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Sutured Heegaard Floer and embedded contact homologies are isomorphic

Vincent Colin, Paolo Ghiggini, Ko Honda

Abstract

We prove the equivalence of the sutured versions of Heegaard Floer homology, monopole Floer homology, and embedded contact homology. As applications we show that the knot versions of Heegaard Floer homology and embedded contact homology are equivalent and that product sutured 3-manifolds are characterized by the fact that they carry an adapted Reeb vector field without periodic orbits.

Sutured Heegaard Floer and embedded contact homologies are isomorphic

Abstract

We prove the equivalence of the sutured versions of Heegaard Floer homology, monopole Floer homology, and embedded contact homology. As applications we show that the knot versions of Heegaard Floer homology and embedded contact homology are equivalent and that product sutured 3-manifolds are characterized by the fact that they carry an adapted Reeb vector field without periodic orbits.
Paper Structure (23 sections, 26 theorems, 78 equations, 3 figures)

This paper contains 23 sections, 26 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sutured manifolds and their Floer homologies
  3. Balanced sutured manifolds
  4. Sutured Floer homology and knot Floer homology
  5. Sutured ECH
  6. Periodic Floer homology and sutured ECH
  7. Proofs of Theorems \ref{['thm: sutured manifold']} and \ref{['thm: sutured monopole']}
  8. Reduction to connected sutures
  9. The contact closure
  10. Proofs of Theorems \ref{['thm: sutured manifold']} and \ref{['thm: sutured monopole']} assuming Theorem \ref{['thm: ECH']}
  11. ECH of the contact closure
  12. Construction of a Reeb vector field
  13. The buffer zone
  14. A Morse-Bott contact form on $Y$
  15. Morse-Bott perturbations and excavating the ball
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $(M,\Gamma,\xi)$ be a sutured contact manifold. Then where $ECH(M,\Gamma,\xi)$ is the sutured ECH of $(M,\Gamma,\xi)$ summed over all homology classes and $SFH(-M,-\Gamma)$ is the sutured Heegaard Floer homology of $(-M,-\Gamma)$ summed over all relative Spin$^c$-structures.

Figures (3)

  • Figure 2: The buffer zone for $s\in[-\epsilon,\epsilon]$. The level sets of the Morse function $a$ are oriented by the projection of the Reeb vector field to the $(t,s)$-annulus.
  • Figure 3: The concave ball $B_i$ in $D_2\times S^1$, obtained by rotating the shaded region about the vertical central axis.
  • Figure 4: The orbits that intersect $S\times\{ 0 \}$, given by pink dots. All the orbits except for $h_0$ intersect $S\times\{0 \}$ once; the orbits besides $h_0,e_{1/n}, h_{1/n}$ are "vertical", i.e., parallel to the $S^1$-fibers; the orbit $h_0$ in pink lies on $S\times\{1/2\}$ and bounds $D_1\times\{ 0 \}$. The downward gradient trajectories of $f$ are given in blue.

Theorems & Definitions (61)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.9
  • Remark 1.11
  • Corollary 1.12
  • Remark 1.14
  • ...and 51 more