A Lagrangian Klein bottle you can't squeeze

Lagrangian Klein bottles in R^{2n}

It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd.

$S^2 \times S^2$ 中的 Lagrangian Klein 瓶

Lagrangian Klein bottles in $S^2 \times S^2$

本文利用 Luttinger 手术证明，当 $S^2$ 因子的辛面积至少为另一因子的两倍时，$S^2 \times S^2$ 中不存在属于该因子 $\mathbb{Z}_2$-同调类的 Lagrangian Klein 瓶。

Homology class of a Lagrangian Klein bottle

It is shown that an embedded Lagrangian Klein bottle represents a non-trivial mod 2 homology class in a compact symplectic four-manifold $(X,ω)$ with $c_1(X)\cdot[ω]>0$. (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.

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

A proof of non-existence of Lagrangian embeddings of the Klein bottle K in \CP^2 is given. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil on \CP^2 and study its monodromy. As the main technical tool, we develop the theory of mapping class groups, considered as quotients of special Artin braid groups, and obtain some new results about combinatorial structure of such groups.

Classification of Lagrangian Fibrations over a Klein Bottle

This paper completes the classification of regular Lagrangian fibratiopns over compact surfaces. \cite{misha} classifies regular Lagrangian fibrations over $\mathbb{T}^2$. The main theorem in \cite{hirsch} is used in order to classify integral affine structures on the Klein bottle $K^2$ and, hence, regular Lagrangian fibrations over this space.

Moduli of flat SU(3)-bundles over a Klein bottle

In this short note, we compute the Betti numbers of the moduli stack of flat SU(3)-bundles over a Klein bottle. We also handle the general compact group case over RP^2. In all cases the cohomology is found to be equivariantly formal, supporting a conjecture from the author's doctoral thesis. Our results also verify conjectural formulas obtained by Ho-Liu using Yang-Mills Morse theory.