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On PFH and HF spectral invariants

Guanheng Chen

Abstract

For a closed symplectic surface, there are two types of spectral invariants: one defined by periodic Floer homology (PFH) and another by quantitative Heegaard Floer homology (QHF). The theme of this paper is to investigate the relationship between these two invariants. We begin by defining intermediate invariants using the cylindrical formulation of QHF, which we call HF spectral invariants. These invariants are shown to be equivalent to the link spectral invariants in the author's previous work. In the case of the sphere, we prove that the homogenized HF spectral invariants at the unit are equal to the homogenized PFH spectral invariants. This result is derived by constructing homomorphisms from quantitative Heegaard Floer homology to periodic Floer homology, which we refer to as open-closed morphisms. In addition, we show that the homogenized PFH spectral invariants are quasi-morphisms.

On PFH and HF spectral invariants

Abstract

For a closed symplectic surface, there are two types of spectral invariants: one defined by periodic Floer homology (PFH) and another by quantitative Heegaard Floer homology (QHF). The theme of this paper is to investigate the relationship between these two invariants. We begin by defining intermediate invariants using the cylindrical formulation of QHF, which we call HF spectral invariants. These invariants are shown to be equivalent to the link spectral invariants in the author's previous work. In the case of the sphere, we prove that the homogenized HF spectral invariants at the unit are equal to the homogenized PFH spectral invariants. This result is derived by constructing homomorphisms from quantitative Heegaard Floer homology to periodic Floer homology, which we refer to as open-closed morphisms. In addition, we show that the homogenized PFH spectral invariants are quasi-morphisms.
Paper Structure (43 sections, 29 theorems, 159 equations, 3 figures)

This paper contains 43 sections, 29 theorems, 159 equations, 3 figures.

Table of Contents

  1. Introduction and Main results
  2. Preliminaries
  3. Periodic Floer homology
  4. Periodic orbits.
  5. ECH index and $J_0$ index.
  6. PFH complex.
  7. Holomorphic curves and holomorphic currents.
  8. Differential on PFH.
  9. Notation.
  10. Grading.
  11. The U-map.
  12. PFH unit.
  13. Cobordism maps on PFH.
  14. Filtered PFH and PFH spectral invariants.
  15. Quantitative Heegaard Floer homology
  16. ...and 28 more sections

Key Result

Theorem 1

Suppose that $\underline{L}$ is a $0$-admissible link on $\mathbb{S}^2$. Then for any Hamiltonian function $H$, we have Moreover, for any $\varphi \in Ham(\mathbb{S}^2, \omega)$, we have where $\mathbf{1}_{\underline{L}} \in HF(\operatorname{Sym}^d \underline{L})$ is the unit of QHF, $\mu_{\underline{L}}, \mu^{link}_{\underline{L}}, \mu_{d}^{pfh}$ are the homogenization of $c_{\underline{L}, \et

Figures (3)

  • Figure 1: The red circles are the admissible link.
  • Figure 2: A picture of the case $m=5$.
  • Figure 3: The open-closed surface

Theorems & Definitions (76)

  • Theorem 1
  • Definition 1.1
  • Lemma 1.2: Lemma 2.4 of GHC2
  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • ...and 66 more