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Foliated Open Books

Joan E. Licata, Vera Vertesi

Abstract

This paper introduces a new type of open book decomposition for a contact three-manifold with a specified characteristic foliation $\mathcal{F}_ξ$ on its boundary. These \textit{foliated open books} offer a finer tool for studying contact manifolds with convex boundary than existing models, as the boundary foliation carries more data than the dividing set. In addition to establishing fundamental results about the uniqueness and existence of foliated open books, we carefully examine their relationship with the partial open books introduced by Honda-Kazez-Matic. Foliated open books have user-friendly cutting and gluing properties, and they arise naturally as submanifolds of classical open books for closed three-manifolds. We define three versions of foliated open books (embedded, Morse, and abstract), and we prove the equivalence of these models as well as a Giroux Correspondence which characterizes the foliated open books associated to a fixed triple $(M, ξ, \mathcal{F})$.

Foliated Open Books

Abstract

This paper introduces a new type of open book decomposition for a contact three-manifold with a specified characteristic foliation on its boundary. These \textit{foliated open books} offer a finer tool for studying contact manifolds with convex boundary than existing models, as the boundary foliation carries more data than the dividing set. In addition to establishing fundamental results about the uniqueness and existence of foliated open books, we carefully examine their relationship with the partial open books introduced by Honda-Kazez-Matic. Foliated open books have user-friendly cutting and gluing properties, and they arise naturally as submanifolds of classical open books for closed three-manifolds. We define three versions of foliated open books (embedded, Morse, and abstract), and we prove the equivalence of these models as well as a Giroux Correspondence which characterizes the foliated open books associated to a fixed triple .

Paper Structure

This paper contains 38 sections, 46 theorems, 25 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Applications and results
  3. Organization of the Paper
  4. Foliations and vector fields on surfaces
  5. Signed foliations
  6. Characteristic foliation
  7. Open book foliations
  8. Gradient-like vector fields
  9. Foliated open books
  10. Embedded foliated open books
  11. Compatible contact structures
  12. Abstract foliated open books
  13. Morse foliated open books
  14. Local Models
  15. Neighborhood Theorem
  16. ...and 23 more sections

Key Result

Theorem 1.2

Suppose that the foliated open books compatible with $(M^L,\xi^L)$ and $(M^R,\xi^R)$ induce the same foliation along their boundary. Then the contact 3-manifold $(M^L\cup M^R,\xi^L\cup\xi^R)$ formed by gluing them has a compatible honest open book decomposition that restricts to each piece as its or

Figures (24)

  • Figure 1: Left: $M=D^2\times S^1$ in $\mathbb{R}^3$. Center: Selected pages $S_c= M\cap \{\theta=c\}$. Right: singular foliation on $\partial M$. The green and blue curves in the three pictures indicate the leaves $\theta=0$ and $\theta=\frac{\pi}{3}$, respectively.
  • Figure 2: Left: hyperbolic point. Center: elliptic point (source). Right: center.
  • Figure 3: The bold curves show $G_{{++}}$ on a tile defined by a positive (left) and negative (right) hyperbolic point, and the red neighborhood is $\Gamma$. That the separatrices are labelled with $u$ and $s$ for unstable and stable, respectively.
  • Figure 4: In this signed foliation on the annulus, boxes represent hyperbolic points and circles represent elliptic points.
  • Figure 5: The flow of $\nabla{\widetilde{\pi}}$ on a tile.
  • ...and 19 more figures

Theorems & Definitions (113)

  • Example 1.1
  • Theorem 1.2: See Proposition \ref{['prop:glue']}
  • Theorem 1.3: See Theorems \ref{['thm:existencecontact']}, \ref{['thm:uniquecontact']} and \ref{['thm:fobexistence']}
  • Theorem 1.4: Giroux Correspondence, see Theorem \ref{['thm:giroux']}
  • Theorem 1.5: See Propositions \ref{['prop:fobpobequiv']} and \ref{['prop:pobfobequiv']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Giroux, Girbif
  • Theorem 2.5: Giroux Gi
  • ...and 103 more