TITLE: Lagrangian Kleinian bottles
QUESTION [7 upvotes]: I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in ${\mathbb C}^2$ for the standard symplectic structure, mentioning that this were the only compact surface for which the problem were open.
Question 1: Where to find references for the other surfaces?
Question 2: What is the current status for Kleinian bottles? Do there exist written proofs now?

REPLY [8 votes]: While working on my thesis, I (by accident) constructed a Lagrangian Klein bottle in $(S^2\times S^2, \omega\oplus\omega)$.  The construction works in $S^2\times D^2$ as long as the area of $D^2$ is strictly greater than half the area of $S^2$.  The method is to pick a Hamiltonian on $S^2$ that rotates $S^2$ halfway around, and realize it as the monodromy around $\partial D^2$, of a symplectic fibration over $D^2$ with fiber $S^2$. There is a circle that goes through the two fixed points of the rotation, which then traces out the Klein bottle.  See McDuff's ``Geometric Variants of the Hofer Norm'' for the precise version of how to realize such a monodromy and the constraints: the area of the base is an easy way to think of it, but it's really a ratio of volume to fiber area.
As I understand it the paper by Nemirovskii called ``homology class of a Lagrangian Klein bottle'', is flawed, because it predates the accepted proofs of non-existence as described by others.  I would love to know if its main claim is true, that a Lagrangian Klein bottle in a symplectic 4-manifold must realize a nontrivial second homology class with mod 2 coefficients. This is true for the example above. (Might need to ask this as a question.)