Algebraic & Geometric Topology

Nonorientable Lagrangian surfaces in rational $4$–manifolds

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

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We show that for any nonzero class A in H2(X;2) in a rational 4manifold X, A is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if P(A)χ(L)(mod4), where P(A) denotes the mod 4 valued Pontryagin square of A.

Article information

Algebr. Geom. Topol., Volume 19, Number 6 (2019), 2837-2854.

Received: 25 August 2017
Revised: 16 December 2018
Accepted: 10 February 2019
First available in Project Euclid: 29 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D12: Lagrangian submanifolds; Maslov index 57Q35: Embeddings and immersions

nonorientable Lagrangian surface Lagrangian blowup


Dai, Bo; Ho, Chung-I; Li, Tian-Jun. Nonorientable Lagrangian surfaces in rational $4$–manifolds. Algebr. Geom. Topol. 19 (2019), no. 6, 2837--2854. doi:10.2140/agt.2019.19.2837.

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  • M Audin, Quelques remarques sur les surfaces lagrangiennes de Givental, J. Geom. Phys. 7 (1990) 583–598
  • M Audin, F Lalonde, L Polterovich, Symplectic rigidity: Lagrangian submanifolds, from “Holomorphic curves in symplectic geometry” (M Audin, J Lafontaine, editors), Progr. Math. 117, Birkhäuser, Basel (1994) 271–321
  • P Biran, Geometry of symplectic intersections, from “Proceedings of the International Congress of Mathematicians” (T Li, editor), volume II, Higher Ed. Press, Beijing (2002) 241–255
  • A B Givental, Lagrangian imbeddings of surfaces and the open Whitney umbrella, Funktsional. Anal. i Prilozhen. 20 (1986) 35–41 In Russian; translated in Funct. Anal. Appl. 20 (1986) 197–203
  • R E Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995) 527–595
  • M Gromov, Partial differential relations, Ergeb. Math. Grenzgeb. 9, Springer (1986)
  • F Lalonde, J-C Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv. 66 (1991) 18–33
  • J A Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976) 217–224
  • T-J Li, W Wu, Lagrangian spheres, symplectic surfaces and the symplectic mapping class group, Geom. Topol. 16 (2012) 1121–1169
  • D McDuff, D Salamon, Introduction to symplectic topology, 2nd edition, Clarendon, New York (1998)
  • J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974)
  • S Y Nemirovskiĭ, The homology class of a Lagrangian Klein bottle, Izv. Ross. Akad. Nauk Ser. Mat. 73 (2009) 37–48 In Russian; translated in Izv. Math. 73 (2009) 689–698
  • L Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991) 198–210
  • A Rieser, Lagrangian blow-ups, blow-downs, and applications to real packing, J. Symplectic Geom. 12 (2014) 725–789
  • V V Shevchishin, Lagrangian embeddings of the Klein bottle and the combinatorial properties of mapping class groups, Izv. Ross. Akad. Nauk Ser. Mat. 73 (2009) 153–224 In Russian; translated in Izv. Math. 73 (2009) 797–859
  • R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17–86
  • E Thomas, A generalization of the Pontrjagin square cohomology operation, Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 266–269