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

Nikolas Adaloglou, Jonathan David Evans

Abstract

We use Luttinger surgery to show that there are no Lagrangian Klein bottles in $S^2\times S^2$ in the $\mathbb{Z}_2$-homology class of an $S^2$-factor if the symplectic area of that factor is at least twice that of the other.

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

Abstract

We use Luttinger surgery to show that there are no Lagrangian Klein bottles in in the -homology class of an -factor if the symplectic area of that factor is at least twice that of the other.

Paper Structure

This paper contains 23 sections, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theorem.
  3. Klein bottles.
  4. Maslov class constraint.
  5. Example.
  6. Outline of the proof.
  7. Klein bottle Luttinger surgery
  8. Lemma.
  9. Lemma.
  10. Corollary.
  11. Lemma.
  12. Lemma.
  13. Corollary.
  14. Chern and Maslov classes.
  15. Lemma.
  16. ...and 8 more sections

Figures (4)

  • Figure 1: The base (left) and fibre (right) of the moment map $\mu\colon X\to \mathbb{R}^2$. A visible Lagrangian Klein bottle lives over the bold line in the base (slope $2$) and intersects each fibre in the bold line shown on the right (slope $-1/2$).
  • Figure 2: The Klein bottle as an identification space.
  • Figure 3: A visible symplectic sphere living over $\ell$ made up of two discs (dotted and dashed) with boundary on the visible Lagrangian $L$.
  • Figure 4: The space $N$ as an identification space.

Theorems & Definitions (12)

  • proof : Proof that $\bm{i}(\partial \Sigma)$ is well-defined modulo $4$
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  • ...and 2 more