On PFH and HF spectral invariants

TL;DR

This paper explores the relationship between PFH and QHF spectral invariants on symplectic surfaces, introducing intermediate invariants and proving their equivalence in certain cases, with implications for quasi-morphisms.

Contribution

It defines HF spectral invariants using cylindrical QHF, establishes their equivalence to link invariants, and constructs open-closed morphisms linking PFH and QHF.

Findings

Homogenized HF spectral invariants at the sphere's unit equal homogenized PFH invariants.

HF spectral invariants are equivalent to link spectral invariants.

Homogenized PFH spectral invariants are quasi-morphisms.

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.