-jet spaces of functions, in the sense of Arnol’d [Lagrange and Legendre cobordisms. I, Funkt. Anal. Prilozhen. 14(3) (1980) 1–13, 96.] — 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.