Lagrangian Klein bottles in $S^2 \times S^2$

TL;DR

This paper proves the non-existence of certain Lagrangian Klein bottles in the symplectic manifold $S^2 imes S^2$ under specific homological and area conditions, using Luttinger surgery techniques.

Contribution

It introduces a novel application of Luttinger surgery to establish non-existence results for Lagrangian Klein bottles in $S^2 imes S^2$.

Findings

No Lagrangian Klein bottles in the specified homology class under given area conditions.

The symplectic area ratio constrains the existence of certain Lagrangian submanifolds.

Luttinger surgery is effective for proving non-existence in symplectic topology.

Abstract

We use Luttinger surgery to show that there are no Lagrangian Klein bottles in $S^2\times S^2$ in the $\mathbb{Z}_2$-homology class of an $S^2$-factor if the symplectic area of that factor is at least twice that of the other.