# How to (symplectically) thread the eye of a (Lagrangian) needle

@article{Mohnke2001HowT, title={How to (symplectically) thread the eye of a (Lagrangian) needle}, author={Klaus Mohnke}, journal={arXiv: Symplectic Geometry}, year={2001} }

We show that there exists no Lagrangian embeddings of the Klein bottle into $\CC^{2}$. Using the same techniques we also give a new proof that any Lagrangian torus in $\CC^2$ is smoothly isotopic to the Clifford torus.

## 17 Citations

A Lagrangian Klein bottle you can’t squeeze

- MathematicsJournal of Fixed Point Theory and Applications
- 2022

Suppose you have a nonorientable Lagrangian surface $$L$$ L in a symplectic 4-manifold. How far can you deform the symplectic form before the smooth isotopy class of $$L$$ L contains no Lagrangians?…

Lefschetz pencils, Morse functions, and Lagrangian embeddings of the Klein bottle

- Mathematics
- 2002

It is shown that the mod 2 homology class represented by a Lagrangian Klein bottle in a complex algebraic surface is non-zero. In particular, the Klein bottle does not admit a Lagrangian embedding…

Lefschetz pencils, Morse functions, and Lagrangian embeddings of the Klein bottle

- Mathematics
- 2001

It is shown that the mod 2 homology class represented by a Lagrangian Klein bottle in a complex algebraic surface is non-zero. In particular, the Klein bottle does not admit a Lagrangian embedding…

Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups

- Mathematics
- 2007

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle in and . We exploit the existence of a special embedding of in a symplectic Lefschetz pencil and study its…

Lagrangian torus invariants using $ECH = SWF$

- MathematicsJournal of Symplectic Geometry
- 2021

We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes.…

Bulky Hamiltonian isotopy of Lagrangian tori with applications

- Mathematics
- 2019

We exhibit an example of a monotone Lagrangian torus inside the standard symplectic four dimensional unit ball which becomes Hamiltonian isotopic to a standard product torus only when considered…

Hamiltonian stability and index of minimal Lagrangian surfaces of the complex projective plane

- Mathematics
- 2005

We show that the Clifford torus and the totally geodesic real projective plane RP 2 in the complex projective plane CP 2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP 2…

Seidel's long exact sequence on Calabi-Yau manifolds

- Mathematics
- 2010

In this paper, we generalize construction of Seidel’s long exact sequence of Lagrangian Floer cohomology to that of compact Lagrangian submanifolds with vanishing Malsov class on general Calabi-Yau…

On the topology of Lagrangian submanifolds, Examples and counter-examples

- Mathematics
- 2005

The main tools we can use to approach these questions are the pseudoholomorphic-curves-Floer-homology on the one hand and the Stein-manifoldssubcritical-polarizations on the other one. Inside these…

Monodromy Groups of Lagrangian Tori in R 4

- Mathematics
- 2009

In this paper we work in the standard symplectic 4-space ( R 4, ω =∑2 j=1 dxj∧dyj) unless otherwise mentioned. Let L ι ↪→ (R4,ω) be an embedded Lagrangian torus with respect to the standard…

## References

SHOWING 1-10 OF 33 REFERENCES

Symplectic rigidity: Lagrangian submanifolds

- Mathematics
- 1994

This chapter is supposed to be a summary of what is known today about Lagrangian embeddings. We emphasise the difference between flexibility results, such as the h-principle of Gromov applied here to…

Lefschetz pencils, Morse functions, and Lagrangian embeddings of the Klein bottle

- Mathematics
- 2001

It is shown that the mod 2 homology class represented by a Lagrangian Klein bottle in a complex algebraic surface is non-zero. In particular, the Klein bottle does not admit a Lagrangian embedding…

Lagrangian embeddings in the complement of symplectic hypersurfaces

- Mathematics
- 2001

For a given embedded Lagrangian in the complement of a complex hypersurface we show the existence of a holomorphic disc in the complement having boundary on that Lagrangian.

Singularities and positivity of intersections of J-holomorphic curves

- Mathematics
- 1994

This chapter is devoted to proving some of the main technical results about J-holomorphic curves which make them such a powerful tool when studying the geometry of symplectic 4-manifolds. We begin by…

Compactness results in Symplectic Field Theory

- Mathematics
- 2003

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic…

First steps in symplectic topology

- Mathematics
- 1986

CONTENTSIntroduction § 1. Is there such a thing as symplectic topology? § 2. Generalizations of the geometric theorem of Poincare § 3. Hyperbolic Morse theory § 4. Intersections of Lagrangian…

J-Holomorphic Curves and Quantum Cohomology

- Mathematics
- 1994

Introduction Local behaviour Moduli spaces and transversality Compactness Compactification of moduli spaces Evaluation maps and transversality Gromov-Witten invariants Quantum cohomology Novikov…

A Morse-Bott approach to contact homology

- Mathematics
- 2002

Contact homology was introduced by Eliashberg, Givental and Hofer; this contact invariant is based on J-holomorphic curves in the symplectization of a contact manifold. We expose an extension of…

A new obstruction to embedding Lagrangian tori

- Mathematics
- 1990

n Let ~2n be endowed with the canonical symplectic form a = ~ i = 1 dxi ^ dY i, and consider a Lagrangian embedding of a compact manifold j : L" --, (R 2", a). A well-known result due to G r o m o v…

Pseudo holomorphic curves in symplectic manifolds

- Mathematics
- 1985

Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called…