Homology class of a Lagrangian Klein bottle

Abstract

It is shown that an embedded Lagrangian Klein bottle represents a non-trivial mod 2 homology class in a compact symplectic four-manifold $(X,\omega )$ with $c_1(X)\cdot [\omega ]>0$. (In versions 1 and 2, the last assumption was missing. A counterexample to this general claim and the first proof of the corrected result have been found by Vsevolod Shevchishin.) As a corollary one obtains that the Klein bottle does not admit a Lagrangian embedding into the standard symplectic four-space.

Version 3 - completely rewritten to correct a mistake; Version 4 - minor edits, added references; AMSLaTeX, 6 pages

References in corpus (2)

• Topological properties of real algebraic varieties: du côté de chez Rokhlin
• Lagrangian Embeddings and Holomorphic Discs in the Complement of Complex Hypersurfaces

Cited by in corpus (8)

• Round handles, logarithmic transforms, and smooth 4-manifolds
• The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow
• Lagrangian Klein bottles in R^{2n}
• Lefschetz fibrations on nonorientable 4-manifolds
• Exact Lagrangians in four-dimensional symplectisations
• Fukaya's work on Lagrangian embeddings
• Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups
• Non-orientable Lagrangian surfaces in rational 4-manifolds