Non-unital monoidal category of contact manifolds and Legendrian correspondence

Abstract

There are two purposes of the present paper which are interrelated. The first goal is to construct the structure of a non-unital monoidal category $\mathfrak{Cont}$ of contact manifolds, not necessarily coorientable, by developing the contact topology \emph{without contact forms}. The non-unital monoidal product is the functorial contact product $\star$, called star product, introduced in \cite{oh:shelukhin-conjecture}. We prove that the product $\star$ is associative and there exist a collection $α= \{α_{X,Y,Z}\}$ of the \emph{associator} isomorphisms $α_{X,Y,Z}: X \star (Y\star Z) \cong (X \star Y) \star Z$ for $X, \, Y, \, Z \in \mathfrak{Cont}$, that satisfy the pentagon axiom, i.e., that the triples $(\mathfrak{Cont}, \star, α)$ form a nonunital monoidal category. The second goal is to develop the calculus of Legendrian correspondences, which are by definition embedded Legendrian submanifolds of the contact product $Q \star Q'$. Legendrian correspondences will play the role of 1-morphisms in the $2$-categorical structure to be equipped with $\mathfrak{Cont}$ whose two morphisms are contact instanton cohomologies $HI(R_{ab},R'_{ab})$ associated to a pair of Legendrian correspondences $R_{ab}, \, R'_{ab} \in \mathfrak{Leg}(Q_a,Q_b)$. With this future application in mind, we define the composition of Legendrian correspondences and prove that the composition of a generic pair is again embedded and hence canonically becomes a Legendrian correspondence.