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

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Pith's one-line read Heegaard Floer homology computes the zeta function of a pseudo-Anosov flow as an Euler characteristic.

desk verdict Strong combinatorial bridge from veering branched surfaces to sutured Floer homology, but the zeta-function categorification currently leans on unpublished flow-box work by Zung. read the letter →

arxiv 2504.15420 v1 pith:GCEZPYGO submitted 2025-04-21 math.GT math.DS

Antonio Alfieri , Chi Cheuk Tsang This is my paper · ORCID
classification math.GTmath.DS MSC 57K1837D2057R30
keywords pseudo-AnosovflowveeringbranchedsurfaceHeegaardFloerhomologysutureddynamicalzetafunctioncategorificationclosedorbitsspin-cgrading
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper bridges two previously separate worlds: the dynamics of pseudo-Anosov flows and Heegaard Floer homology. For a pseudo-Anosov flow on a 3-manifold, with a suitable collection of closed orbits (no perfect fits), the authors construct a chain complex from the flow's veering branched surface. The generators are exactly certain closed multi-orbits of the blown-up flow, and the homology is the sutured Floer homology of the flow's complement. Their main result gives a Z/2-grading on that homology whose Euler characteristic is the reciprocal of the flow's zeta function, so the homology categorifies the zeta function.

What carries the argument

The central object is the veering branched surface B associated to the flow, which carries the blown-up unstable foliation. From B the authors construct a canonical balanced Heegaard diagram (Σ, α, β) whose chain complex is SFC(φ,C). The key combinatorial step is encoding Heegaard states as embedded multi-loops in the augmented dual graph G^+, which in turn correspond to closed multi-orbits of the flow. The categorification is achieved by a Z/2-grading ν defined on generators by the number of side corners plus the sign of the induced permutation; this grading is shown to be a homological grading, i.e., the differential lowers it by 1 mod 2, and the resulting Euler characteristic computes the anti-veering polynomial A, which is identified with the reciprocal zeta function.

What would settle it

Compute for a specific pseudo-Anosov flow with b1≥2, e.g., the geodesic flow on the unit tangent bundle of a hyperbolic surface, the Euler characteristic of SFH with the ν-grading and compare it to the reciprocal of the zeta function. Any difference would disprove the main theorem; alternatively, check whether the flow admits a flow box decomposition satisfying the conditions of Proposition 6.4.

Watch

Extended reading notes

Core claim

The paper establishes that the sutured Floer homology of the complement of a pseudo-Anosov flow, equipped with a natural Z/2-grading defined from the flow's veering branched surface, has Euler characteristic equal to the reciprocal of the dynamical zeta function of the blown-up flow, provided the first Betti number of the complement is at least 2. In the course of the proof, the authors build a canonical Heegaard diagram from the veering branched surface and show its generators correspond to closed multi-orbits, with their spin-c grading given by their homology classes. They also identify two distinguished generators, top and bottom, that give nontrivial homology classes in the flow's spin-c grading and its opposite.

Load-bearing premise

The identification of the zeta function with the anti-veering polynomial relies on a flow box decomposition (due to Zung, in preparation) whose double ungluing has zeta function equal to the reciprocal of the veering polynomial; if this decomposition does not exist, the categorification computes the anti-veering polynomial but not the dynamical zeta function.

Editorial extensions

If this is right

  • The sutured Floer homology of a pseudo-Anosov flow's complement carries a natural Z/2-grading that recovers the flow's zeta function, giving a Floer-theoretic invariant sensitive to closed orbit counts.
  • The top and bottom generators yield nontrivial classes in SFH, providing a Floer-theoretic proof that the flow's spin-c grading is supported in at least two gradings.
  • For suspension flows of pseudo-Anosov maps, the computation of SFH in the flow's spin-c grading gives the fiberedness detection theorem and hints at an orbit-counting refinement.
  • The construction suggests that other dynamical invariants, such as the Fried polytope, might be categorified by sutured Floer homology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One might expect a refinement of the Z/2-grading to a Z-grading emanating from the connection between the ν-grading and the Maslov index, which could yield a richer categorification that tracks more detailed orbit data.
  • The correspondence between generators and closed multi-orbits may allow one to define Floer-theoretic counts of periodic orbits, potentially giving lower bounds on orbit numbers via the rank of SFH.
  • If the flow box decomposition of Zung is made fully constructive, the main theorem could be made algorithmic for veering triangulations, producing explicit chain complexes for many hyperbolic 3-manifolds.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs a canonical balanced sutured Heegaard diagram from a veering branched surface B associated to a pseudo-Anosov flow with no perfect fits relative to a collection C of closed orbits. It defines a chain complex SFC(phi,C) whose generators are identified with certain closed multi-orbits of the blown-up flow, whose homology is the sutured Floer homology SFH(Y^7), and in which the spin^c-grading is computed from the homology class of the associated multi-orbit. The paper proves that the top and bottom generators are nonvanishing and then defines a Z/2-grading nu whose Euler characteristic categorifies the anti-veering polynomial of B. Under the further identification of that polynomial with the reciprocal of the dynamical zeta function zeta_{phi^7}, this yields the advertised equality chi_nu(SFH(Y^7)) = zeta^{-1}_{phi^7}.

Significance. If the zeta-function identifications hold, this is a striking new bridge between pseudo-Anosov dynamics and Heegaard Floer homology: the Floer chain complex is generated by multi-orbits of the flow, and a natural grading categorifies a dynamical zeta function. The manuscript contains substantial and largely self-contained combinatorial work: admissibility of the canonical diagram, the bijection between Heegaard states and embedded multi-loops, the nonvanishing of the top and bottom generators, and the full proof of the factorization theorem for the anti-veering polynomial in Section 7. The proof that the nu-grading is a homological grading (Theorem 8.3) is detailed and appears to be a genuine contribution. However, the final step from the anti-veering polynomial to the zeta function is not fully proved in this preprint: it rests on Propositions 6.4 and 6.5, parts of which are cited to published work but key items are attributed to unpublished work of Zung. This limits the currently checkable scope of the central theorem.

major comments (2)
  1. [§6.3, Propositions 6.4 and 6.5; Eqs. (6.3)–(6.4)] Theorem 1.7 depends on the chain of equalities chi_nu(SFH(Y^7)) = A = zeta^{-1}_{phi^7}, where A is the anti-veering polynomial. The first equality is proved in Section 8, and Theorem 6.3 is proved in Section 7. The second equality, however, rests on Propositions 6.4 and 6.5. Proposition 6.4(1) and (2) are not proved in the manuscript but are cited to [AT24, Section 5] and [Tsa23c, Chapter 2], and Proposition 6.5 is only sketched: the assertions that the orbits in D_uC_s are precisely the ones homotopic to the anti-branch loops, and that an orbit is orientation-preserving exactly when the corresponding anti-branch loop is orientation-preserving, are not written out. These assertions are load-bearing because Equation (6.4) turns them into the factors prod_{pres}(1-[gamma])^{-1} prod_{rev}(1+[gamma])^{-1}; a single sign error would replace factors (1-[gamma])^{-1} by (1+[gamma])^{-1} and change the categorified polynomial. Since [Zun] is listed as in preparation, the full flow-box-to-polynomial machinery is not publicly checkable at present. The manuscript should either include complete proofs of these statements or state Theorem 1.7 explicitly as conditional on Zung's results.
  2. [§6.3, Proof of Proposition 6.5] The sentence 'since B carries the blown-up unstable foliation (Theorem 2.5(i)), such an orbit is orientation-preserving if and only if the corresponding anti-branch loop is orientation-preserving' is a nontrivial geometric correspondence, not a formal consequence of the carrying statement. The proof of Proposition 6.5 uses this assertion to apply Equation (6.4) with the same orientation dichotomy as Theorem 6.1, so this is exactly where the sign information enters. No argument or reference is supplied for the equivalence of the two orientation notions, and it should be proved in detail before the zeta-function identification is used as an unconditional ingredient of Theorem 1.7.
minor comments (4)
  1. [§8, Theorem 8.2 and §1, Theorem 1.7] Theorem 8.2 states equality with the anti-veering polynomial only up to multiplication by a unit of Z[G], while Theorem 1.7 states an exact equality chi_nu(SFH(Y^7)) = zeta^{-1}_{phi^7}. Since zeta^{-1}_{phi^7} has constant term 1 and x_bot is the unique generator in the spin^c-structure s_{phi^7} with nu(x_bot)=0, the unit can presumably be normalized to 1, but this normalization argument is not written. Please add it so that the statement of Theorem 1.7 follows literally from Theorem 8.2.
  2. [§4.5, Proposition 4.19] The proof of Proposition 4.19 asserts that the properties of being tangent to the sectors of B and positively transverse to the branch locus characterize the homotopy class of the vector field X_{phi^7}; this uniqueness is not demonstrated. Since Proposition 4.19 underpins Theorems 1.2 and 1.3, a more detailed argument would be helpful.
  3. [§6.3, definition of zeta function] The zeta function is defined as an infinite product in Z[[G]] 'provided that the coefficient of each g converges'. For flows with infinitely many primitive closed orbits in a single homology class this is not automatic, and the phrase should be clarified or replaced by a precise convergence argument, possibly citing the relevant results from [JZ24].
  4. [§1, Corollary 1.5] The statement of Corollary 1.5 contains a typo: the two displayed dimensions should refer to s_{phi^7} and its conjugate, not to the same spin^c-structure twice.

Circularity Check

0 steps flagged · score 0.0 of 10

No constructional circularity: the categorifying grading and anti-veering polynomial are defined independently of the zeta function.

full rationale

The derivation chain is not circular. The Z/2-grading ν in Definition 8.1 is defined purely combinatorially from the veering branched surface (number of side corners plus sign of the associated permutation), and the anti-veering polynomial A in Section 6.2.3 is defined as a Fitting invariant from the anti-tetrahedron relations; neither is fitted to orbit counts or to ζ. Theorem 8.2 computes χν(SFH) as the signed determinant expansion of the anti-tetrahedron matrix, i.e. as A, and Proposition 6.6 separately identifies A with ζ^{-1}_{φ^7} using Zung's flow-box decomposition and the factorization Theorem 6.3. No equation in the paper is identical to its input by construction: the equalities run through independent algebraic (Fitting invariant, determinant expansion) and dynamical (flow graph, zeta function) computations. The reliance of Proposition 6.4(1)-(2) on [AT24], [Tsa23c], and the in-preparation [Zun] is a verification gap and a correctness risk, but not a circular reduction, because those cited results are not being used to define the objects they are supposed to prove equal.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or fitted constants appear; this is a pure mathematics construction. The central claim rests on standard theorems in Heegaard Floer theory and veering branched surfaces, on the no-perfect-fits hypothesis, and on the unpublished flow box decomposition results of Zung (Propositions 6.4 and 6.5), which are the least externally grounded inputs.

assumptions (7)
  • domain assumption phi is a pseudo-Anosov flow on a closed oriented 3-manifold Y with no perfect fits relative to a finite collection C of closed orbits containing the singularity locus.
    Standing hypothesis of Theorems 1.1 through 1.7; introduced in Section 1.2 and Section 2.1, it guarantees the Agol-Gueritaud correspondence and the generator/orbit identifications.
  • standard math Agol-Gueritaud correspondence between such flows and veering branched surfaces, including that the dual graph loops are homotopic to closed orbits of the blown-up flow.
    Theorem 2.5 assembles this from [Tsa23b], [LMT23], and [SS19-SS23]; it is the foundation of the Heegaard diagram construction in Section 4.
  • standard math Juhasz sutured Heegaard Floer homology is well-defined and invariant for balanced sutured manifolds, and admissible Heegaard diagrams compute it.
    Invoked in Section 3 (Theorems 3.7 and 3.8) to assert that CF(Sigma, alpha, beta) computes SFH(M, Gamma) independently of diagram choices.
  • standard math Lipshitz index formula mu(D) = e(D) + n_x(D) + n_y(D), and the effectiveness obstruction for holomorphic disks.
    Used throughout Sections 5 and 8 to identify Maslov index 1 domains and to prove nonvanishing and parity results.
  • standard math Landry-Minsky-Taylor results: sweep-equivalence classes of loops in the dual graph biject with closed orbits of phi^7, and the veering polynomial is the image of the Perron polynomial of the flow graph.
    These published results are used in Section 4.4 to attach closed multi-orbits to generators and in Proposition 6.4(3) to identify zeta functions with polynomial invariants.
  • ad hoc to paper Zung's flow box decomposition propositions: existence of F with unstable spine orbits C, D_u C_s orbits homotopic to anti-branch loops, and double-ungluing zeta function equal to the inverse veering polynomial.
    Propositions 6.4 and 6.5 are attributed to Zung with reference [Zun] 'In preparation'; items (1)-(2) cite [AT24, Section 5] and [Tsa23c]. These are load-bearing for Theorem 1.7 and are not fully proved in this preprint.
  • standard math Landry-Minsky-Taylor factorization of the veering polynomial V = Theta * product(1 +/- [a_i]) for anti-branch loops.
    Theorem 6.1, cited to [LMT24a, Theorem 6.1 and Lemma 6.2], is used in Proposition 6.5 to pass from the veering polynomial to the taut polynomial.

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Pith. "Pith review of Heegaard Floer theory and pseudo-Anosov flows I: Generators and categorification of the zeta function." pith.science (2026). https://pith.science/paper/GCEZPYGO

@misc{pith2026250415420,
  author       = {Pith},
  title        = {Pith review of: Heegaard Floer theory and pseudo-Anosov flows I: Generators and categorification of the zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GCEZPYGO}},
  note         = {Machine review of arXiv:2504.15420}
}
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 $\phi$ 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 Gu\'eritaud 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 $\phi$. 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 $\phi$.

Figures

Figures reproduced from arXiv: 2504.15420 by the authors.

Figure 1
Figure 1. Local picture of a pseudo-Anosov flow. Left: Near a singular orbit. Right: Away from a singular orbit. 2. Pseudo-Anosov flows and veering branched surfaces In this section, we recall some facts about pseudo-Anosov flows and veering branched surfaces, including the correspondence theorem between these objects. A general reference for this material is [Tsa23c, Chapters 1 and 2]. 2.1. Pseudo-Anosov flows. For the purpo… view at source ↗
Figure 2
Figure 2. Blowing up along [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. A perfect fit rectangle. The dynamical significance of the no perfect fit condition is recorded by the following proposition. Proposition 2.1. Let ϕ be a pseudo-Anosov flow with no perfect fits relative to a collection of closed orbits C. Let ϕ 7 be the blow-up of ϕ along C. Then two distinct closed orbits γ1 and γ2 of ϕ 7 are homotopic in Y 7 only if they lie on the same boundary component. Proof. This is implied b… view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: The local model for a branched surface. The arrows indicate the maw coorientation. 2.2. Veering branched surfaces. Let M be a compact oriented 3-manifold with torus boundary components. A branched surface in M is a 2-complex B Ă M where every point in B has a neighborh…
Figure 5
Figure 5. Figure 5: A local picture around a triple point in a branched surface satisfying Definition 2.4(3). The edges of brlocpBq also inherit orientations from the branch loops. This upgrades the structure of brlocpBq as a 4-valent graph to a p2, 2q-valent directed graph, i.e. a direct…
Figure 6
Figure 6. Figure 6: Building a triangulation ∆ out of tetrahedra tR corresponding to maximal rectangles R. More concretely, we can perform the construction of ∆ within Yr. However, note that ∆ is not a triangulation of Yr, since each flow line in Cr is ‘collapsed’ down to a point in ∆. In…
Figure 7
Figure 7. Figure 7: Approximating FĂu within each tetrahedron by a branched surface. Conversely, given a loop c in Γ, consider a lift rc of c in Yr. The sequence of vertices passed through by rc corresponds to a sequence of tetrahedra of ∆r, which in turn corresponds to a sequence of maxi…
Figure 8
Figure 8. Figure 8: The sutured manifold structure on a tubular neighborhood of a branched surface. The two main sources of sutured manifolds we shall consider in this paper are summarized below. Example 3.2 (Neighbourhoods of branched surfaces). Suppose B is a branched surface in a close…
Figure 9
Figure 9. Figure 9: Some examples of domains that connect a state x to a state y. Here the coefficient of each shaded effective domain is 1. The Maslov indices of these domain can be computed using Equation (3.3). Dj Di Dk Dl p [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: A pictorial key for Equation (3.2). Lemma 3.12. Suppose D is a domain connecting a state x to a state y. Let p P α X β be an intersection point between an α- and a β-curve. Then we have that ni ´ nj ` nk ´ nl “ $ ’& ’% 1 if p P xzy ´1 if p P yzx 0 otherwise (3.2) wher…
Figure 11
Figure 11. Figure 11: Constructing the spinc -structure spxq associated to a Hee￾gaard state x. extended to a non-vanishing vector field over B1Y¨ ¨ ¨YBd. We define spxq P Spinc pM, Γq as the homotopy class of Vx. See [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: The α-curves are placed in correspondence of the triple points of B. 4.1. The Heegaard diagram. Let B be a veering branched surface on a 3-manifold with torus boundary M. We shall denote by v1, . . . , vn the triple points of B, and by C1, . . . , Ck, the complementar…
Figure 13
Figure 13. Figure 13: The 3-manifold obtained from pΣ0, α, βq is the submanifold M0 Ă M with complement MzM0 a collection of solid tori tT1, . . . , Tℓu, one for each cusp curve on BM. In M0, we have a collection of annuli Ai , each with one boundary component along the core of Ti and the …
Figure 14
Figure 14. Figure 14: The annuli Ai intersect Σ0 away from the α- and the β-curves. Furthermore, we observe that the annuli Ai intersect Σ0 away from the α- and β-curves. See [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: Each sector of a veering branched surface is a diamond. that 0 “ χpMq “ χpBq “ χpGq ` m. On the other hand, as shown in Lemma 4.1 we have that χpGq “ ´n. Thus 0 “ χpΓq ` m “ m ´ n. □ 4.2. Heegaard states. Summarizing, we can canonically associate to each veering branc…
Figure 16
Figure 16. Figure 16: Each β-curve βi has exactly four intersection points with the α-curves, corresponding to the four corners of the sector Si . Definition 4.5 (Corners). As suggested by [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: Picking the edges ex,i, the union of which is the multi-loop µx associated to x. Proposition 4.8. The assignments x top and x bot define two Heegaard states. Proof. We should verify that there is no triple point that is picked up twice by x top . In Proposition 4.3 we…
Figure 18
Figure 18. Figure 18: Each edge of G`zG can be strum into a path of G in two ways. Proposition 4.14. There is a bijection F from the sweep-equivalence classes of loops in G` to the closed orbits of ϕ 7 , such that Fpcq is homotopic to c in M for each loop c. Proof. The results of [LMT23] i…
Figure 19
Figure 19. Figure 19: Reasoning that the spinc -grading of x top is sϕ. Proof of Theorem 1.3. In Example 4.13 we showed that µxbot “ ∅. This implies that γxbot “ ∅. In Proposition 4.19 we showed that spx botq “ sϕ7 . For any other state y, combining Lemma 4.17 and Lemma 4.18, we have spyq …
Figure 20
Figure 20. Figure 20: The color of a triple point is determined by the local form of the branched surface. Here the orientation on M is used to distinguish between the two local pictures. Note that in [PITH_FULL_IMAGE:figures/full_fig_p035_20.png]
Figure 21
Figure 21. Figure 21: An example of the local configuration of sectors in a veering branched surface. 5.2. Structure of the Heegaard diagram: local picture. In this section we relate the structure of the Heegaard diagram associated to a veering branched surface B to the combinatorics of th…
Figure 22
Figure 22. Figure 22: The local combinatorics of the Heegaard diagram near a triple point v depends on the color of v. while the point x bot j lies on the upper left β-arc. See the purple and yellow dots on [PITH_FULL_IMAGE:figures/full_fig_p038_22.png]
Figure 23
Figure 23. Figure 23: Reading the combinatorics of Σzα off from the colors of triple points lying on the corresponding branch loop. Based on the conventions we set in Definition 5.1 the vertices of the dual graph G can be colored blue or red. The portion of the Heegaard diagram pΣ, α, βq l…
Figure 24
Figure 24. Figure 24: The relations used to define the taut, veering, and anti￾veering polynomials. Over the next few subsections, we will define some relations in S. These relations will in turn define modules whose Fitting ideal gives the respective polynomial invariants. 6.2.1. The face…
Figure 25
Figure 25. Figure 25: A flow box decomposition. Left: An unstable spine with an unstable spine orbit. Middle: The flow boxes. Right: A stable spine with a stable spine orbit. Let G “ H1pXq{Torsion. We define the zeta function of ϕ to be ζϕ “ ź primitive γ p1 ´ rγsq´1 P ZrrGss where the pro…
Figure 26
Figure 26. Figure 26: Left: A flow box decomposition F near an unstable spine orbit. Right: The unstable ungluing of F. Fix a flow box decomposition F. A stable spine is a component of the union of stable rectangles in Y . See [PITH_FULL_IMAGE:figures/full_fig_p049_26.png]
Figure 27
Figure 27. Figure 27: Left: The unstable ungluing of a flow box decomposition F. Right: The double ungluing of F. map D2F Ñ DuF obtained by regluing the stable rectangles. Pulling back the flow lines of Duϕ, we have a partial flow D2ϕ. We denote the collection of primitive closed orbits of…
Figure 28
Figure 28. Figure 28: Choosing lifts for vertices and edges for the proof of Lemma 7.3. This is a rankpT q ˆ rankpT q “: t ˆ t submatrix of rDΘ ‘ DAs βE\βT βS . Thus Θ “ FitpDΘ ‘ DAq | detprDΘ ‘ DAs β 1 E\β 2 T βS q (7.4) Meanwhile, consider the matrix rD 3 s “ ” p1 ´ h1q ¨ DΘpep1q ¨ ¨ ¨ p…
Figure 29
Figure 29. Figure 29: Dividing the domain D into D1 and D2 in order to apply the induction hypothesis. There are three cases depending on how many endpoints of c lie on concave corners of D. Furthermore, in each case c can either be an α- or β-arc. For simplicity in this picture we only dr…

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Pith tools

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