We show that for any nonzero class in in a rational manifold , is represented by a nonorientable embedded Lagrangian surface (for some symplectic structure) if and only if , where denotes the mod valued Pontryagin square of .
"Nonorientable Lagrangian surfaces in rational $4$–manifolds." Algebr. Geom. Topol. 19 (6) 2837 - 2854, 2019. https://doi.org/10.2140/agt.2019.19.2837