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Properties of contact toric structures and concave boundaries of linear plumbings

Aleksandra Marinković, Jo Nelson, Ana Rechtman, Laura Starkston, Shira Tanny, Luya Wang

TL;DR

The paper studies concave boundaries of linear plumbings through the lens of contact toric geometry and embedded contact homology (ECH). It provides a complete criterion to detect overtwistedness via moment-cone data, and constructs a global concave toric structure on linear plumbings, linking tightness to normal Euler numbers. An extensive ECH toolkit is developed to compute the contact invariant and algebraic torsion for these concave boundaries, with explicit Morse–Bott to nondegenerate perturbation analyses and construction of pseudoholomorphic planes. The results yield concrete torsion and fillability/detectability statements for concave boundaries of linear plumbings and lay groundwork for extending to more general plumbings and singular curve realizability questions.

Abstract

We consider plumbings of symplectic disk bundles over spheres admitting concave contact boundary, with the goal of understanding the geometric properties of the boundary contact structure in terms of the data of the plumbing. We focus on the linear plumbing case in this article. We study the properties of the contact structure using two different sets of tools. First, we prove that all such contact manifolds have a global contact toric structure, and use tools from toric geometry to identify when the contact structure is tight versus overtwisted. Second, we study algebraic torsion measurements from embedded contact homology for these concavely induced contact manifolds, which has largely been unexplored. We develop a toolkit establishing existence and constraints of pseudoholomorphic curves adapted to the Morse-Bott Reeb dynamics of these plumbing examples, to provide the algebraic torsion and contact invariant calculations for the concave boundaries of linear plumbings.

Properties of contact toric structures and concave boundaries of linear plumbings

TL;DR

The paper studies concave boundaries of linear plumbings through the lens of contact toric geometry and embedded contact homology (ECH). It provides a complete criterion to detect overtwistedness via moment-cone data, and constructs a global concave toric structure on linear plumbings, linking tightness to normal Euler numbers. An extensive ECH toolkit is developed to compute the contact invariant and algebraic torsion for these concave boundaries, with explicit Morse–Bott to nondegenerate perturbation analyses and construction of pseudoholomorphic planes. The results yield concrete torsion and fillability/detectability statements for concave boundaries of linear plumbings and lay groundwork for extending to more general plumbings and singular curve realizability questions.

Abstract

We consider plumbings of symplectic disk bundles over spheres admitting concave contact boundary, with the goal of understanding the geometric properties of the boundary contact structure in terms of the data of the plumbing. We focus on the linear plumbing case in this article. We study the properties of the contact structure using two different sets of tools. First, we prove that all such contact manifolds have a global contact toric structure, and use tools from toric geometry to identify when the contact structure is tight versus overtwisted. Second, we study algebraic torsion measurements from embedded contact homology for these concavely induced contact manifolds, which has largely been unexplored. We develop a toolkit establishing existence and constraints of pseudoholomorphic curves adapted to the Morse-Bott Reeb dynamics of these plumbing examples, to provide the algebraic torsion and contact invariant calculations for the concave boundaries of linear plumbings.
Paper Structure (35 sections, 44 theorems, 191 equations, 25 figures)

This paper contains 35 sections, 44 theorems, 191 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Concave plumbings
  3. Contact toric 3-manifolds
  4. Algebraic torsion
  5. Organization of the paper
  6. Preliminaries on plumbings and symplectic toric geometry
  7. Plumbings
  8. Symplectic toric manifolds
  9. Symplectic toric 4-manifolds
  10. Contact toric manifolds
  11. Classification of contact toric 3-manifolds with singular orbits
  12. Tight and overtwisted contact toric 3-manifolds with singular orbits
  13. Constructions of linear plumbings with concave boundary using toric models
  14. Plumbing $(s_1,s_2)$, with $s_1\geq0$ or $s_{2}\geq0$
  15. Linear plumbing $(s_1,\ldots,s_n)$, with at least one $s_i\geq0$
  16. ...and 20 more sections

Key Result

Lemma 3.1

A toric action on a symplectic manifold $(W^{2n},\omega)$ with a convex (resp., concave) contact boundary $(Y,\xi)$ preserves the contact structure $\xi$.

Figures (25)

  • Figure 1: The moment map images of the plumbings $(-2,1,0,-1)$ (left), $(-2,1,0,-2)$ (middle) and $(-2,1,0,-3)$ (right).
  • Figure 2: The standard moment cone as the moment map image of $(\mathbb{C}^2,\omega_{st})$.
  • Figure 3: A moment map image of a neighborhood of a symplectic sphere with self-intersection number $s=v_1u_2-v_2u_1$.
  • Figure 4: A Hirzebruch trapesoid
  • Figure 5: A moment map image before and after a blow up of the fixed point, with the changes in the self-intersection numbers.
  • ...and 20 more figures

Theorems & Definitions (125)

  • Definition 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Example 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 3.5
  • Remark 3.6
  • Theorem 3.7
  • ...and 115 more