A product structure on Generating Family Cohomology for Legendrian Submanifolds

Abstract

One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces $J^{1}M$, 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.

50 pages, 4 figures, minor changes

References in corpus (4)

• Generating families and Legendrian contact homology in the standard contact space
• Generating family invariants for Legendrian links of unknots
• A combinatorial DGA for Legendrian knots from generating families
• Bilinearised Legendrian contact homology and the augmentation category