  A proof of non-existence of Lagrangian embeddings of the Klein bottle K in
\CP^2 is given. We exploit the existence of a special embedding of K in a
symplectic Lefschetz pencil on \CP^2 and study its monodromy. As the main
technical tool, we develop the theory of mapping class groups, considered as
quotients of special Artin braid groups, and obtain some new results about
combinatorial structure of such groups.
