A product structure on Generating Family Cohomology for Legendrian Submanifolds

Abstract

One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces $J^1M$, equipped with the standard contact structure, is through the Morse theoretic technique of generating families. This paper extends the invariant of generating family cohomology by giving it a product $\mu_2$. To define the product, moduli spaces of flow trees are constructed and shown to have the structure of a smooth manifold with corners. These spaces consist of intersecting half-infinite gradient trajectories of functions whose critical points correspond to Reeb chords of the Legendrian. This paper lays the foundation for an $A_\infty$ algebra which will show, in particular, that $\mu_2$ is associative and thus gives generating family cohomology a ring structure.