Nonorientable Lagrangian surfaces in rational $4$–manifolds

Abstract

We show that for any nonzero class $A$ in $H_2\left(X;{\mathbb{Z}}_2\right)$ in a rational $4-$manifold $X$, $A$ is represented by a nonorientable embedded Lagrangian surface $L$ (for some symplectic structure) if and only if $\mathcal{P}\left(A\right)\equiv \chi \left(L\right)\left(mod4\right)$, where $\mathcal{P}\left(A\right)$ denotes the mod $4$ valued Pontryagin square of $A$.