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Symplectic foliated fillings of sphere cotangent bundles

Francisco Presas, Sushmita Venugopalan

Abstract

We classify symplectically foliated fillings of certain foliated manifolds with a contact structure on the leaves. We show that for the foliated sphere cotangent bundle of the Reeb foliation on the three-sphere, the corresponding foliated disk cotangent bundle is the unique strong symplectic foliated filling up to blowups and symplectic deformation equivalence. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with an almost horizontal foliation. In this case, the foliated filling of the foliated sphere cotangent bundle is not unique. We show that any such filling is a foliated Lefschetz fibration, and is determined up to symplectic deformation equivalence, by combinatorial invariants arising from the singular locus of the Lefschetz fibration.

Symplectic foliated fillings of sphere cotangent bundles

Abstract

We classify symplectically foliated fillings of certain foliated manifolds with a contact structure on the leaves. We show that for the foliated sphere cotangent bundle of the Reeb foliation on the three-sphere, the corresponding foliated disk cotangent bundle is the unique strong symplectic foliated filling up to blowups and symplectic deformation equivalence. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with an almost horizontal foliation. In this case, the foliated filling of the foliated sphere cotangent bundle is not unique. We show that any such filling is a foliated Lefschetz fibration, and is determined up to symplectic deformation equivalence, by combinatorial invariants arising from the singular locus of the Lefschetz fibration.

Paper Structure

This paper contains 26 sections, 39 theorems, 159 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivating the problem
  3. Statement of results
  4. Background
  5. Symplectic and contact foliations
  6. Symplectic foliated filling
  7. Foliated cotangent bundle
  8. Foliated blow-ups and blow-downs
  9. Hamiltonian perturbation of a foliated symplectic form
  10. Holomorphic foliations
  11. Holomorphic curves
  12. Holomorphic curves in symplectizations
  13. Holomorphic curves in the filling
  14. Model fillings
  15. Model filling of $\mathbb{T}^3$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(\mathbb{S}^3,\mathcal{F}_{Reeb})$ be the Reeb foliation. Any minimal strong symplectic filling of the sphere cotangent bundle $\mathbb{S}(T^*\mathcal{F}_{Reeb})$ is symplectic deformation equivalent to the disk cotangent bundle $\mathbb{D} (T^*\mathcal{F}_{Reeb})$.

Figures (4)

  • Figure 1: Left: Reeb foliation on $\mathbb{R} \times \mathbb{D}^2$. This foliation is $\mathbb{R}$-invariant, and descends to a foliation on the solid torus $\mathbb{S}^1 \times \mathbb{D}^2$. Right: Reeb foliation on $\mathbb{S}^1 \times \mathbb{D}^2$ restricted to equatorial annulus $A^2$. The Reeb foliation on $\mathbb{S}^3$ has a compact leaf, which is the Clifford torus. It divides $\mathbb{S}^3$ into two solid tori, each of which has the Reeb foliation.
  • Figure 2: Left: Almost horizontal foliation on the strip $\mathbb{R} \times [0,1]$. Right: Almost horizontal foliation on the annulus $A^2\simeq \mathbb{S}^1 \times [0,1]$ obtained by quotienting the foliated strip on the left.
  • Figure 3: The Reeb foliation in a part of the torus, with transversal $\gamma$, and a neighbourhood $U_{Reeb}$
  • Figure 4: Left: A cross-section of a Reeb-foliated solid torus. Right: A cross-section of an almost-horizontally foliated $A^2 \times S^1$.

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • ...and 92 more