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Construction of symplectic flexible links

Johan Björklund, Georgios Dimitroglou Rizell

TL;DR

The work addresses representing any smooth link in $\mathbb{R}P^3$ as the fixed-point set of an $I$-invariant symplectic surface in $\mathbb{C}P^3$, showing that the surface degree can be taken as $d\in\{1,2\}$ depending on the link's homology class. It develops a framework using conical submanifolds, cobordisms, and push-offs within the cotangent bundle and the symplectisation to assemble the needed invariant surfaces, including explicit genus control via Type I/II complements. The main contribution is a constructive existence result: any link in $\mathbb{R}P^3$ admits an $I$-invariant symplectic representative of degree $1$ or $2$, with the complement genus adjustable through connected sums and cobordism techniques. The findings have potential implications for understanding the interplay between real algebraic geometry, symplectic topology, and the topology of links in projective spaces, offering a flexible topological route to symplectic realizations that respect complex conjugation.

Abstract

We show that any smooth one-dimensional link in the real projective three-plane is the fixed-point locus of a smooth symplectic surface in the complex projective three-plane which is invariant under complex conjugation. The degree of the surface can be taken to be either one or two, depending on the homology class of the link. In other words, there are no obstructions to finding a symplectic representative of a flexible link beyond the classical topology.

Construction of symplectic flexible links

TL;DR

The work addresses representing any smooth link in as the fixed-point set of an -invariant symplectic surface in , showing that the surface degree can be taken as depending on the link's homology class. It develops a framework using conical submanifolds, cobordisms, and push-offs within the cotangent bundle and the symplectisation to assemble the needed invariant surfaces, including explicit genus control via Type I/II complements. The main contribution is a constructive existence result: any link in admits an -invariant symplectic representative of degree or , with the complement genus adjustable through connected sums and cobordism techniques. The findings have potential implications for understanding the interplay between real algebraic geometry, symplectic topology, and the topology of links in projective spaces, offering a flexible topological route to symplectic realizations that respect complex conjugation.

Abstract

We show that any smooth one-dimensional link in the real projective three-plane is the fixed-point locus of a smooth symplectic surface in the complex projective three-plane which is invariant under complex conjugation. The degree of the surface can be taken to be either one or two, depending on the homology class of the link. In other words, there are no obstructions to finding a symplectic representative of a flexible link beyond the classical topology.
Paper Structure (18 sections, 12 theorems, 39 equations)

This paper contains 18 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction and results
  2. Background on symplectic and contact geometry
  3. Symplectic manifolds
  4. Contact manifolds
  5. Cotangent bundles
  6. Isotropic, Lagrangian, and symplectic cobordisms
  7. Symplectic push-offs of Lagrangian cobordisms
  8. Conical submanifolds of cotangent bundles
  9. Deformation of conical symplectic surfaces
  10. Construction of two-dimensional symplectic cobordisms
  11. Fillings of transverse knots by genus-$g$ surfaces
  12. Connected $I_0$-invariant filling without fixed points
  13. Pairs of boundary connected sums via invariant pairs of one-handles
  14. Conical surfaces in the complex projective space
  15. The symplectic structure on the complement of a smooth quadric
  16. ...and 3 more sections

Key Result

Theorem 1.2

Any smooth link $L \subset \mathbb{R}P^3$ can be realised as the real-part $L=\Sigma_\mathbb{R}$ of a connected flexible-symplectic link of degree $d \in \{1,2\}$ of the same parity as $[L] \in H_1(\mathbb{R}P^3)=\mathbb{Z}_2$. Furthermore, the surface $\Sigma \setminus L$ can be either of Type I or

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Example 2.1
  • Example 2.2
  • Theorem 2.3: The Hamiltonian Thom Isotopy Lemma; e.g. HamiltonianThom
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Proposition 2.7: Eliashberg:IntroductionH
  • ...and 22 more