A product structure on Generating Family Cohomology for Legendrian Submanifolds

TL;DR

This paper extends generating family cohomology for Legendrian submanifolds by defining a product structure using moduli spaces of flow trees, paving the way for an $A_$ algebra and ring structure.

Contribution

It introduces a product $$ on generating family cohomology via flow trees, establishing a foundation for an $A_$ algebra and ring structure.

Findings

Constructed moduli spaces of flow trees with smooth manifold structure.

Defined a product $$ on generating family cohomology.

Laid groundwork for an $A_$ algebra and ring structure.

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.