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 R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 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]∈H2(X,Z2) is trivial, the monodromy of pr:X→S2 must be of a special form. Finally, we show that such a monodromy cannot be realized in CP2.