Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symplectic embeddings of toric domains with boundary a lens space

Jonathan Trejos

Abstract

We give a combinatorial description of the embedded contact complex (ECC) of a certain family of contact toric lens spaces that we call concave lens spaces. We also define a notion of a concave toric domain that generalizes the usual concave toric domain in a way that possesses a singularity point and has a boundary a lens space. After desingularization these toric domains include the unitary cotangent bundle of $\mathbb{S}^2$ and the unitary cotangent bundle of $\mathbb{R}P^2$. We use the combinatorial expression of the ECC to compute the ECH capacities of these toric domains. Furthermore, for certain concave toric domains we describe a packing of symplectic manifolds that recovers their ECH capacities.

Symplectic embeddings of toric domains with boundary a lens space

Abstract

We give a combinatorial description of the embedded contact complex (ECC) of a certain family of contact toric lens spaces that we call concave lens spaces. We also define a notion of a concave toric domain that generalizes the usual concave toric domain in a way that possesses a singularity point and has a boundary a lens space. After desingularization these toric domains include the unitary cotangent bundle of and the unitary cotangent bundle of . We use the combinatorial expression of the ECC to compute the ECH capacities of these toric domains. Furthermore, for certain concave toric domains we describe a packing of symplectic manifolds that recovers their ECH capacities.
Paper Structure (38 sections, 20 theorems, 90 equations, 8 figures)

This paper contains 38 sections, 20 theorems, 90 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Symplectic Toric Orbifolds
  3. Construction of the toric orbifold $M(n,m)$.
  4. Visible submanifolds of $M(n,m)$.
  5. ECH Capacities of toric domains in $M(n,m)$
  6. ECH capacities of Ellipsoids with singularities.
  7. ECH capacities of concave toric domains on $M(n,m)$
  8. Ball Packing
  9. Weight Expansions
  10. Packing Theorem
  11. Idea of the proof of Theorem \ref{['concavecapacities']}
  12. Foundations of Embedded Contact Homology
  13. The ECH index
  14. Conley-Zenhder index
  15. Relative first Chern class
  16. ...and 23 more sections

Key Result

Lemma 1.4

Let $a:[0,1]\rightarrow \bar{V}_{n,m}$ be a smooth curve such that $a(0)$ lies in the ray $\{t(n,m):t>0\}$ and $a(1)$ lies in the ray $\{t(0,1):t>0\}$ then

Figures (8)

  • Figure 1: Example of a Toric Concave Domain
  • Figure 2: Examples of decorated $(n,m)$-polygonal paths.
  • Figure 3: Some examples of generators for $L(3,2)$
  • Figure 4: A convex region $R_\alpha$ in $L(2,1)$.
  • Figure 5: Examples of corounding the corner in $L(2,1)$. The paths $P_\alpha$ and $P_\beta$ are represented by blue and red respectively.
  • ...and 3 more figures

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Lemma 1.8
  • Example 1.9
  • ...and 47 more