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Closed-open morphisms on periodic Floer homology

Guanheng Chen

TL;DR

This work introduces closed-open morphisms from PFH to a cylindrical reformulation of quantitative HF, enabling direct comparison between PFH and HF spectral invariants. It develops a HF-type homology parallel to PFH, proves an index-compatible equivalence between the two formulations, and constructs nonvanishing closed-open maps that preserve units. By analyzing energy, ECH index, and partition conditions in PFH–HF cobordisms, the paper derives a partial inequality bounding HF spectral invariants by PFH invariants, revealing dynamics–Lagrangian interactions in a unified framework. The results pave a path toward comparing PFH and HF spectral theories and have implications for understanding Hofer-type and Calabi properties in a Floer-theoretic context.

Abstract

In this note, we investigate homomorphisms from the periodic Floer homology (PFH) to the quantitative Heegaard Floer homology. We call the homomorphisms closed-open morphisms. Under certain assumptions on the Lagrangian link, we first follow R. Lipshitz's idea to give a cylindrical formulation of the quantitative Heegaard Floer homology. Then we construct the closed-open morphisms from the PFH to the quantitative Heegaard Floer homology. Moreover, we show that the morphisms are non-vanishing. As an application, we deduce a relation between the PFH-spectral invariants and the HF-spectral invariants.

Closed-open morphisms on periodic Floer homology

TL;DR

This work introduces closed-open morphisms from PFH to a cylindrical reformulation of quantitative HF, enabling direct comparison between PFH and HF spectral invariants. It develops a HF-type homology parallel to PFH, proves an index-compatible equivalence between the two formulations, and constructs nonvanishing closed-open maps that preserve units. By analyzing energy, ECH index, and partition conditions in PFH–HF cobordisms, the paper derives a partial inequality bounding HF spectral invariants by PFH invariants, revealing dynamics–Lagrangian interactions in a unified framework. The results pave a path toward comparing PFH and HF spectral theories and have implications for understanding Hofer-type and Calabi properties in a Floer-theoretic context.

Abstract

In this note, we investigate homomorphisms from the periodic Floer homology (PFH) to the quantitative Heegaard Floer homology. We call the homomorphisms closed-open morphisms. Under certain assumptions on the Lagrangian link, we first follow R. Lipshitz's idea to give a cylindrical formulation of the quantitative Heegaard Floer homology. Then we construct the closed-open morphisms from the PFH to the quantitative Heegaard Floer homology. Moreover, we show that the morphisms are non-vanishing. As an application, we deduce a relation between the PFH-spectral invariants and the HF-spectral invariants.
Paper Structure (54 sections, 47 theorems, 246 equations, 5 figures)

This paper contains 54 sections, 47 theorems, 246 equations, 5 figures.

Table of Contents

  1. Introduction and Main results
  2. Coefficient
  3. Twisted periodic Floer homology
  4. Holomorphic curves and holomorphic currents
  5. The ECH index
  6. Fredholm index
  7. $J_0$ index
  8. Partition conditions
  9. Definition of periodic Floer homology
  10. Cobordism maps
  11. Invariance of PFH
  12. PFH-spectral invariants
  13. Quantitative Heegaard Floer homology
  14. Invariance of quantitative Heegaard Floer homology
  15. Action functional and link spectral invariants
  16. ...and 39 more sections

Key Result

Lemma 1.3

Let $J \in \mathcal{J}(Y_{\varphi}, \omega_{\varphi})$ be an admissible almost complex structure in the symplectization of $\mathbb{R} \times Y_{\varphi}$. Let $\mathcal{C} \in \mathcal{M}^{J}(\alpha_+, \alpha_-)$ be a $J$-holomorphic current in $\mathbb{R} \times Y_{\varphi}$ without closed compone

Figures (5)

  • Figure 1: The red circles are the links. Here the first two pictures are examples of $\eta$-admissible links. The third one is a non-example. However, it satisfies the monotone conditions in Definition 1.12 of CHMSS.
  • Figure 2: The small red circles are the boundary punctures. If the holomorphic curve degenerates along the blue dashed arc, then a disk bubble is created as shown in the figure on the right.
  • Figure 3: If the curve degenerates along the pink dashed arcs in Figure 2, then the curve splits into two levels as depicted in the figure to the left. If the curve degenerates along the purple dashed arcs in Figure \ref{['figure5']}, then an extra boundary is created as shown in the picture on the right.
  • Figure 4:
  • Figure 5: The red points are the index 2 critical points. The green points are saddle points. The blue points are index 0 critical points. The union of the red circles is the link. The function $f$ is a small perturbation of the height function.

Theorems & Definitions (121)

  • Definition 1.1
  • Definition 1.2: Section 3.1 of H4
  • Lemma 1.3
  • proof
  • Remark 1.1
  • Definition 1.4
  • Definition 1.5
  • Remark 1.2
  • Remark 1.3
  • Definition 1.6
  • ...and 111 more