We show that for any nonzero class A in H2(X;ℤ2) in a rational 4-manifold X, A is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if P(A) ≡ ξ(L)(mod 4), where P(A) denotes the mod 4 valued Pontryagin square of A.
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Acknowledgements Li would like to thank Banghe Li for useful discussions on Proposition 1.1. The research of Dai is partially supported by NSFC 11771232 and 11431001. The research of Ho is partially supported by MOST 105-2115-M-017-005-MY2. The research of Li is partially supported by the NSF.
- Lagrangian blowup
- Nonorientable Lagrangian surface