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Embedded contact homology and open book decompositions

Vincent Colin, Paolo Ghiggini, Ko Honda

TL;DR

This work develops a robust bridge between embedded contact homology (ECH) of a closed 3-manifold M and a relative ECH on the complement N of a binding of an adapted open book, under precise Reeb dynamics constraints. It constructs and analyzes relative ECH groups ECH(N,∂N,α) and their hat variants, using Morse–Bott theory, exact symplectic cobordisms, and carefully designed contact forms on torus and solid-torus neighborhoods. Central to the argument is a filtration-based comparison that yields an isomorphism between ECH(M) and ECH(N,∂N,α), with the U-map commuting across the equivalence; this also underpins the first steps toward the broader equivalence between ECH and Heegaard Floer homology. The paper further applies these techniques to sutured ECH, proving invariance properties and linking the solid-torus computation to the broader sutured framework, thereby strengthening the conceptual and computational toolkit for ECH in 3-manifolds with boundary and their open-book decompositions.

Abstract

This is the first of a series of papers devoted to proving the equivalence of Heegaard Floer homology and embedded contact homology (abbreviated ECH). In this paper we prove that, given a closed, oriented, contact $3$-manifold, there is an equivalence between ECH of the closed $3$-manifold and a version of ECH, defined on the complement of the binding of an adapted open book decomposition. In the appendix we give a full proof of the Morse-Bott gluing result that we need in this article and in the subsequent ones of the series proving the isomorphism between Heegaard Floer homology and ECH. V.8: we fixed a mistake in the appendix and added Yuan Yao as a coauthor.

Embedded contact homology and open book decompositions

TL;DR

This work develops a robust bridge between embedded contact homology (ECH) of a closed 3-manifold M and a relative ECH on the complement N of a binding of an adapted open book, under precise Reeb dynamics constraints. It constructs and analyzes relative ECH groups ECH(N,∂N,α) and their hat variants, using Morse–Bott theory, exact symplectic cobordisms, and carefully designed contact forms on torus and solid-torus neighborhoods. Central to the argument is a filtration-based comparison that yields an isomorphism between ECH(M) and ECH(N,∂N,α), with the U-map commuting across the equivalence; this also underpins the first steps toward the broader equivalence between ECH and Heegaard Floer homology. The paper further applies these techniques to sutured ECH, proving invariance properties and linking the solid-torus computation to the broader sutured framework, thereby strengthening the conceptual and computational toolkit for ECH in 3-manifolds with boundary and their open-book decompositions.

Abstract

This is the first of a series of papers devoted to proving the equivalence of Heegaard Floer homology and embedded contact homology (abbreviated ECH). In this paper we prove that, given a closed, oriented, contact -manifold, there is an equivalence between ECH of the closed -manifold and a version of ECH, defined on the complement of the binding of an adapted open book decomposition. In the appendix we give a full proof of the Morse-Bott gluing result that we need in this article and in the subsequent ones of the series proving the isomorphism between Heegaard Floer homology and ECH. V.8: we fixed a mistake in the appendix and added Yuan Yao as a coauthor.

Paper Structure

This paper contains 70 sections, 95 theorems, 352 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Review of embedded contact homology
  3. Generators of the ECH chain complex
  4. Moduli spaces
  5. The ECH index
  6. The ECH differential
  7. Definition of $\widehat{ECH}(M)$
  8. Cobordism maps and direct limits
  9. Maps induced by cobordisms
  10. Direct limits
  11. Morse-Bott theory
  12. Morse-Bott contact forms
  13. Morse-Bott buildings
  14. ECH and Fredholm indices
  15. Morse-Bott chain complex
  16. ...and 55 more sections

Key Result

Theorem 1.1.1

Let $N \subset M$ be the complement of a tubular neighborhood $int(V)$ of a null-homologous knot $K$, where $V \simeq K\times D^2$, $\xi$ a contact form on $M$ which is transverse to the foliation $K\times \{ *\}$ on $V$ and $\alpha$ a contact form on $N$ for the contact structure $\xi|_N$. If the R Moreover, the first isomorphism is compatible with the $U$-maps on both sides.

Figures (8)

  • Figure 1: Trajectory of $(f_\delta(y),g_\delta(y))$.
  • Figure 2: Trajectory of $(f(\rho),g(\rho))$. Here the arrow points in the positive $\rho$-direction.
  • Figure 3: Trajectory of $(f(\rho),g(\rho))$. The arrow is in the direction of increasing $\rho$.
  • Figure 4: Trajectory of $(f(y),g(y))$. The $f$-axis and $g$-axis do not necessarily intersect at $(0,0)$ in this figure.
  • Figure 5: The Reeb vector field $R'_i$ on $N_0(K)=(T^2 \times [1,2])\cup V$. The top and the bottom are identified.
  • ...and 3 more figures

Theorems & Definitions (231)

  • Theorem 1.1.1
  • Remark 1.1.2
  • Definition 2.2.1
  • Definition 2.3.1: Hu
  • Remark 2.3.2
  • Theorem 2.3.3: Hu2
  • Lemma 2.4.1: HT1
  • Definition 2.4.3
  • Definition 2.5.1
  • Definition 3.1.1
  • ...and 221 more