Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle $K$ in $\mathbb{R}^4$ and $\mathbb{C}\mathbb{P}^2$. We exploit the existence of a special embedding of $K$ in a symplectic Lefschetz pencil $\operatorname{pr}\colon X \to S^2$ and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results obtained enable us to show that in the case when the class $[K]\in\mathsf{H}_2(X,\mathbb{Z}_2)$ is trivial, the monodromy of $\operatorname{pr}\colon X\to S^2$ must be of a special form. Finally, we show that such a monodromy cannot be realized in $\mathbb{C}\mathbb{P}^2$.