#### Volume 19, issue 6 (2019)

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Nonorientable Lagrangian surfaces in rational $4$–manifolds

### Bo Dai, Chung-I Ho and Tian-Jun Li

Algebraic & Geometric Topology 19 (2019) 2837–2854
##### Abstract

We show that for any nonzero class $A$ in ${H}_{2}\left(X;{ℤ}_{2}\right)$ in a rational $4-$manifold $X\phantom{\rule{0.3em}{0ex}}$, $A$ is represented by a nonorientable embedded Lagrangian surface $L$ (for some symplectic structure) if and only if $\mathsc{P}\left(A\right)\equiv \chi \left(L\right)\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$, where $\mathsc{P}\left(A\right)$ denotes the mod $4$ valued Pontryagin square of $A$.

##### Keywords
nonorientable Lagrangian surface, Lagrangian blowup
##### Mathematical Subject Classification 2010
Primary: 53D12, 57Q35
##### Publication
Received: 25 August 2017
Revised: 16 December 2018
Accepted: 10 February 2019
Published: 20 October 2019
##### Authors
 Bo Dai School of Mathematical Sciences Peking University Beijing China Chung-I Ho Department of Mathematics National Kaohsiung Normal University Kaohsiung Taiwan Tian-Jun Li Department of Mathematics University of Minnesota Minneapolis, MN United States