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

TL;DR

This paper proves that the Klein bottle cannot be Lagrangian embedded in complex projective plane, using symplectic topology and new insights into the structure of certain mapping class groups.

Contribution

It introduces a novel approach combining symplectic geometry and the combinatorial analysis of mapping class groups to address embedding problems.

Findings

Non-existence of Lagrangian Klein bottle in  proved

Develops new results on the structure of mapping class groups as quotients of Artin braid groups

Establishes connections between symplectic embeddings and group theory

Abstract

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.