# On Lagrangian embeddings of closed non-orientable 3-manifolds

## TL;DR

This paper demonstrates that certain 3-manifolds with boundary can be embedded as Lagrangian submanifolds in symplectic 6-space, with minimal Maslov number equal to 1, expanding understanding of Lagrangian embeddings.

## Contribution

It provides a construction of Lagrangian embeddings for specific 3-manifolds with boundary, including the Klein bottle, into symplectic 6-space, with minimal Maslov number 1.

## Key findings

- Lagrangian embeddings exist for the specified 3-manifolds.
- The minimal Maslov number of these embeddings is 1.
- The embeddings are into standard symplectic 6-space.

## 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, minimal Maslov number of the Lagrangian embedding is equal to 1.