Product structures in Floer theory for Lagrangian cobordisms

We construct a product on the Floer complex associated to a pair of Lagrangian cobordisms. More precisely, given three exact pairwise transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map $\mathfrak{m}_2$ by a count of rigid pseudoholomorphic disks with boundary on the cobordisms and having punctures asymptotic to intersection points and Reeb chords of the negative Legendrian ends of the cobordisms. More generally, to a $(d + 1)$-tuple of exact transverse Lagrangian cobordisms we associate a map md such that the family $(\mathfrak{m}_d)_{d \geq 1}$ are maps satisfying the $A_\infty$ equations. Finally, we extend the Ekholm-Seidel isomorphism to an $A_\infty$-morphism, giving in particular that it is a ring isomorphism.

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Received 4 July 2018

Accepted 24 February 2020

Published 2 February 2021