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Unlinking and unknottedness of monotone Lagrangian submanifolds

Georgios Dimitroglou Rizell, Jonathan David Evans

Abstract

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

Unlinking and unknottedness of monotone Lagrangian submanifolds

Abstract

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

Paper Structure

This paper contains 26 sections, 33 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Basic definitions
  4. Context
  5. Luttinger surgery
  6. Perturbing the symplectic form
  7. Filling with discs
  8. Holomorphic foliations
  9. This paper
  10. Proofs
  11. Holomorphic discs which avoid a Lagrangian
  12. Unlinking of Lagrangians
  13. Unknottedness of Lagrangians
  14. Statement of results
  15. Haefliger-Hirsch theory
  16. ...and 11 more sections

Key Result

Theorem A

Let $X$ be a subcritical Stein manifold and let $0<K_1\leq K_2$ be real numbers. An $K_1$-monotone Lagrangian torus embedded in the complement $C$ of an embedded $K_2$-monotone Lagrangian torus must be homologically trivial in $C$. In particular, two embedded $K$-monotone Lagrangian tori are each nu

Figures (3)

  • Figure 1: The Gromov-Hofer limit must contain a finite-energy plane as a component of $u_V$. This plane (shaded in the figure) has large area since it can be considered as a disc with boundary on $L_2$. This contradicts the conservation of area in the limit.
  • Figure 2: Part of the $E_{n-1}$-page of the spectral sequence for the fibration $SO(n-1)\to SO(2n)\to V_{2n,n+1}$. The coefficient field $\mathbf{F}$ is either $\mathbf{Q}$ if $n$ is odd or $\mathbf{Z}/2$ if $n$ is even.
  • Figure 3: The $E_1$-page of the Biran-Cornea spectral sequence for a monotone Maslov 4 Lagrangian $S^1\times S^3$ in $\mathbf{C}^2$ with $E_1$ (horizontal) and $E_2$ (knight's move) differentials indicated.

Theorems & Definitions (66)

  • Theorem A
  • Theorem B
  • Corollary C
  • Theorem D
  • Corollary E
  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2: Gromov Gro, Lees Lee
  • Definition 2.4
  • Theorem 2.5: EvaKed
  • ...and 56 more