Homology class of a Lagrangian Klein bottle

Question

Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H_2(M; \mathbb Z/2\mathbb Z)$. Unfortunately the paper is known to be flawed, as it would imply (as explained in the paper) that there are no Lagrangian Klein bottles in $\mathbb R^4$, and this predates the accepted proofs. I haven't seen it claimed in another reference, but I am also not sure which references to trust. So my question is: does anyone know if this theorem is true and where it is proven?

A secondary question to which I would love an answer just as much is: can anyone make more examples of Lagrangian Klein bottles in four dimensions? I have just one: a construction of Lagrangian Klein bottle in $S^2\times D^2\subset S^2\times S^2$ with some area constraints, I briefly described it in an answer to Lagrangian Kleinian bottles and then realized I should ask this as a question. This example does represent the nontrivial second homology class of $S^2$ with $\mathbb Z/2\mathbb Z$ coefficients. I would love to know of more consructions.

Answer 1

The homology class of a Lagrangian Klein bottle is non-zero in any ruled symplectic four-manifold, e.g. in $S^2\times S^2$ with a product symplectic form. This was first proved by Shevchishin (Izvestia Math., 2009, 73:4, 797-859) and then a shorter proof was given by Nemirovski (Izvestia Math., 2009, 73:4, 689-698).

Nemirovski's earlier claim (Izvestia Math., 2002, 66:1, 151-164) that this holds for all closed symplectic manifolds is known to be false. An example of a Lagrangian Klein bottle with trivial homology class in a non-ruled symplectic four-manifold can be found in Section 1.6 of Shevchishin's paper.