On Lagrangian embeddings of closed nonorientable $3$–manifolds

Abstract

We prove that for any compact orientable connected $3$–manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open $2$–disk removed admits a Lagrangian embedding into the standard symplectic $6$–space. Moreover, the minimal Maslov number of the Lagrangian embedding is equal to $1$.