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.

arXiv commentAdd the following context: 1. U map on QHF and its computation 2. Coefficient extension of the open-closed morphisms and its computations in the sphere case 3. Discussion on the correspondence between open-closed morphisms and quantum inclusions. These new content arise from addressing the referee's comments. The author is grateful to the anonymous referees for their helpful suggestions