Heegaard Floer theory and pseudo-Anosov flows II: Differential and Fried pants

TL;DR

This paper explores the structure of Heegaard Floer chain complexes associated with pseudo-Anosov flows, introducing a refined grading to analyze differentials and connecting Floer homology to periodic orbit counts.

Contribution

It introduces a refined spin^c grading for Heegaard Floer complexes related to pseudo-Anosov flows, enabling explicit analysis of differentials and homology in this context.

Findings

Refined the spin^c grading to obstruct certain differentials.

Explicitly described subcomplexes representing irreducible multi-orbits.

Showed that homology of these subcomplexes is 1-dimensional.

Abstract

In earlier work, relying on work of Agol-Gu\'eritaud and Landry-Minsky-Taylor, we showed that given a pseudo-Anosov flow $(Y,\phi)$ and a collection of closed orbits $\mathcal{C}$ satisfying the `no perfect fit' condition, one can construct a special Heegaard diagram for the link complement $Y^\sharp= Y \setminus \nu (\mathcal{C})$ framed by the degeneracy curves. In this paper, we demonstrate how the special combinatorics of this diagram can be used to understand the differential of the associated Heegaard Floer chain complex. More specifically, we introduce a refinement of the $\text{spin}^\text{c}$-grading obstructing two Heegaard states from being connected by an effective domain. We describe explicitly the subcomplexes in the refined gradings that represent irreducible multi-orbits, in the sense that they contain states corresponding to multi-orbits which cannot be resolved along Fried pants. In particular we show that the homology of these subcomplexes are 1-dimensional. When specialized to the case of suspension flows our arguments prove some results in the spirit of Ni, Ghiggini, and Spano: the next-to-top non-zero sutured Floer group counts the number of periodic points of least period.