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End Khovanov homology and exotic Lagrangian planes

Yikai Teng

TL;DR

This work defines end and co-end Khovanov homology for properly embedded noncompact surfaces in $\mathbb{R}^4$ with finite Euler characteristic, using a compact exhaustion to form direct and inverse limits of link Kh groups. It proves these invariants are preserved under ambient diffeomorphisms and applies them to detect exotic planes: by constructing a specific surface $\Sigma$ and showing its end Kh has a nontrivial class in bigrading $(-2,-3)$, the plane is topologically standard but smoothly nonstandard. Furthermore, the authors show $\Sigma$ can be smoothly isotoped to a Lagrangian (and, with a symplectic analogue, to a symplectic) submanifold in $(\mathbb{R}^4,\omega_{std})$ by stacking compact Lagrangian cobordisms between Legendrian links, thereby producing the first known exotic Lagrangian and symplectic planes in 4-space. The paper also discusses connections to abstract Khovanov theory, spectral sequences, and potential links to Heegaard Floer homology, outlining directions for future work in distinguishing exotic planes further.

Abstract

In this paper, we define the end Khovanov homology, which is an invariant of properly embedded surfaces in 4-space up to ambient diffeomorphism. Moreover, we apply this invariant to detect the first known examples of exotic Lagrangian and symplectic planes in 4-space.

End Khovanov homology and exotic Lagrangian planes

TL;DR

This work defines end and co-end Khovanov homology for properly embedded noncompact surfaces in with finite Euler characteristic, using a compact exhaustion to form direct and inverse limits of link Kh groups. It proves these invariants are preserved under ambient diffeomorphisms and applies them to detect exotic planes: by constructing a specific surface and showing its end Kh has a nontrivial class in bigrading , the plane is topologically standard but smoothly nonstandard. Furthermore, the authors show can be smoothly isotoped to a Lagrangian (and, with a symplectic analogue, to a symplectic) submanifold in by stacking compact Lagrangian cobordisms between Legendrian links, thereby producing the first known exotic Lagrangian and symplectic planes in 4-space. The paper also discusses connections to abstract Khovanov theory, spectral sequences, and potential links to Heegaard Floer homology, outlining directions for future work in distinguishing exotic planes further.

Abstract

In this paper, we define the end Khovanov homology, which is an invariant of properly embedded surfaces in 4-space up to ambient diffeomorphism. Moreover, we apply this invariant to detect the first known examples of exotic Lagrangian and symplectic planes in 4-space.

Paper Structure

This paper contains 16 sections, 16 theorems, 19 equations, 36 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Acknowledgments
  4. Background on Khovanov homology
  5. Abstract Khovanov homology
  6. End Khovanov Homologies
  7. The plane is exotic
  8. The plane is topologically standard
  9. $\pi_1(\mathbb{R}^4\backslash {\Sigma})\cong \mathbb{Z}$
  10. $\pi_1^\infty(\mathbb{R}^4\backslash \Sigma)\cong \mathbb{Z}$
  11. The plane is smoothly non-standard
  12. Defining a class in $Kh(L_1)$.
  13. It survives the limiting process.
  14. The Lagrangian and symplectic setting
  15. The symplectic setting
  16. ...and 1 more sections

Key Result

Theorem A

There exists a proper embedding of a plane in $\mathbb{R}^4$ that is topologically but not smoothly isotopic to the standardly embedded plane. Moreover, this surface can be smoothly isotoped to a Lagrangian submanifold of $\mathbb{R}^4$ equipped with the standard symplectic structure.

Figures (36)

  • Figure 1: An exotic plane $\Sigma$ that can be smoothly isotoped to be Lagrangian.
  • Figure 2: 0- and 1- resolutions of a crossing.
  • Figure 3: A schematic for the surface $\Sigma\subset \mathbb{R}^4$.
  • Figure 4: Left: The first stage $\Sigma_1$ of the exotic plane, which is a disjoint union of two disks. Right: The second stage $\Sigma_2$ of the exotic plane.
  • Figure 5: The third stage $\Sigma_3$ of the exotic plane.
  • ...and 31 more figures

Theorems & Definitions (35)

  • Theorem A
  • Theorem B
  • Lemma 2.1: Proposition 3.2 of Ell10
  • Lemma 2.2
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['isotopy-inv']}
  • Remark 3.4
  • ...and 25 more