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 $\alpha =\{{\alpha }_{X,Y,Z}\}$ of the \emph{associator} isomorphisms ${\alpha }_{X,Y,Z}:X\star (Y\star Z)\cong (X\star Y)\star Z$ for $X,Y,Z\in \mathfrak{Cont}$, that satisfy the pentagon axiom, i.e., that the triples $(\mathfrak{Cont},\star ,\alpha )$ 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 $Q\star Q^{\prime }$. 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 $HI(R_{ab},R_{ab}^{\prime })$ associated to a pair of Legendrian correspondences $R_{ab},R_{ab}^{\prime }\in \mathfrak{Leg}(Q_a,Q_b)$. 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.

67 pages, 3 figures; v2) 80 pages, exposition much improved