Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symplectic forms on trisected 4-manifolds

Peter Lambert-Cole

Abstract

Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,ω)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the converse direction and give explicit criteria on a trisection of a closed, smooth 4-manifold $X$ that allows one to construct a symplectic structure on $X$. Combined, these give a new characterization of 4-manifolds that admit symplectic structures. This construction motivates several problems on taut foliations, the Thurston norm and contact geometry in 3-dimensions by connecting them to questions about the existence, classification and uniqueness of symplectic structures on 4-manifolds.

Symplectic forms on trisected 4-manifolds

Abstract

Previously work of the author with Meier and Starkston showed that every closed symplectic manifold with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the converse direction and give explicit criteria on a trisection of a closed, smooth 4-manifold that allows one to construct a symplectic structure on . Combined, these give a new characterization of 4-manifolds that admit symplectic structures. This construction motivates several problems on taut foliations, the Thurston norm and contact geometry in 3-dimensions by connecting them to questions about the existence, classification and uniqueness of symplectic structures on 4-manifolds.
Paper Structure (24 sections, 28 theorems, 123 equations)

This paper contains 24 sections, 28 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Representing cohomology via cocycles
  3. Main theorem
  4. How can the construction fail?
  5. Symplectic structures and trisection diagrams
  6. Grafted contact structures
  7. Symplectic cone
  8. Classification of symplectic structures
  9. Inspiration
  10. The Fubini-Study form on $\mathbb{CP}^2$
  11. Branched covering
  12. Handlebody setup
  13. Morse form foliations
  14. Tautness of the foliation
  15. Thurston norm
  16. ...and 9 more sections

Key Result

Theorem 1.4

Let $X$ be a closed, oriented 4-manifold and let $\mathcal{T}$ be a trisection of $X$ with central surface $\Sigma$. Let $(\widetilde{\beta}_1,\widetilde{\beta}_2,\widetilde{\beta}_3)$ be a symplectic 1-cocycle for $(X,\mathcal{T})$. Then

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.8
  • proof
  • Theorem 1.10: LMS
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 44 more