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 transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map $\mathfrak{m}_2$ by a count of rigid pseudo-holomorphic 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 $\mathfrak{m}_d$ such that the family $(\mathfrak{m}_d)_{d\geq1}$ are $A_\infty$-maps. Finally, we extend the Ekholm-Seidel isomorphism to an $A_\infty$-morphism, giving in particular that it is a ring isomorphism.
Comment: 70 pages, 38 figures. Accepted version. Quite a lot of modifications: more rigorous statment of the results, remark 11 turned into a corollary with proof, addition of an example of computation (section 6), modification of notation for some moduli spaces, correction of the definition of (unfinished) pseudo-holomorphic buildings and equivalence relation, section 7 partly rewritten