Abstract:
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 $\mathbb{C}\mathbb{P}^2$.
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