Heegaard Floer theory and pseudo-Anosov flows I: Generators and categorification of the zeta function

TL;DR

This work develops a direct bridge between pseudo-Anosov dynamics and Heegaard Floer theory by constructing a combinatorial chain complex $SFC(\phi,\mathcal{C})$ from veering branched surfaces associated to a blown-up flow $\phi^{\sharp}$, whose homology recovers sutured Floer homology $SFH(Y^{\sharp})$ and whose generators correspond to closed multi-orbits of the dynamical system. It identifies two canonical generators, top and bottom, with nontrivial Spin$^c$-gradings, and proves a detailed Spin$^c$-grading framework arising from the flow data, including a gradation that categorifies a normalization of the zeta function of the blown-up flow. The paper then defines and analyzes polynomial invariants (taut, veering, anti-veering) from veering branched surfaces, relates these to dynamical zeta functions via Floer-theoretic constructions, and proves that the anti-veering polynomial admits a categorification by SFH with a suitably defined $\nu$-grading. The results yield a robust, combinatorial pathway to extract dynamical information from Floer theoretic data and provide a new perspective on the relation between flow dynamics and 3-manifold invariants. The framework sets the stage for the subsequent development of a full dynamical zeta categorification within Heegaard Floer theory and suggests broader applications to L-spaces, taut foliations, and polytope invariants in 3-manifold topology.

Abstract

We bring to light a new connection between dynamics and Heegaard Floer homology. On a closed 3-manifold $Y$ we consider a pseudo-Anosov flow $φ$ with no perfect fits with respect to its singularity locus $L \subset Y$, or perhaps a larger collection of closed orbits. Using work of Agol and Guéritaud on veering branched surfaces we produce a chain complex computing the link Floer homology of $L$ in the framing specified by the degeneracy curves of the flow. Using work of Landry, Minsky, and Taylor we show that the generators of the chain complex correspond to certain closed multi-orbits of $φ$. We prove that two canonical generators $\mathbf{x}^\mathrm{top}$ and $\mathbf{x}^\mathrm{bot}$ determine non-trivial homology classes located in the $\text{spin}^\text{c}$-grading of the flow, and its opposite. Finally, we observe that our specific model of the chain complex for link Floer homology naturally supports a grading with dynamical significance. This grading, a modification of the regular Maslov grading, is shown to categorify a suitable normalization of the zeta function associated to $φ$.

Figures (29)

• Figure 1: Local picture of a pseudo-Anosov flow. Left: Near a singular orbit. Right: Away from a singular orbit.
• Figure 2: Blowing up along \ref{['fig:paflow']} left.
• Figure 3: A perfect fit rectangle.
• Figure 4: The local model for a branched surface. The arrows indicate the maw coorientation.
• Figure 5: A local picture around a triple point in a branched surface satisfying \ref{['defn:vbs']}(3).

Theorems & Definitions (126)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Proposition 2.2: Tsa24a